A249489 -id:A249489 - cp's OEIS frontend

A249459 a(n) = Sum_{k=0..n} k^(2*n).

1, 1, 17, 794, 72354, 10874275, 2438235715, 762963987380, 317685943157892, 169842891165484965, 113394131858832552133, 92465351109879998121806, 90431265068257318469676710, 104479466717230437574945525959, 140782828210237288756752539959687

Offset: 0

Vaclav Kotesovec, Oct 29 2014

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..214

Cf. A031971, A224899, A249489.

Table[Sum[k^(2*n),{k,1,n}],{n,1,20}]
Table[n!*SeriesCoefficient[Sum[Exp[k^2*x], {k, 1, n}],{x,0,n}], {n,1,20}]

a(n)=n!*polcoeff(sum(k=0, n, exp(k*x+x*O(x^n))^k), n);
for(n=1, 20, print1(a(n), ", "))

my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k^2*x)^k/(1-k^2*x))) \\ Seiichi Manyama, Dec 03 2021

E.g.f.: Sum_{n>=0} exp(n^2*x).
a(n) ~ exp(2)/(exp(2)-1) * n^(2*n).
G.f.: Sum_{k>=0} (k^2 * x)^k/(1 - k^2 * x). - Seiichi Manyama, Dec 03 2021

a(0)=1 prepended by Seiichi Manyama, Dec 03 2021

A224899 E.g.f.: Sum_{n>=0} sinh(n*x)^n.

Original entry on oeis.org

1, 1, 8, 163, 6272, 389581, 35560448, 4479975823, 744707981312, 157897753198201, 41585725184933888, 13318468253704790683, 5097100004294081380352, 2297277197389011910783621, 1204339195916670860817072128, 726625952070893090583192860743

Offset: 0

Paul D. Hanna, Jul 24 2013

Conjecture: Let p be prime. The sequence obtained by reducing a(n) modulo p for n >= 1 is purely periodic with period p - 1. For example, modulo 7 the sequence becomes [1, 1, 2, 0, 3, 0, 1, 1, 2, 0, 3, 0, 1, 1, 2, 0, 3, 0, ...], with an apparent period of 6. Cf. A245322. - Peter Bala, May 29 2022

			E.g.f.: A(x) = 1 + x + 8*x^2/2! + 163*x^3/3! + 6272*x^4/4! +...
where
A(x) = 1 + sinh(x) + sinh(2*x)^2 + sinh(3*x)^3 + sinh(4*x)^4 +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..225

Cf. A122399, A249489, A245322, A220181, A221077, A221078, A198513, A220181, A249459, A195415, A245322, A338040.

Flatten[{1,Table[Sum[Sum[Binomial[k,j] * (-1)^j * k^n*(k-2*j)^n / 2^k,{j,0,k}],{k,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, Oct 29 2014 *)
Join[{1},Rest[With[{nn=20},CoefficientList[Series[Sum[Sinh[n*x]^n,{n,nn}],{x,0,nn}],x] Range[0,nn]!]]] (* Harvey P. Dale, May 18 2018 *)

{a(n)=n!*polcoeff(sum(k=0, n, sinh(k*x+x*O(x^n))^k), n)}
for(n=0, 20, print1(a(n), ", "))

E.g.f.: Sum_{n>=0} exp(-n^2*x) * (exp(2*n*x) - 1)^n / 2^n.

a(n) ~ sqrt(Pi) * 2^(2*n+1) * n^(2*n+1/2) / (sqrt(3-2*log(2)) * 3^(n+1/2) * exp(2*n) * (log(2))^(2*n+1)). - Vaclav Kotesovec, Oct 28 2014

A221077 E.g.f.: Sum_{n>=0} tanh(n*x)^n.

Original entry on oeis.org

1, 1, 8, 160, 5888, 345856, 29677568, 3502489600, 544181977088, 107675615297536, 26435436140822528, 7885689342279024640, 2809177794704769548288, 1177952320402008693538816, 574318105367992485583781888, 322156963576521588458420961280, 206009256195720974104252003647488

Offset: 0

Paul D. Hanna, Dec 31 2012

nonn

Conjecture: Let p be prime. The sequence obtained by reducing a(n) modulo p for n >= 1 is purely periodic with period p - 1. For example, modulo 7 the sequence becomes [1, 1, 6, 1, 0, 4, 1, 1, 6, 1, 0, 4, 1, 1, 6, 1, 0, 4 ...], with an apparent period of 6. - Peter Bala, Jun 01 2022

			E.g.f.: A(x) = 1 + x + 8*x^2/2! + 160*x^3/3! + 5888*x^4/4! + 345856*x^5/5! +...
where
A(x) = 1 + tanh(x) + tanh(2*x)^2 + tanh(3*x)^3 + tanh(4*x)^4 + tanh(5*x)^5 +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..220

Cf. A122399, A195415, A220181, A221078, A221079, A224899, A249489, A245322, A338040.

nmax = 20; CoefficientList[Series[1 + Sum[Tanh[k*x]^k, {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, May 31 2022 *)
Join[{1}, Table[Sum[2^n * k^n * Sum[(-1)^j * Binomial[k, j] * Sum[(-1)^m * Binomial[j + m - 1, m] * StirlingS2[n, m] * m! / 2^m, {m, 1, n}], {j, 0, k}], {k, 0, n}], {n, 1, 20}]] (* Vaclav Kotesovec, Jun 01 2022 *)

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

{a(n)=local(X=x+x*O(x^n), Egf); Egf=sum(m=0, n, (exp(2*m*X)-1)^m/(exp(2*m*X)+1)^m); n!*polcoeff(Egf, n)}
for(n=0,20,print1(a(n),", ") )

E.g.f.: Sum_{n>=0} (exp(2*n*x) - 1)^n / (exp(2*n*x) + 1)^n.

a(n) ~ c * 2^n * (n!)^2 / (sqrt(n) * (log(1+sqrt(2)))^(2*n)), where c = 0.521427744491499132141002572969819345522922990165233786929882335275903215... - Vaclav Kotesovec, Nov 05 2014, updated Jun 02 2022

A221078 E.g.f.: Sum_{n>=0} tan(n*x)^n.

Original entry on oeis.org

1, 1, 8, 164, 6400, 404176, 37541888, 4814990144, 815074508800, 176018678814976, 47223034903789568, 15407438848482919424, 6007522256082907955200, 2758698201106509138251776, 1473586749521302260021198848, 905915791153129699969076117504

Offset: 0

Paul D. Hanna, Dec 31 2012

nonn

Conjecture: Let p be prime. The sequence obtained by reducing a(n) modulo p for n >= 1 is purely periodic. If p = 4*m + 1 the period appears to be p - 1, while if p = 4*m + 3 the period appears to be 2*(p - 1). Cf. A245322. - Peter Bala, Jun 01 2022

			E.g.f.: A(x) = 1 + x + 8*x^2/2! + 164*x^3/3! + 6400*x^4/4! + 404176*x^5/5! +...
where
A(x) = 1 + tan(x) + tan(2*x)^2 + tan(3*x)^3 + tan(4*x)^4 + tan(5*x)^5 +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..224

Cf. A122399, A195415, A220181, A221077, A221079, A245322, A224899, A249489, A338040.

nmax = 20; CoefficientList[Series[1 + Sum[Tan[k*x]^k, {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, May 31 2022 *)
Join[{1}, Table[Sum[(-1)^((n-k)/2) * 2^n * k^n * Sum[(-1)^j * Binomial[k, j] * Sum[(-1)^m * Binomial[j + m - 1, m] * StirlingS2[n, m] * m! / 2^m, {m, 1, n}], {j, 0, k}], {k, 0, n}], {n, 1, 20}]] (* Vaclav Kotesovec, Jun 01 2022 *)

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

a(n) ~ c * d^n * (n!)^2 / sqrt(n), where d = 2.82830192319144609189890882712268369027077465204866199572119508594067235975..., c = 0.3460492649810724519960613805096579760009441161242336020188358769124140... - Vaclav Kotesovec, Nov 05 2014, updated Jun 02 2022

A245322 E.g.f.: Sum_{n>=0} sin(n*x)^n.

Original entry on oeis.org

1, 1, 8, 161, 6016, 360421, 31628288, 3823725821, 609263681536, 123729353398441, 31195066498285568, 9560281195915697081, 3500145542231863853056, 1508772905238685631514061, 756360258034794813559144448, 436312320288025061112662937941, 286966475921556619941746443288576

Offset: 0

Vaclav Kotesovec, Nov 05 2014

nonn

It appears that for n >= 1, a(2*n) is even and a(2*n-1) is odd. Conjecture: Let p be prime. The sequence obtained by reducing a(n) modulo p for n >= 1 is purely periodic. If p = 4*m + 1 the period appears to be p - 1, while if p = 4*m + 3 the period appears to be 2*(p - 1). Cf. A224899 and A221078. - Peter Bala, May 31 2022

Vaclav Kotesovec, Table of n, a(n) for n = 0..200

Cf. A122399, A195415, A220181, A221077, A221078, A224899, A249489, A338040.

nmax=20; Flatten[{1,Rest[CoefficientList[Series[Sum[Sin[k*x]^k, {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!]}]
Flatten[{1,Table[Sum[(-1)^k * (n-2*k)^n * 2^(2*k-n) * Sum[Binomial[n-2*k,j] * (-1)^j * (n-2*k-2*j)^n,{j,0,n-2*k}],{k,0,n/2}],{n,1,20}]}]

{a(n)=n!*polcoeff(sum(k=0, n, sin(k*x+x*O(x^n))^k), n)}
for(n=0, 20, print1(a(n), ", "))

a(n) ~ c * d^n * (n!)^2 / sqrt(n), where d = 2.6508143537621057095493599669955786931108630276472035393383790812849064745..., c = 0.447880926276318254580767843378566025547642779941081708311676940459098... - Vaclav Kotesovec, Nov 05 2014, updated Jun 02 2022

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

Original entry on oeis.org

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...

cp's OEIS Frontend

A249459 a(n) = Sum_{k=0..n} k^(2*n).

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A224899 E.g.f.: Sum_{n>=0} sinh(n*x)^n.

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A221077 E.g.f.: Sum_{n>=0} tanh(n*x)^n.

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A221078 E.g.f.: Sum_{n>=0} tan(n*x)^n.

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A245322 E.g.f.: Sum_{n>=0} sin(n*x)^n.

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

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