A249459 -id:A249459 - cp's OEIS frontend

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

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

A249489 a(n) = [x^n/n!] Sum_{k=0..n} cosh(k*x)^k.

Original entry on oeis.org

1, 0, 9, 0, 12070, 0, 126447741, 0, 5100496997940, 0, 562605048135059545, 0, 138523311740417986721274, 0, 66543520389763227261554370645, 0, 56664734898911130799849838608991176, 0, 79610326854782816434044397510470501877041

Offset: 0

Vaclav Kotesovec, Oct 30 2014

nonn

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

Cf. A224899, A249459, A245322, A221077, A221078.

Flatten[{1,Table[Sum[Sum[Binomial[k,j] * k^n*(k-2*j)^n / 2^k,{j,0,k}],{k,0,n}],{n,1,20}]}]

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

{a(n) = sum(k=0,n,sum(j=0,k, binomial(k, j) * k^n*(k-2*j)^n / 2^k ))}
for(n=0, 20, print1(a(n), ", ")) \\ Paul D. Hanna, Oct 15 2018, using Vaclav's formula.

a(n) = Sum_{k=0..n} Sum_{j=0..k} binomial(k, j) * k^n * (k-2*j)^n / 2^k. [Explicitly stating Vaclav's formula in Mma program - Paul D. Hanna, Oct 15 2018]

If n is even, then a(n) ~ c * (1-2*r)^n * n^(2*n) / (2^n * exp(n) * (r*(1-r))^(n/2)), where r = 0.0832217201995176507819192648878903254298041... is the root of the equation (r/(1-r))^(1-2*r) = exp(-2), and c = 2.09233700490262732901066903251002074102409436600891921766318742438...

Name clarified by Paul D. Hanna, Oct 15 2018

A349886 a(n) = Sum_{k=0..n} k^(k*n).

Original entry on oeis.org

1, 2, 18, 19749, 4295498995, 298024323402930834, 10314425729813391637014599924, 256923578002288684397369021397408936103993, 6277101735598268377660667072561845282166297358613176925573

Offset: 0

Seiichi Manyama, Dec 03 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..26

Cf. A031971, A062970, A249459.

Table[1 + Sum[k^(k*n), {k, 1, n}], {n, 0, 10}] (* Vaclav Kotesovec, Dec 04 2021 *)
a[n_] := Sum[If[k == 0, 1, k^(k*n)], {k, 0, n}]; Array[a, 9, 0] (* Amiram Eldar, Dec 04 2021 *)

PARI
```
a(n) = sum(k=0, n, k^(k*n));
```

my(N=10, x='x+O('x^N)); Vec(sum(k=0, N, k^k^2*x^k/(1-k^k*x)))

G.f.: Sum_{k>=0} k^(k^2) * x^k/(1 - k^k * x).

a(n) ~ n^(n^2). - Vaclav Kotesovec, Dec 04 2021

A352983 a(n) = Sum_{k=0..floor(n/2)} k^(2*n).

Original entry on oeis.org

1, 0, 1, 1, 257, 1025, 535538, 4799354, 4338079554, 69107159370, 96470431101379, 2401809362313955, 4798267740520031875, 172076350440523281571, 466164803742660494432996, 22761346686115003736962100, 80340572151131167125889902852

Offset: 0

Seiichi Manyama, Apr 13 2022

Seiichi Manyama, Table of n, a(n) for n = 0..240

Cf. A249459, A352984.

a[0] = 1; a[n_] := Sum[k^(2*n), {k, 0, Floor[n/2]}]; Array[a, 17, 0] (* Amiram Eldar, Apr 13 2022 *)

PARI
```
a(n) = sum(k=0, n\2, k^(2*n));
```

my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k^2*x)^(2*k)/(1-k^2*x)))

G.f.: Sum_{k>=0} (k^2 * x)^(2 * k) / (1 - k^2 * x).

a(n) ~ exp(3 + (-1)^n) * (n/2)^(2*n) / (exp(4) - 1). - Vaclav Kotesovec, Apr 14 2022

A349836 Expansion of Sum_{k>=0} (k * x)^k/(1 - k^2 * x).

Original entry on oeis.org

1, 1, 5, 44, 564, 9665, 211025, 5686104, 184813048, 7118824417, 320295658577, 16626717667348, 985178854556524, 66005199079345025, 4958773228726876257, 414664315430994701616, 38344259607889223269168, 3898112616839310343827009

Offset: 0

Seiichi Manyama, Dec 03 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..299

Cf. A031971, A349883.

Cf. A249459, A349884, A368505.

Join[{1}, Table[Sum[k^(2*n - k), {k, 0, n}], {n, 1, 20}]] (* Vaclav Kotesovec, Dec 04 2021 *)

a(n, t=2) = sum(k=0, n, k^(t*(n-k)+k));

my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k*x)^k/(1-k^2*x)))

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

a(n) ~ sqrt(Pi) * 2^(1 + 2*n - 2*n/LambertW(2*exp(1)*n)) * (n/LambertW(2*exp(1)*n))^(1/2 + 2*n - 2*n/LambertW(2*exp(1)*n)) / sqrt(1 + LambertW(2*exp(1)*n)). - Vaclav Kotesovec, Dec 04 2021

A349963 a(n) = Sum_{k=0..n} (2*k)^n.

Original entry on oeis.org

1, 2, 20, 288, 5664, 141600, 4298944, 153638912, 6319260672, 294044152320, 15272286131200, 875880428003328, 54976337351106560, 3748609104907476992, 275924407293425336320, 21806398621389422592000, 1841661678145084557099008, 165530736067119754944577536

Offset: 0

Seiichi Manyama, Dec 07 2021

nonn

Cf. A031971, A185393, A249459, A349970.

a[n_] := Sum[If[k == n == 0, 1, (2*k)^n], {k, 0, n}]; Array[a, 18, 0] (* Amiram Eldar, Dec 07 2021 *)

PARI
```
a(n) = sum(k=0, n, (2*k)^n);
```

my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (2*k*x)^k/(1-2*k*x)))

G.f.: Sum_{k>=0} (2*k * x)^k/(1 - 2*k * x).
a(n) = 2^n * A031971(n).
a(n) ~ c * 2^n * n^n, where c = 1/(1 - 1/exp(1)) = A185393. - Vaclav Kotesovec, Dec 07 2021

A349882 Expansion of Sum_{k>=0} k^2 * x^k/(1 - k^2 * x).

Original entry on oeis.org

0, 1, 5, 26, 162, 1267, 12343, 145652, 2036148, 33192789, 622384729, 13263528350, 318121600694, 8517247764135, 252725694989611, 8258153081400856, 295515712276222952, 11523986940937975401, 487562536078882116717, 22291094729329088403298

Offset: 0

Seiichi Manyama, Dec 03 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..339

Cf. A003101, A234568, A249459, A349863, A359659.

a[n_] := Sum[If[k == n - k + 1 == 0, 1, k^(2*(n - k + 1))], {k, 0, n}]; Array[a, 20, 0] (* Amiram Eldar, Dec 04 2021 *)

a(n, s=2, t=2) = sum(k=0, n, k^(t*(n-k)+s));

my(N=20, x='x+O('x^N)); concat(0, Vec(sum(k=0, N, k^2*x^k/(1-k^2*x))))

my(N=20, x='x+O('x^N)); concat(0, Vec(sum(k=1, N, x^k/(1-(k+1)^2*x)))) \\ Seiichi Manyama, Jan 12 2023

a(n) = Sum_{k=0..n} k^(2*(n-k+1)).
a(n) = A234568(n+1) - 1. - Hugo Pfoertner, Dec 04 2021
a(n) ~ sqrt(Pi) * ((n+1)/LambertW(exp(1)*(n+1)))^(5/2 + 2*n - 2*(n+1)/LambertW(exp(1)*(n+1))) / sqrt(1 + LambertW(exp(1)*(n+1))). - Vaclav Kotesovec, Dec 04 2021
G.f.: Sum_{k>=1} x^k/(1 - (k+1)^2 * x). - Seiichi Manyama, Jan 12 2023

A349889 a(n) = Sum_{k=0..n} (-1)^(n-k) * k^(2*n).

Original entry on oeis.org

1, 1, 15, 666, 59230, 8775075, 1948891581, 605698755508, 250914820143996, 133610836793706405, 88919025666286620475, 72317513878698256697166, 70571883548815735717843290, 81383769918571603591381635271

Offset: 0

Seiichi Manyama, Dec 04 2021

nonn

Cf. A120485, A249459, A349884, A349891, A349902.

Join[{1},Table[Sum[(-1)^(n-k) k^(2n),{k,0,n}],{n,20}]] (* Harvey P. Dale, Nov 19 2023 *)

a(n) = sum(k=0, n, (-1)^(n-k)*k^(2*n));

my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k^2*x)^k/(1+k^2*x)))

G.f.: Sum_{k>=0} (k^2 * x)^k/(1 + k^2 * x).

a(n) ~ 1/(1 + exp(-2)) * n^(2*n). - Vaclav Kotesovec, Dec 10 2021

A349901 a(n) = Sum_{k=0..n} k^(3*n).

Original entry on oeis.org

1, 1, 65, 20196, 17312754, 31605701625, 105443761093411, 580964060390826448, 4918745981990731659972, 60634331963604550954204425, 1043651859661187698792930519525, 24256699178432730349549665042311076, 740737411098120942914045235001015624310

Offset: 0

Seiichi Manyama, Dec 05 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..152

Cf. A249459, A349883, A349886.

PARI
```
a(n) = sum(k=0, n, k^(3*n));
```

my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k^3*x)^k/(1-k^3*x)))

G.f.: Sum_{k>=0} (k^3 * x)^k/(1 - k^3 * x).

a(n) ~ exp(3)/(exp(3)-1) * n^(3*n). - Vaclav Kotesovec, Dec 05 2021

A356687 a(n) = n! * Sum_{k=0..n} k^(2*n)/k!.

Original entry on oeis.org

1, 1, 18, 927, 94876, 16251045, 4210190766, 1543550310211, 764096247603480, 493254380867214249, 404269328278061434810, 411862088865696890314311, 512690851568229926690616948, 768775988931240685277619894157

Offset: 0

Seiichi Manyama, Aug 23 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..196

Cf. A256016, A356688.

Cf. A030297, A249459, A356672.

a[n_] := n! * Sum[k^(2*n)/k!, {k, 0, n}]; a[0] = 1; Array[a, 14, 0] (* Amiram Eldar, Aug 23 2022 *)

PARI
```
a(n) = n!*sum(k=0, n, k^(2*n)/k!);
```

my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k^2*x)^k/(k!*(1-k^2*x)))))

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

cp's OEIS Frontend

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

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A249489 a(n) = [x^n/n!] Sum_{k=0..n} cosh(k*x)^k.

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A349886 a(n) = Sum_{k=0..n} k^(k*n).

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A352983 a(n) = Sum_{k=0..floor(n/2)} k^(2*n).

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A349836 Expansion of Sum_{k>=0} (k * x)^k/(1 - k^2 * x).

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A349963 a(n) = Sum_{k=0..n} (2*k)^n.

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A349882 Expansion of Sum_{k>=0} k^2 * x^k/(1 - k^2 * x).

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