A351136 -id:A351136 - cp's OEIS frontend

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

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

A351137 a(n) = Sum_{k=0..n} (-1)^(n-k) * k! * k^(3n) Stirling1(n,k).

Original entry on oeis.org

1, 1, 129, 121172, 421875178, 3922823960054, 80130334773241142, 3156849112458066440568, 218554371053209725986724984, 24795129220015277612148345850896, 4365539219231132131300647267518575008, 1141930521329052244894253748456776246166288

Offset: 0

Seiichi Manyama, Feb 02 2022

nonn

In general, for m >= 0, Sum_{k=0..n} (-1)^(n-k) * k! * k^(m*n) * Stirling1(n,k) ~ c * r^(m*n) * (1 + r*exp(m/r))^n * n^((m+1)*n + 1/2) / exp((m+1)*n), where r is the real root of the equation LambertW(-1, -r*exp(-r)) = -r - exp(-m/r) and c is a constant (depending only on m). - Vaclav Kotesovec, Feb 04 2022

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

Cf. A007840 (m=0), A320096 (m=1), A351136 (m=2).

Cf. A203798, A351134, A351138.

a[0] = 1; a[n_] := Sum[(-1)^(n - k) * 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, (-1)^(n-k)*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 * r^(3*n) * (1 + r*exp(3/r))^n * n^(4*n + 1/2) / exp(4*n), where r = 0.97698437755148201976772582981871258235824532360125531194... is the real root of the equation LambertW(-1, -r*exp(-r)) = -r - exp(-3/r) and c = 2.3655154360078103511101518906595610482889989819... - Vaclav Kotesovec, Feb 04 2022

A351138 a(n) = Sum_{k=0..n} (-1)^(n-k) * k! * k^(kn) Stirling1(n,k).

Original entry on oeis.org

1, 1, 33, 118484, 103098352618, 35763050751038414134, 7426387531294394110580641088438, 1294894837982331434068068403253026516109577144, 253092742000650212462862632240661689524832716838851180353875064

Offset: 0

Seiichi Manyama, Feb 02 2022

nonn

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

Cf. A007840, A320096, A351136, A351137.

Cf. A249584, A351135.

a[0] = 1; a[n_] := Sum[(-1)^(n - k) * 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, (-1)^(n-k)*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

A373856 a(n) = Sum_{k=1..n} k! * k^(2n-1) |Stirling1(n,k)|.

Original entry on oeis.org

0, 1, 17, 1652, 474770, 301474214, 357901156354, 712632435944568, 2204970751341231816, 10017874331177386762512, 63973486554110386836270096, 554598491512901862814742673168, 6344773703149123365957506715989568, 93563015826037060521986513216617599504

Offset: 0

Seiichi Manyama, Jun 19 2024

nonn

Cf. A003713, A373855.
Cf. A220179, A242228, A373858.
Cf. A351136, A373861.

nmax=13; Range[0,nmax]!CoefficientList[Series[Sum[(-Log[1 - k^2*x])^k / k,{k,nmax}],{x,0,nmax}],x] (* Stefano Spezia, Jun 19 2024 *)

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

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

cp's OEIS Frontend

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

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A351137 a(n) = Sum_{k=0..n} (-1)^(n-k) * k! * k^(3*n) * Stirling1(n,k).

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A351138 a(n) = Sum_{k=0..n} (-1)^(n-k) * k! * k^(k*n) * Stirling1(n,k).

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PARI

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A373856 a(n) = Sum_{k=1..n} k! * k^(2*n-1) * |Stirling1(n,k)|.

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Formula

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

A351137 a(n) = Sum_{k=0..n} (-1)^(n-k) * k! * k^(3n) Stirling1(n,k).

A351138 a(n) = Sum_{k=0..n} (-1)^(n-k) * k! * k^(kn) Stirling1(n,k).

A373856 a(n) = Sum_{k=1..n} k! * k^(2n-1) |Stirling1(n,k)|.