A320083 -id:A320083 - cp's OEIS frontend

A122399 a(n) = Sum_{k=0..n} k^n * k! * Stirling2(n,k).

1, 1, 9, 211, 9285, 658171, 68504709, 9837380491, 1863598406805, 450247033371451, 135111441590583909, 49300373690091496171, 21495577955682021043125, 11037123350952586270549531, 6591700149366720366704735109

Offset: 0

Vladeta Jovovic, Aug 31 2006

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, 2, 1, 3, 3, 0, 1, 2, 1, 3, 3, 0, ...], with an apparent period of 6. Cf. A338040. - Peter Bala, May 31 2022

			E.g.f.: A(x) = 1 + x + 9*x^2/2! + 211*x^3/3! + 9285*x^4/4! + 658171*x^5/5! + ...
such that
A(x) = 1 + (exp(x)-1) + (exp(2*x)-1)^2 + (exp(3*x)-1)^3 + (exp(4*x)-1)^4 + ...
The e.g.f. is also given by the series:
A(x) = 1/2 + exp(x)/(1+exp(x))^2 + exp(4*x)/(1+exp(2*x))^3 + exp(9*x)/(1+exp(3*x))^4 + exp(16*x)/(1+exp(4*x))^5 + exp(25*x)/(1+exp(5*x))^6 + ...
or, equivalently,
A(x) = 1/2 + exp(-x)/(1+exp(-x))^2 + exp(-2*x)/(1+exp(-2*x))^3 + exp(-3*x)/(1+exp(-3*x))^4 + exp(-4*x)/(1+exp(-4*x))^5 + exp(-5*x)/(1+exp(-5*x))^6 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A195415, A122400, A220181, A224899, A245322, A320083, A338040.

a := n -> add(k^n*k!*combinat[stirling2](n,k),k=0..n); # Max Alekseyev, Feb 01 2007

Flatten[{1,Table[Sum[k^n*k!*StirlingS2[n,k],{k,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, Jun 21 2013 *)

{a(n)=polcoeff(sum(m=0, n, m^m*m!*x^m/prod(k=1, m, 1-m*k*x+x*O(x^n))), n)}
for(n=0, 20, print1(a(n), ", ")) \\ Paul D. Hanna, Jan 05 2013

{a(n)=n!*polcoeff(sum(k=0, n, (exp(k*x +x*O(x^n)) - 1)^k), n)}
for(n=0,25,print1(a(n),", ")) \\ Paul D. Hanna, Oct 26 2014

/* From e.g.f. infinite series: */
\p100 \\ set precision
{A=Vec(serlaplace(sum(n=0, 500, 1.*exp(n^2*x +O(x^26))/(1 + exp(n*x +O(x^26)))^(n+1)) ))}
for(n=0, #A-1, print1(round(A[n+1]), ", ")) \\ Paul D. Hanna, Oct 30 2014

E.g.f.: Sum_{n >= 0} (exp(n*x) - 1)^n. - Vladeta Jovovic, Sep 03 2006
E.g.f.: Sum_{n>=0} exp(n^2*x) / (1 + exp(n*x))^(n+1). - Paul D. Hanna, Oct 26 2014
E.g.f.: Sum_{n>=0} exp(-n*x) / (1 + exp(-n*x))^(n+1). - Paul D. Hanna, Oct 30 2014
O.g.f.: Sum_{n>=0} n^n * n! * x^n / Product_{k=1..n} (1 - n*k*x). - Paul D. Hanna, Jan 05 2013
Limit n->infinity (a(n)/n!)^(1/n)/n = ((1+exp(1/r))*r^2)/exp(1) = A317855/exp(1) = 1.162899527477400818845..., where r = 0.87370243323966833... is the root of the equation 1/(1+exp(-1/r)) = -r*LambertW(-exp(-1/r)/r). - Vaclav Kotesovec, Jun 21 2013
a(n) ~ c * A317855^n * (n!)^2 / sqrt(n), where c = 0.327628285569869481442286492410507030710253054522608... - Vaclav Kotesovec, Aug 09 2018
Let A(x) = 1 + x + 9*x^2/2! + 211*x^3/3! + ... denote the e.g.f. of the sequence. Let F(x) denote the series reversion of A(x) - 1 = x - 9*x^2/2 + 16*x^3/3 - 205*x^4/4 - 2714*x^5/5 - .... Then both dF/dx = 1 - 9*x + 16*x^2 - 205*x^3 - 2714*x^4 - ... and exp(F(x)) = 1 + x - 4*x^2 + x^3 - 38*x^4 - 606*x^5 - ... have integer coefficients. Note that 1 + series reversion(exp(F(x)) - 1) is the o.g.f. for A122400. - Peter Bala, Aug 09 2022

More terms from Max Alekseyev, Feb 01 2007

A320096 a(n) = Sum_{k=1..n} (-1)^(n-k) * Stirling1(n,k) * k! * k^n, with a(0)=1.

Original entry on oeis.org

1, 1, 9, 212, 9418, 675014, 71092502, 10334690232, 1982433606264, 485065343565072, 147433546709109408, 54493722609862927632, 24069397682825072219040, 12520250948941157091235344, 7575515622713954399390221008, 5275250174853125498317783254528

Offset: 0

Vaclav Kotesovec, Oct 05 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..229

Cf. A007840, A220181, A320083.

Flatten[{1, Table[Sum[(-1)^(n-k)*StirlingS1[n, k]*k!*k^n, {k, 1, n}], {n, 1, 20}]}]
nmax = 20; CoefficientList[Series[1 + Sum[(-Log[1 - k*x])^k, {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jun 04 2022 *)

a(n) = sum(k=0, n, (-1)^(n-k)*k!*k^n*stirling(n, k, 1)); \\ Seiichi Manyama, Feb 02 2022

my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (-log(1-k*x))^k))) \\ Seiichi Manyama, Feb 02 2022

a(n) ~ c * d^n * n^(2*n + 1/2), where
w = -LambertW(-1, -exp(-r)*r) = 1.1628296650659469964248518258036278907318113...
r = 0.8531304407911771560472963194514988627832723535823134189532... is the real root of the equation w = r + exp(-1/r)
d = exp(-1)*r*w*(w-r)^(r-1) = 0.433513333588184444899487502412976956849408575992...
c = 1.959633090979666812031505093625147349925787002426082...
E.g.f.: Sum_{k>=0} (-log(1 - k*x))^k. - Seiichi Manyama, Feb 02 2022

A350721 a(n) = Sum_{k=0..n} k! * k^(k+n) * Stirling1(n,k).

Original entry on oeis.org

1, 1, 31, 4184, 1495534, 1110325474, 1481505320078, 3225820132807320, 10696978730747904696, 51287741246274865567776, 341442095880058160040860592, 3055472627228313328903357352784, 35788671820468495762774011478900032

Offset: 0

Seiichi Manyama, Feb 03 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..161

Cf. A320083, A350719, A350720.

Cf. A350722, A351135.

a[0] = 1; a[n_] := Sum[k! * k^(k+n) * StirlingS1[n, k], {k, 1, n}]; Array[a, 13, 0] (* Amiram Eldar, Feb 03 2022 *)

a(n) = sum(k=0, n, k!*k^(k+n)*stirling(n, k, 1));

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

E.g.f.: Sum_{k>=0} (k * log(1 + k*x))^k.

a(n) ~ exp(-exp(-2)/2) * n! * n^(2*n). - Vaclav Kotesovec, Feb 04 2022

A351133 a(n) = Sum_{k=0..n} k! * k^(2n) Stirling1(n,k).

Original entry on oeis.org

1, 1, 31, 3992, 1342294, 932514674, 1161340476698, 2356863300156504, 7278091701243797640, 32477694155566998880608, 201155980661221409458717152, 1674230688936725338278370413264, 18235249164492209082483584810706528

Offset: 0

Seiichi Manyama, Feb 02 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..162

Cf. A006252, A320083, A351134.

Cf. A229260, A351135, A351136.

a[0] = 1; a[n_] := Sum[k! * k^(2*n) * StirlingS1[n, k], {k, 1, n}]; Array[a, 13, 0] (* Amiram Eldar, Feb 02 2022 *)

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

first(n)=my(x='x+O('x^(n+1))); Vec(serlaplace(sum(k=0, n, log(1+k^2*x)^k)))

E.g.f.: Sum_{k>=0} log(1 + k^2*x)^k.

a(n) ~ c * d^n * n^(3*n + 1/2), where d = 0.3417329834649268103028466896966197580428514873775849996969994420891... and c = 2.92355271092039591960355156784704285135358... - Vaclav Kotesovec, Feb 03 2022

A351135 a(n) = Sum_{k=0..n} k! * k^(kn) Stirling1(n,k).

Original entry on oeis.org

1, 1, 31, 117716, 103060088854, 35762522985456876854, 7426384178533125493811949517898, 1294894823429942179301223205449027573956692920, 253092741940931724343266089700550691376738432767085871485096840

Offset: 0

Seiichi Manyama, Feb 02 2022

nonn

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

Cf. A006252, A320083, A351133, A351134.

Cf. A350721, A351117, A351138.

a[0] = 1; a[n_] := Sum[k! * k^(k*n) * StirlingS1[n, k], {k, 1, n}]; Array[a, 9, 0] (* Amiram Eldar, Feb 02 2022 *)

a(n) = sum(k=0, n, k!*k^(k*n)*stirling(n, k, 1));

my(N=10, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, log(1+k^k*x)^k)))

E.g.f.: Sum_{k>=0} log(1 + k^k*x)^k.

a(n) ~ n! * n^(n^2). - Vaclav Kotesovec, Feb 03 2022

A351134 a(n) = Sum_{k=0..n} k! * k^(3n) Stirling1(n,k).

Original entry on oeis.org

1, 1, 127, 115028, 383611414, 3407421330934, 66396378581670602, 2493320561997330821496, 164454446238949941359354760, 17769323863754938530919641304080, 2978930835291629440372517431365668448, 741834782450714229554166000654848368247568

Offset: 0

Seiichi Manyama, Feb 02 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..125

Cf. A006252, A320083, A351133.

Cf. A242229, A351135, A351137.

a[0] = 1; a[n_] := Sum[k! * k^(3*n) * StirlingS1[n, k], {k, 1, n}]; Array[a, 12, 0] (* Amiram Eldar, Feb 02 2022 *)

a(n) = sum(k=0, n, k!*k^(3*n)*stirling(n, k, 1));

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

E.g.f.: Sum_{k>=0} log(1 + k^3*x)^k.

a(n) ~ c * d^n * n^(4*n + 1/2), where d = 0.358437102792682941192966771107499325675345706113923587904567864366079667... and c = 2.68150179193269103258189978938660205530269361522513... - Vaclav Kotesovec, Feb 04 2022

A351280 a(n) = Sum_{k=0..n} k! * k^k * Stirling1(n,k).

Original entry on oeis.org

1, 1, 7, 140, 5254, 318854, 28455182, 3506576856, 570360248856, 118356589567440, 30512901324706608, 9566812017770347152, 3584662956711860108352, 1581905384865801328253712, 812047187127758913474118032, 479763784808095613489811245568

Offset: 0

Seiichi Manyama, Feb 06 2022

nonn

Cf. A006252, A305819, A320083, A350721, A350725, A351281.

a[0] = 1; a[n_] := Sum[k! * k^k * StirlingS1[n, k], {k, 1, n}]; Array[a, 16, 0] (* Amiram Eldar, Feb 06 2022 *)

a(n) = sum(k=0, n, k!*k^k*stirling(n, k, 1));

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

E.g.f.: Sum_{k>=0} (k * log(1+x))^k.

a(n) ~ exp(-exp(-1)/2) * n! * n^n. - Vaclav Kotesovec, Feb 06 2022

A373869 a(n) = Sum_{k=1..n} k! * k^(n-3) * Stirling1(n,k).

Original entry on oeis.org

0, 1, 0, 2, 26, 674, 28894, 1848216, 165229560, 19698788448, 3022496261616, 580460752264656, 136441193196585408, 38540172064949405616, 12883204327833557091984, 5030833813902039858261504, 2269484487197629285690675584, 1171368942033975021150888242304

Offset: 0

Seiichi Manyama, Jun 20 2024

nonn

Cf. A373870, A373871, A373872.

Cf. A320083, A373857, A373874.

a(n) = sum(k=1, n, k!*k^(n-3)*stirling(n, k, 1));

E.g.f.: Sum_{k>=1} log(1 + k*x)^k / k^3.

A350719 a(n) = Sum_{k=0..n} k! * 2^k * k^n * Stirling1(n,k).

Original entry on oeis.org

1, 2, 30, 1108, 76372, 8463328, 1375868768, 308440047648, 91189383264864, 34376022491122368, 16093445542120281792, 9160424435706947112576, 6230035512106223752576896, 4989402076922846372194268160, 4647526704475074504983564884992

Offset: 0

Seiichi Manyama, Feb 03 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..221

Cf. A320083, A350720.

Cf. A088501, A195005, A350721.

a[0] = 1; a[n_] := Sum[k! * 2^k * k^n * StirlingS1[n, k], {k, 1, n}]; Array[a, 15, 0] (* Amiram Eldar, Feb 03 2022 *)

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

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

E.g.f.: Sum_{k>=0} (2 * log(1 + k*x))^k.

A350720 a(n) = Sum_{k=0..n} k! * 3^k * k^n * Stirling1(n,k).

Original entry on oeis.org

1, 3, 69, 3948, 422082, 72567522, 18304992558, 6367730357160, 2921446409138136, 1709074810258369776, 1241694104839498851552, 1096850187800368469477424, 1157691464039682741551221152, 1438880771284303822650674399664

Offset: 0

Seiichi Manyama, Feb 03 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..213

Cf. A320083, A350719.

Cf. A195263, A335531, A350721.

a[0] = 1; a[n_] := Sum[k! * 3^k * k^n * StirlingS1[n, k], {k, 1, n}]; Array[a, 14, 0] (* Amiram Eldar, Feb 03 2022 *)

a(n) = sum(k=0, n, k!*3^k*k^n*stirling(n, k, 1));

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

E.g.f.: Sum_{k>=0} (3 * log(1 + k*x))^k.

cp's OEIS Frontend

A122399 a(n) = Sum_{k=0..n} k^n * k! * Stirling2(n,k).

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A320096 a(n) = Sum_{k=1..n} (-1)^(n-k) * Stirling1(n,k) * k! * k^n, with a(0)=1.

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A350721 a(n) = Sum_{k=0..n} k! * k^(k+n) * Stirling1(n,k).

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A351133 a(n) = Sum_{k=0..n} k! * k^(2*n) * Stirling1(n,k).

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A351135 a(n) = Sum_{k=0..n} k! * k^(k*n) * Stirling1(n,k).

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A351134 a(n) = Sum_{k=0..n} k! * k^(3*n) * Stirling1(n,k).

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A351280 a(n) = Sum_{k=0..n} k! * k^k * Stirling1(n,k).

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A351133 a(n) = Sum_{k=0..n} k! * k^(2n) Stirling1(n,k).

A351135 a(n) = Sum_{k=0..n} k! * k^(kn) Stirling1(n,k).

A351134 a(n) = Sum_{k=0..n} k! * k^(3n) Stirling1(n,k).