> AA:=n->[16*n+2,8*n+1,9*n+1,10*n+1]; > BB:=n->[11*n+2,1,16*n+2,16*n+2]; > AAstar:=n->[0,8*n+1,9*n+1,10*n+1]; > BBstar:=n->[0,0,16*n+2,16*n+2]; AA := n -> [16 n + 2, 8 n + 1, 9 n + 1, 10 n + 1] BB := n -> [11 n + 2, 1, 16 n + 2, 16 n + 2] AAstar := n -> [0, 8 n + 1, 9 n + 1, 10 n + 1] BBstar := n -> [0, 0, 16 n + 2, 16 n + 2] > RR:=(n,t)->(BB(n)[3]-AA(n)[3]-1)!*(BB(n)[4]-AA(n)[4]-1)!/((AA(n)[1]-BB > (n)[1])!*(AA(n)[2]-BB(n)[2])!)*product(2*t+j,j=BB(n)[1]..AA(n)[1]-1)*p > roduct(t+j,j=BB(n)[2]..AA(n)[2]-1)/(product(t+j,j=AA(n)[3]..BB(n)[3]-1 > )*product(t+j,j=AA(n)[4]..BB(n)[4]-1)); / | | RR := (n, t) -> | |factorial(BB(n)[3] - AA(n)[3] - 1) factorial(BB(n)[4] | \ /AA(n)[1] - 1 \ /AA(n)[2] - 1 \\// | ,--------' | | ,--------' || | | | | | | | | || | | | | | | | | || | - AA(n)[4] - 1) | | | (2 t + j)| | | | (t + j)|| |factorial( | | | | | | | || | \j = BB(n)[1] / \j = BB(n)[2] // \ /BB(n)[3] - 1 \ /Product( | ,--------' | | | | | | | | | | | | AA(n)[1] - BB(n)[1]) factorial(AA(n)[2] - BB(n)[2]) | | | (t + j)| | | | | | | \j = AA(n)[3] / \ (t + j), j = AA(n)[4] .. BB(n)[4] - 1)\\ || || || || || // > RR(n,t); (factorial(7 n) factorial(6 n) GAMMA(2 t + 16 n + 2) GAMMA(t + 8 n + 1) GAMMA(t + 9 n + 1) GAMMA(t + 10 n + 1))// \factorial(5 n) factorial(8 n) GAMMA(2 t 2\ + 11 n + 2) GAMMA(t + 1) GAMMA(t + 16 n + 2) / > dd:=n->AA(n)[1]+AA(n)[2]+AA(n)[3]+AA(n)[4]-BB(n)[1]-BB(n)[2]; dd := n -> AA(n)[1] + AA(n)[2] + AA(n)[3] + AA(n)[4] - BB(n)[1] - BB(n)[2] > dd(n); 32 n + 2 > a:=(n,k)->(-1)^dd(n)*binomial(2*k-BB(n)[1],2*k-AA(n)[1])*binomial(k-BB > (n)[2],k-AA(n)[2])*binomial(BB(n)[3]-AA(n)[3]-1,k-AA(n)[3])*binomial(B > B(n)[4]-AA(n)[4]-1,k-AA(n)[4]); dd(n) a := (n, k) -> (-1) binomial(2 k - BB(n)[1], 2 k - AA(n)[1]) binomial( k - BB(n)[2], k - AA(n)[2]) binomial(BB(n)[3] - AA(n)[3] - 1, k - AA(n)[3]) binomial(BB(n)[4] - AA(n)[4] - 1, k - AA(n)[4]) > a(n,k); (32 n + 2) (-1) binomial(2 k - 11 n - 2, 2 k - 16 n - 2) binomial(k - 1, k - 8 n - 1) binomial(7 n, k - 9 n - 1) binomial(6 n, k - 10 n - 1) > aa:=n->min(ceil(AA(n)[1]/2),AAstar(n)[2]); / /1 \ \ aa := n -> min|ceil|- AA(n)[1]|, AAstar(n)[2]| \ \2 / / > aa(3); 25 > b:=(n,k)->unapply(diff(RR(n,t)*(t+k)^2,t),t)(-k); / d / 2\ \ b := (n, k) -> unapply|--- \RR(n, t) (t + k) /, t|(-k) \ dt / > check:=proc(n) > Q:=sum(a(n,k)/(t+k)^2,k=AAstar(n)[4]..BBstar(n)[3]-1); > QQ:=seq(b(n,k)/(t+k),k=AAstar(n)[3]..BBstar(n)[4]-1); > S:=sum(QQ[k],k=1..BBstar(n)[4]-AAstar(n)[3]); > is(simplify(Q+S)=simplify(RR(n,t))); > end proc; # Warning, `Q` is implicitly declared local to procedure `check` # Warning, `QQ` is implicitly declared local to procedure `check` # Warning, `S` is implicitly declared local to procedure `check` check := proc(n) local Q, QQ, S; Q := sum((a(n, k))/((t + k)^2), k = AAstar(n)[4] .. BBstar(n)[3] - 1); QQ := seq((b(n, k))/(t + k), k = AAstar(n)[3] .. BBstar(n)[4] - 1); S := sum(QQ[k], k = 1 .. BBstar(n)[4] - AAstar(n)[3]); is(simplify(Q + S) = simplify(RR(n, t))); end proc; > check(1); true > check(2); true > check(3); true > check(4); true > pp:=proc(n) > X:=seq(a(n,k)*sum(1/l^3,l=1..k-aa(n)),k=AAstar(n)[4]..BBstar(n)[3]-1); > Y:=seq(b(n,k)*sum(1/l^2,l=1..k-aa(n)),k=AAstar(n)[3]..BBstar(n)[4]-1); > x:=(-1)^dd(n)*sum(X[u],u=1..BBstar(n)[3]-AAstar(n)[4]); > y:=(-1)^dd(n)/2*sum(Y[v],v=1..BBstar(n)[4]-AAstar(n)[3]); > simplify(x+y); > end proc; # Warning, `X` is implicitly declared local to procedure `pp` # Warning, `Y` is implicitly declared local to procedure `pp` # Warning, `x` is implicitly declared local to procedure `pp` # Warning, `y` is implicitly declared local to procedure `pp` pp := proc(n) local X, Y, x, y; X := seq(a(n, k)*(sum((1)/(l^3), l = 1 .. k - aa(n))), k = AAstar(n)[4] .. BBstar(n)[3] - 1); Y := seq(b(n, k)*(sum((1)/(l^2), l = 1 .. k - aa(n))), k = AAstar(n)[3] .. BBstar(n)[4] - 1); x := (-1)^dd(n)*(sum(X[u], u = 1 .. BBstar(n)[3] - AAstar(n)[4])); y := 1/2*(-1)^dd(n)*(sum(Y[v], v = 1 .. BBstar(n)[4] - AAstar(n)[3])); simplify(x + y); end proc; > pp(1); 7953492001094261 ---------------- 537600 > pp(2); 37762843816152998347068580008855083 ----------------------------------- 5540664729600 > pp(3); 71177637042478935679933724271874209441285543475044377 ----------------------------------------------------- 15002859781367377920 > pp(4); 19641131680482401221797951183607031311026133902035745863015672448388709/50077\ 26658621606993920000 > pp1:=7953492001094261/537600; 7953492001094261 pp1 := ---------------- 537600 > pp2:=37762843816152998347068580008855083/5540664729600; 37762843816152998347068580008855083 pp2 := ----------------------------------- 5540664729600 > pp3:=71177637042478935679933724271874209441285543475044377/15002859781 > 367377920; 71177637042478935679933724271874209441285543475044377 pp3 := ----------------------------------------------------- 15002859781367377920 > pp4:=19641131680482401221797951183607031311026133902035745863015672448 > 388709/5007726658621606993920000; pp4 := 19641131680482401221797951183607031311026133902035745863015672448388709/5007726658621606993920000 > DD:=proc(n) > if n=1 then 1 else lcm(DD(n-1),n) fi > end proc; DD := proc(n) if n = 1 then 1 else lcm(DD(n - 1), n) end if; end proc; > cc1:=n->max(AA(n)[1]-BB(n)[1],AA(n)[2]-BB(n)[2],BBstar(n)[4]-AA(n)[3]- > 1,BBstar(n)[4]-AA(n)[4]-1,BBstar(n)[3]-ceil(AA(n)[1]/2)-1,BBstar(n)[3] > -AAstar(n)[2]-1); / cc1 := n -> max|AA(n)[1] - BB(n)[1], AA(n)[2] - BB(n)[2], \ BBstar(n)[4] - AA(n)[3] - 1, BBstar(n)[4] - AA(n)[4] - 1, /1 \ \ BBstar(n)[3] - ceil|- AA(n)[1]| - 1, BBstar(n)[3] - AAstar(n)[2] - 1| \2 / / > cc2:=n->max(BBstar(n)[4]-ceil(AA(n)[1]/2)-1,BBstar(n)[4]-AAstar(n)[2]- > 1); / /1 \ cc2 := n -> max|BBstar(n)[4] - ceil|- AA(n)[1]| - 1, \ \2 / \ BBstar(n)[4] - AAstar(n)[2] - 1| / > type(2*DD(cc1(1))*DD(cc2(1))^2*pp1,integer); true > type(2*DD(cc1(2))*DD(cc2(2))^2*pp2,integer); true > type(2*DD(cc1(3))*DD(cc2(3))^2*pp3,integer); true