A006252 -id:A006252 - cp's OEIS frontend

A302547 Expansion of e.g.f. -log(1 - log(1 + x))/(1 - log(1 + x)).

0, 1, 2, 4, 11, 33, 131, 516, 2810, 12934, 97870, 447940, 5308112, 16394116, 450505844, -315178912, 60774618672, -394330113648, 12662225550288, -157622647720032, 3766647294946944, -64679214198647520, 1475157821754785184, -30431206030329719424, 719032203373502252160

Offset: 0

Ilya Gutkovskiy, Jun 20 2018

sign

			E.g.f.: A(x) = x + 2*x^2/2! + 4*x^3/3! + 11*x^4/4! + 33*x^5/5! + 131*x^6/6! + ...

Alois P. Heinz, Table of n, a(n) for n = 0..451

Cf. A000254, A001008, A002805, A006252, A073596, A089064, A222058, A300490, A302548.

H:= proc(n) H(n):= 1/n +`if`(n=1, 0, H(n-1)) end:
a:= n-> add(Stirling1(n, k)*H(k)*k!, k=1..n):
seq(a(n), n=0..27);  # Alois P. Heinz, Jun 21 2018

nmax = 24; CoefficientList[Series[-Log[1 - Log[1 + x]]/(1 - Log[1 + x]), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS1[n, k] HarmonicNumber[k] k!, {k, 0, n}], {n, 0, 24}]

a(n) = Sum_{k=1..n} Stirling1(n,k)*H(k)*k!, where H(k) is the k-th harmonic number.

A317280 Expansion of e.g.f. 1/(1 - log(1 + x))^2.

Original entry on oeis.org

1, 2, 4, 10, 30, 108, 444, 2112, 11040, 65712, 414816, 2992944, 21876816, 188936928, 1527813216, 15991733376, 133364903040, 1794144752640, 13329036288000, 270750383400960, 1167153128110080, 57074973648030720, -103080839984916480, 17319631144046423040, -171982551742151685120

Offset: 0

Ilya Gutkovskiy, Jul 25 2018

sign

Exponential self-convolution of A006252.

Seiichi Manyama, Table of n, a(n) for n = 0..451
N. J. A. Sloane, Transforms

Cf. A005444, A005649, A006252, A052801, A089064.

Cf. A006252, A354120, A354121.

a:=series(1/(1 - log(1 + x))^2, x=0, 25): seq(n!*coeff(a, x, n), n=0..24); # Paolo P. Lava, Mar 26 2019

nmax = 24; CoefficientList[Series[1/(1 - Log[1 + x])^2, {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS1[n, k] (k + 1)!, {k, 0, n}], {n, 0, 24}]

a(n) = Sum_{k=0..n} Stirling1(n,k)*(k + 1)!.
a(n) ~ n! * 2 * (-1)^(n+1) / (n * log(n)^3) * (1 - 3*(gamma+1) / log(n) + (6*gamma^2 + 12*gamma + 6 - Pi^2) / log(n)^2), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, May 15 2022
a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k-1) * (k/n + 1) * (k-1)! * binomial(n,k) * a(n-k). - Seiichi Manyama, Nov 19 2023

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

A352139 Expansion of e.g.f. 1/(exp(x) - log(1 - x)).

Original entry on oeis.org

1, -2, 6, -27, 161, -1205, 10799, -113043, 1351461, -18183781, 271784079, -4469044657, 80160267791, -1557710354083, 32597642189657, -730897865864471, 17480390183397209, -444198879957594857, 11951585821669838395, -339434402344422296117

Offset: 0

Seiichi Manyama, Mar 06 2022

sign

Seiichi Manyama, Table of n, a(n) for n = 0..421

Cf. A352138, A352146, A352147.

Cf. A006252, A038507.

m = 19; Range[0, m]! * CoefficientList[Series[1/(Exp[x] - Log[1 - x]), {x, 0, m}], x] (* Amiram Eldar, Mar 06 2022 *)

my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(exp(x)-log(1-x))))

a(n) = if(n==0, 1, -sum(k=1, n, ((k-1)!+1)*binomial(n, k)*a(n-k)));

a(0) = 1; a(n) = -Sum_{k=1..n} ((k-1)! + 1) * binomial(n,k) * a(n-k).

A354120 Expansion of e.g.f. 1/(1 - log(1 + x))^3.

Original entry on oeis.org

1, 3, 9, 30, 114, 492, 2388, 12912, 77016, 503112, 3570552, 27399600, 225729360, 1991996640, 18690559200, 186620451840, 1963991600640, 21914748541440, 255336518292480, 3155705206364160, 40209018105116160, 547746803311864320, 7525926332189130240

Offset: 0

Seiichi Manyama, May 17 2022

sign

a(34) is negative. - Vaclav Kotesovec, Jun 04 2022

Vaclav Kotesovec, Table of n, a(n) for n = 0..449

Cf. A006252, A317280, A354121.

Cf. A226515, A354122.

Table[Sum[(k+2)! * StirlingS1[n,k], {k,0,n}]/2, {n,0,35}] (* Vaclav Kotesovec, Jun 04 2022 *)
With[{nn=30},CoefficientList[Series[1/(1-Log[1+x])^3,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, May 16 2025 *)

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-log(1+x))^3))

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

a(n) = (1/2) * Sum_{k=0..n} (k + 2)! * Stirling1(n,k).

a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k-1) * (2 * k/n + 1) * (k-1)! * binomial(n,k) * a(n-k). - Seiichi Manyama, Nov 19 2023

A354121 Expansion of e.g.f. 1/(1 - log(1 + x))^4.

Original entry on oeis.org

1, 4, 16, 68, 316, 1616, 9080, 55800, 373080, 2699520, 21035040, 175708320, 1566916320, 14862171840, 149429426880, 1587766126080, 17779538050560, 209295747832320, 2583920845209600, 33389139008678400, 450642388471395840, 6342869733912760320

Offset: 0

Seiichi Manyama, May 17 2022

sign

a(46) is negative. - Vaclav Kotesovec, Jun 04 2022

It appears that a(n) is negative for even n >= 46. - Felix Fröhlich, Jun 04 2022

Vaclav Kotesovec, Table of n, a(n) for n = 0..450

Cf. A006252, A317280, A354120.

Cf. A226738, A354123.

Table[Sum[(k+3)! * StirlingS1[n,k], {k,0,n}]/6, {n,0,20}] (* Vaclav Kotesovec, Jun 04 2022 *)

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-log(1+x))^4))

a(n) = sum(k=0, n, (k+3)!*stirling(n, k, 1))/6;

a(n) = (1/6) * Sum_{k=0..n} (k + 3)! * Stirling1(n,k).

a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k-1) * (3 * k/n + 1) * (k-1)! * binomial(n,k) * a(n-k). - Seiichi Manyama, Nov 19 2023

A217033 Expansion of e.g.f. 1/(1 - log(1 - log(1-x))).

Original entry on oeis.org

1, 1, 2, 7, 33, 198, 1432, 12136, 117772, 1287718, 15658052, 209568126, 3061140398, 48454548452, 826155841924, 15094511153752, 294206836405288, 6093273074402848, 133628182522968752, 3093469935389714928, 75384936371166307872, 1928960833317580172688

Offset: 0

Paul D. Hanna, Sep 24 2012

nonn

			E.g.f.: A(x) = 1 + x + 2*x^2/2! + 7*x^3/3! + 33*x^4/4! + 198*x^5/5! +...

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

Cf. A089064, A305323.

CoefficientList[Series[1/(1-Log[1-Log[1-x]]), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Feb 12 2013 *)

{a(n)=n!*polcoeff(1/(1-log(1-log(1-x +x*O(x^n)))),n)}
for(n=0,25,print1(a(n),", "))

a(n) ~ n! * exp(2-exp(1))/(1-exp(1-exp(1)))^(n+1). - Vaclav Kotesovec, Feb 12 2013

a(n) = Sum_{k=0..n} |Stirling1(n,k)| * A006252(k). - Seiichi Manyama, May 11 2023

A347020 Expansion of e.g.f. 1 / (1 - 3 * log(1 + x))^(1/3).

Original entry on oeis.org

1, 1, 3, 18, 150, 1644, 22116, 353856, 6554376, 138001896, 3254445144, 84979363248, 2433814616592, 75858381808416, 2556180134677152, 92597465283789312, 3588434497019272320, 148134619713440384640, 6489652665043455707520, 300712023388466713739520

Offset: 0

Ilya Gutkovskiy, Aug 11 2021

nonn

Cf. A006252, A007559, A320343, A335531, A346982, A347015, A347021, A347022, A347023.

nmax = 19; CoefficientList[Series[1/(1 - 3 Log[1 + x])^(1/3), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS1[n, k] 3^k Pochhammer[1/3, k], {k, 0, n}], {n, 0, 19}]

a(n) = Sum_{k=0..n} Stirling1(n,k) * A007559(k).
a(n) ~ n! * exp(1/9) / (Gamma(1/3) * 3^(1/3) * n^(2/3) * (exp(1/3) - 1)^(n + 1/3)). - Vaclav Kotesovec, Aug 14 2021
a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k-1) * (3 - 2*k/n) * (k-1)! * binomial(n,k) * a(n-k). - Seiichi Manyama, Sep 11 2023

A347021 Expansion of e.g.f. 1 / (1 - 4 * log(1 + x))^(1/4).

Original entry on oeis.org

1, 1, 4, 32, 364, 5444, 100520, 2210760, 56406240, 1637877600, 53327583360, 1924096475520, 76198487927040, 3285955396558080, 153273199794071040, 7689131281851770880, 412809183978447306240, 23616192920003184176640, 1434201753814306170808320

Offset: 0

Ilya Gutkovskiy, Aug 11 2021

nonn

Cf. A006252, A007696, A320343, A346983, A347016, A347020, A347022, A347023.

nmax = 18; CoefficientList[Series[1/(1 - 4 Log[1 + x])^(1/4), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS1[n, k] 4^k Pochhammer[1/4, k], {k, 0, n}], {n, 0, 18}]

a(n) = Sum_{k=0..n} Stirling1(n,k) * A007696(k).
a(n) ~ n! * exp(1/16) / (Gamma(1/4) * 2^(1/2) * n^(3/4) * (exp(1/4) - 1)^(n + 1/4)). - Vaclav Kotesovec, Aug 14 2021
a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k-1) * (4 - 3*k/n) * (k-1)! * binomial(n,k) * a(n-k). - Seiichi Manyama, Sep 11 2023

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

cp's OEIS Frontend

A302547 Expansion of e.g.f. -log(1 - log(1 + x))/(1 - log(1 + x)).

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A317280 Expansion of e.g.f. 1/(1 - log(1 + x))^2.

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

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A352139 Expansion of e.g.f. 1/(exp(x) - log(1 - x)).

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A354120 Expansion of e.g.f. 1/(1 - log(1 + x))^3.

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A354121 Expansion of e.g.f. 1/(1 - log(1 + x))^4.

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A217033 Expansion of e.g.f. 1/(1 - log(1 - log(1-x))).

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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).