A242228 -id:A242228 - cp's OEIS frontend

A220179 E.g.f.: Sum_{n>=1} (1 - exp(-n^2*x))^n / n.

1, 15, 1267, 316275, 174397531, 179770837155, 310789895286907, 834906367019076675, 3293344593080631993211, 18259284528276047000517795, 137429981152689382429349060347, 1365009985652048448232840864764675, 17475885712645599218827214639383437691

Offset: 1

Paul D. Hanna, Dec 06 2012

nonn

Compare to the trivial identity: x = Sum_{n>=1} (1 - exp(-x))^n/n.

Compare to the e.g.f. of A092552: Sum_{n>=1} (1 - exp(-n*x))^n/n.

			E.g.f.: A(x) = x + 15*x^2/2! + 1267*x^3/3! + 316275*x^4/4! + 174397531*x^5/5! +...
where
A(x) = (1-exp(-x)) + (1-exp(-4*x))^2/2 + (1-exp(-9*x))^3/3 + (1-exp(-16*x))^4/4 + (1-exp(-25*x))^5/5 +...

Seiichi Manyama, Table of n, a(n) for n = 1..162

Cf. A092552, A187755, A220181, A242228.

Table[Sum[(-1)^(n-k) * k^(2*n-1) * k! * StirlingS2[n,k], {k,1,n}], {n,1,20}] (* Vaclav Kotesovec, May 08 2014 *)

a(n)=n!*polcoeff(sum(k=1, n, (1-exp(-k^2*x+x*O(x^n)))^k/k), n)
for(n=1,20,print1(a(n),", "))

a(n)=polcoeff(sum(m=1, n, m^(2*m-1)*m!*x^m/prod(k=1, m, 1+m^2*k*x+x*O(x^n))), n) \\ Paul D. Hanna, Jan 05 2013
for(n=1,20,print1(a(n),", "))

{a(n)=sum(k=1, n, (-1)^(n-k)*k^(2*n-1)*k!*stirling(n, k, 2))}
for(n=1, 20, print1(a(n), ", ")) \\ Paul D. Hanna, Jan 05 2013

O.g.f.: Sum_{n>=1} n^(2*n-1) * n! * x^n / Product_{k=1..n} (1 - n^2*k*x). - Paul D. Hanna, Jan 05 2013
a(n) = Sum_{k=1..n} (-1)^(n-k) * k^(2*n-1) * k! * Stirling2(n,k). - Paul D. Hanna, Jan 05 2013
a(n) ~ c * d^n * (n!)^3 / n^2, where d = 6.8312860494079582446988970296645779575650627187418208311407895492635... and c = 0.175744118254830086361220160145768507562830495967... . - Vaclav Kotesovec, May 08 2014

A242229 a(n) = Sum_{k=0..n} k^(3n) k! * StirlingS2(n,k).

Original entry on oeis.org

1, 1, 129, 121171, 421842405, 3921960731851, 80097035486409669, 3154805675402432477371, 218356776433458097793841045, 24765902586563160053438320367371, 4359137561016969073655241431827801509, 1139916274502953599866121961715757905518171

Offset: 0

Vaclav Kotesovec, May 08 2014

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

Cf. A203798, A242228

Cf. A000670, A122399, A229260.

Table[Sum[k^(3*n) * k! * StirlingS2[n,k], {k,0,n}], {n,0,20}]

a(n) = sum(k=0, n, k!*k^(3*n)*stirling(n, k, 2)); \\ Seiichi Manyama, Feb 01 2022

my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (exp(k^3*x)-1)^k))) \\ Seiichi Manyama, Feb 01 2022

my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, k!*(k^3*x)^k/prod(j=1, k, 1-k^3*j*x))) \\ Seiichi Manyama, Feb 01 2022

a(n) ~ c * d^n * (n!)^4 / n^(3/2), where d = 20.5647332000203822461493845960846630764635... = r^4*(1+exp(3/r)), r = 0.97762267432285928683132021521727105447350... is the root of the equation (1+exp(-3/r))*LambertW(-exp(-1/r)/r) = -1/r, and c = 0.0600744446309702764688382302731840300640714536...
E.g.f.: Sum_{k>=0} (exp(k^3*x) - 1)^k. - Seiichi Manyama, Feb 01 2022
G.f.: Sum_{k>=0} k! * (k^3*x)^k/Product_{j=1..k} (1 - k^3*j*x). - Seiichi Manyama, Feb 01 2022

a(0)=1 prepended by Seiichi Manyama, Feb 01 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.

A373858 a(n) = Sum_{k=1..n} k! * k^(2n-1) Stirling1(n,k).

Original entry on oeis.org

0, 1, 15, 1268, 317294, 175542694, 181641609214, 315309390376056, 850661260866748728, 3370191684116333977872, 18768704088141613880906736, 141902519646656406912522712848, 1415862822521619228707500717132224, 18210234893009450819658863637633454608

Offset: 0

Seiichi Manyama, Jun 19 2024

nonn

Cf. A220179, A242228, A373856.

Cf. A351133, A351183.

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)*stirling(n, k, 1));

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

cp's OEIS Frontend

A220179 E.g.f.: Sum_{n>=1} (1 - exp(-n^2*x))^n / n.

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A242229 a(n) = Sum_{k=0..n} k^(3*n) * k! * StirlingS2(n,k).

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

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Formula

A373858 a(n) = Sum_{k=1..n} k! * k^(2*n-1) * Stirling1(n,k).

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Author

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Crossrefs

Programs

Mathematica

PARI

Formula

A242229 a(n) = Sum_{k=0..n} k^(3n) k! * StirlingS2(n,k).

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

A373858 a(n) = Sum_{k=1..n} k! * k^(2n-1) Stirling1(n,k).