A177387 -id:A177387 - cp's OEIS frontend

A249698 E.g.f.: Sum_{n>=0} Product_{k=1..n} cosh(k*x).

1, 0, 6, 0, 2426, 0, 7553776, 0, 90192976308, 0, 2939813898295990, 0, 213701821328573755046, 0, 30292525174041077292043440, 0, 7609302838629919155170452856136, 0, 3152886110080180503361685427596189430, 0, 2038143533263759863560759054752335955960482

Offset: 0

Vaclav Kotesovec, Nov 04 2014

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..234

Cf. A177385, A224899, A249489, A177387, A249737.

Table[n!*SeriesCoefficient[Sum[Product[Cosh[k*x],{k,1,j}],{j,0,n}],{x,0,n}],{n,0,20}]
nn=20; tab = ConstantArray[0,nn]; tab[[1]] = Series[Cosh[x],{x,0,nn}]; Do[tab[[k]] = Series[tab[[k-1]]*Cosh[k*x],{x,0,nn}],{k,2,nn}]; Flatten[{1,Table[kk!*Sum[Coefficient[tab[[k]],x^kk],{k,1,kk}],{kk,1,nn}]}] (* more efficient *)

{a(n)=local(X=x+x*O(x^n), Egf); Egf=sum(m=0, n, prod(k=1, m, cosh(k*X))); n!*polcoeff(Egf, n)}
for(n=0, 20, print1(a(n), ", "))

If n is even a(n) ~ c * d^n * n^(2*n) / (2^(2*n-2) * exp(2*n)), where d = 8.9061971328050809899679389417314..., c = 1.243878632396819914960247452516...

A177388 O.g.f.: Sum_{n>=0} Product_{k=1..n} sin(k*arcsin(2x)).

Original entry on oeis.org

1, 2, 8, 48, 368, 3488, 39408, 517536, 7747552, 130224448, 2428303280, 49745334816, 1110519910176, 26832869319872, 697671946188128, 19422303020653632, 576390053072381888, 18164695560213480064

Offset: 0

Paul D. Hanna, May 15 2010

nonn

			O.g.f.: A(x) = 1 + 2*x + 8*x^2 + 48*x^3 + 368*x^4 + 3488*x^5 + ...
Let G(x) be the e.g.f. of A177387:
G(x) = 1 + x + 4*x^2/2! + 35*x^3/3! + 536*x^4/4! + ...
then A(x) = G(arcsin(2*x)).

Vaclav Kotesovec, Table of n, a(n) for n = 0..260

Cf. A177387.

{a(n)=local(X=x+x*O(x^n),Ogf);Ogf=sum(m=0,n,prod(k=1,m,sin(k*asin(2*X))));polcoeff(Ogf,n)}

O.g.f.: A(x) = G(arcsin(2x)) where G(x) = e.g.f. of A177387.

a(n) ~ c * (4/(Pi*log(2)))^n * n! * n^(1/6), where c = 1.01529686... . - Vaclav Kotesovec, Nov 04 2014

A249737 E.g.f.: Product_{k=1..n} cosh(k*x).

Original entry on oeis.org

1, 0, 5, 0, 1992, 0, 6167551, 0, 73432708224, 0, 2389444075877425, 0, 173496878823412858880, 0, 24573448663070711791073155, 0, 6168942712247503719875933929472, 0, 2554865971518520622455831203134760669, 0, 1650933998542152349112398040415912949710848, 0

Offset: 0

Vaclav Kotesovec, Nov 04 2014

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..235

Cf. A249698, A177385, A177387, A001044 (e.g.f.: product_{k=1..n} sinh(k*x)).

Table[n!*SeriesCoefficient[Product[Cosh[k*x],{k,1,n}],{x,0,n}],{n,0,20}]
nn = 20; tab = ConstantArray[0,nn]; tab[[1]] = Series[Cosh[x],{x,0,nn}]; Do[tab[[k]] = Series[tab[[k-1]]*Cosh[k*x],{x,0,nn}],{k,2,nn}]; Flatten[{1,Table[k!*Coefficient[tab[[k]],x^k],{k,1,nn}]}] (* more efficient *)

If n is even a(n) ~ c * d^n * n^(2*n) / (2^(2*n-2) * exp(2*n)), where d = 8.9061971328050809899679389417314..., c = 1.004120096780056350248342856...

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A249698 E.g.f.: Sum_{n>=0} Product_{k=1..n} cosh(k*x).

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A177388 O.g.f.: Sum_{n>=0} Product_{k=1..n} sin(k*arcsin(2x)).

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A249737 E.g.f.: Product_{k=1..n} cosh(k*x).

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