A000482 -id:A000482 - cp's OEIS frontend

A243570 Unsigned Stirling numbers of the first kind s(n,9).

1, 45, 1320, 32670, 749463, 16669653, 368411615, 8207628000, 185953177553, 4308105301929, 102417740732658, 2503858755467550, 63030812099294896, 1634980697246583456, 43714229649594412832

Offset: 9

Stanislav Sykora, Jun 06 2014

Stanislav Sykora, Table of n, a(n) for n = 9..308
Wikipedia, Stirling numbers of the first kind

Cf. A000254, A000399, A000454, A000482, A001233, A001234, A008275, A243569.

Drop[Table[Abs[StirlingS1[n, 9]], {n, 0, 20}], 9] (* Vaclav Kotesovec, Jun 06 2014 *)

PARI
```
abs(stirling(n,9))
```

A346924 Expansion of e.g.f. 1 / (1 + log(1 - x)^5 / 5!).

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 15, 175, 1960, 22449, 269577, 3430790, 46480830, 671260876, 10329270952, 169125055736, 2940784282800, 54182845939104, 1055291277366108, 21674715826211532, 468366193441002564, 10624074081842024496, 252432685158931968768, 6270222495850552958004

Offset: 0

Ilya Gutkovskiy, Aug 07 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..444

Cf. A007840, A346921, A346922, A346923.

Cf. A000482, A347004, A353200.

nmax = 23; CoefficientList[Series[1/(1 + Log[1 - x]^5/5!), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] Abs[StirlingS1[k, 5]] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 23}]

my(x='x+O('x^25)); Vec(serlaplace(1/(1+log(1-x)^5/5!))) \\ Michel Marcus, Aug 07 2021

a(n) = sum(k=0, n\5, (5*k)!*abs(stirling(n, 5*k, 1))/120^k); \\ Seiichi Manyama, May 06 2022

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * |Stirling1(k,5)| * a(n-k).
a(n) ~ n! * 2^(3/5) * 3^(1/5) * exp(2^(3/5)*15^(1/5)*n) / (5^(4/5) * (exp(2^(3/5)*15^(1/5)) - 1)^(n+1)). - Vaclav Kotesovec, Aug 08 2021
a(n) = Sum_{k=0..floor(n/5)} (5*k)! * |Stirling1(n,5*k)|/120^k. - Seiichi Manyama, May 06 2022

A347004 Expansion of e.g.f. exp( -log(1 - x)^5 / 5! ).

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 15, 175, 1960, 22449, 269451, 3423860, 46238280, 664233856, 10143487354, 164423078456, 2823768543960, 51272283444264, 982177492263750, 19807082824819374, 419629806223448346, 9320808413229618816, 216645165604679499072, 5259724543984442886486

Offset: 0

Ilya Gutkovskiy, Aug 10 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..448

Cf. A000482, A327506, A346924, A347001, A347002, A347003.

nmax = 23; CoefficientList[Series[Exp[-Log[1 - x]^5/5!], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] Abs[StirlingS1[k, 5]] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 23}]

a(n) = sum(k=0, n\5, (5*k)!*abs(stirling(n, 5*k, 1))/(120^k*k!)); \\ Seiichi Manyama, May 06 2022

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * |Stirling1(k,5)| * a(n-k).

a(n) = Sum_{k=0..floor(n/5)} (5*k)! * |Stirling1(n,5*k)|/(120^k * k!). - Seiichi Manyama, May 06 2022

A348065 Coefficient of x^4 in expansion of n!* Sum_{k=0..n} binomial(x,k).

Original entry on oeis.org

1, -5, 55, -350, 3969, -31563, 408050, -3920950, 58206676, -657328100, 11111159696, -144321864960, 2747845864464, -40364369180016, 856755330487200, -14042902728462624, 329258021171239296, -5956512800554963584, 153050034289602269952, -3028534064042216488704, 84691080748928315003904

Offset: 4

Seiichi Manyama, Sep 26 2021

sign

Column k=4 of A190782.

Cf. A000482, A054651, A028341, A347987, A348063, A348064, A348068.

a(n) = n!*polcoef(sum(k=4, n, binomial(x, k)), 4);

N=40; x='x+O('x^N); Vec(serlaplace(log(1+x)^4/(24*(1-x))))

from sympy.abc import x
from sympy import ff, expand
def A348065(n): return sum(ff(n,n-k)*expand(ff(x,k)).coeff(x**4) for k in range(4,n+1)) # Chai Wah Wu, Sep 27 2021

E.g.f.: (log(1 + x))^4/(24 * (1 - x)).

A348068 Coefficient of x^5 in expansion of n!* Sum_{k=0..n} binomial(x,k).

Original entry on oeis.org

1, -9, 112, -1064, 12873, -140595, 1870385, -23551110, 351042406, -5043110072, 84074954600, -1361614072000, 25218570009424, -455365645674480, 9298765013106384, -185409487083100320, 4144212593899945056, -90492302454898284864, 2199399908894486591040

Offset: 5

Seiichi Manyama, Sep 27 2021

sign

Column k=5 of A190782.

Cf. A000482, A001233, A054651, A347987, A348063, A348064, A348065.

a(n) = n!*polcoef(sum(k=5, n, binomial(x, k)), 5);

N=40; x='x+O('x^N); Vec(serlaplace(log(1+x)^5/(120*(1-x))))

from sympy.abc import x
from sympy import ff, expand
def A348068(n): return sum(ff(n,n-k)*expand(ff(x,k)).coeff(x**5) for k in range(5,n+1)) # Chai Wah Wu, Sep 27 2021

E.g.f.: (log(1 + x))^5/(120 * (1 - x)).

A346947 Expansion of e.g.f. log( 1 + log(1 + x)^5 / 5! ).

Original entry on oeis.org

1, -15, 175, -1960, 22449, -269451, 3423860, -46238280, 664233856, -10143487354, 164423204582, -2823783679080, 51273355515264, -982236541934430, 19809898439192946, -419752648063849626, 9325875631405818996, -216846992855331506052, 5267598064689049209252

Offset: 5

Ilya Gutkovskiy, Aug 08 2021

sign

Cf. A000482, A003713, A346944, A346945, A346946.

nmax = 23; CoefficientList[Series[Log[1 + Log[1 + x]^5/5!], {x, 0, nmax}], x] Range[0, nmax]! // Drop[#, 5] &
a[n_] := a[n] = StirlingS1[n, 5] - (1/n) Sum[Binomial[n, k] StirlingS1[n - k, 5] k a[k], {k, 1, n - 1}]; Table[a[n], {n, 5, 23}]

a(n) = Stirling1(n,5) - (1/n) * Sum_{k=1..n-1} binomial(n,k) * Stirling1(n-k,5) * k * a(k).

a(n) = Sum_{k=1..floor(n/5)} (-1)^(k-1) * (5*k)! * Stirling1(n,5*k)/(k * 120^k). - Seiichi Manyama, Jan 23 2025

A052767 Expansion of e.g.f.: -(log(1-x))^5.

Original entry on oeis.org

0, 0, 0, 0, 0, 120, 1800, 21000, 235200, 2693880, 32319000, 410031600, 5519487600, 78864820320, 1194924450720, 19166592681600, 324817601472000, 5803921108010880, 109115988701293440, 2154085473710580480, 44566174481427360000, 964537418717406213120, 21799797542483649131520

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Andrew Howroyd, Table of n, a(n) for n = 0..200
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 724

Column k=5 of A225479.

Cf. A000482, A052517.

spec := [S,{B=Cycle(Z),S=Prod(B,B,B,B,B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

With[{nn=20},CoefficientList[Series[-(Log[1-x])^5,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Oct 14 2019 *)

a(n) = {5!*stirling(n,5,1)*(-1)^(n+1)} \\ Andrew Howroyd, Jul 27 2020

E.g.f.: log(-1/(-1+x))^5.
Recurrence: a(1)=0, a(0)=0, a(2)=0, a(4)=0, a(3)=0, (-1-5*n-10*n^2-10*n^3-5*n^4-n^5)*a(n+1) + (31+5*n^4+70*n^2+30*n^3+75*n)*a(n+2) + (-125*n-90-60*n^2-10*n^3)*a(n+3) + (10*n^2+65+50*n)*a(n+4) + (-15-5*n)*a(n+5) + a(n+6)=0, a(5)=120.
a(n) = 120*A000482(n) = 5!*Stirling1(n,5)*(-1)^(n+1). - Andrew Howroyd, Jul 27 2020

Definition clarified by Harvey P. Dale, Oct 14 2019

Terms a(20) and beyond from Andrew Howroyd, Jul 27 2020

A126678 Product_{i=5..n} |Stirling_1(i,5)|.

Original entry on oeis.org

1, 15, 2625, 5145000, 115500105000, 31107065779125000, 106290666272665586250000, 4888916787397632685446712500000, 3213029533312882851092617168726650000000, 31994396252008620689512942684899448442297400000000, 5110196342132990821731075942773950194550115556089232000000000

Offset: 5

N. J. A. Sloane, Feb 13 2007

nonn

Partial products of A000482.

A372973 Triangle read by rows: the exponential almost-Riordan array ( 1/(1-x) | 1/(1-x), log(1/(1-x)) ).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 6, 2, 3, 1, 24, 6, 11, 6, 1, 120, 24, 50, 35, 10, 1, 720, 120, 274, 225, 85, 15, 1, 5040, 720, 1764, 1624, 735, 175, 21, 1, 40320, 5040, 13068, 13132, 6769, 1960, 322, 28, 1, 362880, 40320, 109584, 118124, 67284, 22449, 4536, 546, 36, 1

Offset: 0

Stefano Spezia, May 26 2024

			The triangle begins:
    1;
    1,   1;
    2,   1,   1;
    6,   2,   3,   1;
   24,   6,  11,   6,  1;
  120,  24,  50,  35, 10,  1;
  720, 120, 274, 225, 85, 15, 1;
  ...

Y. Alp and E. G. Kocer, Exponential Almost-Riordan Arrays, Results Math 79, 173 (2024). See page 6.

Cf. A000012 (right diagonal), A000254, A000399 (k=3), A000454 (k=4), A000482 (k=5), A001233 (k=6), A001234 (k=7), A098558 (row sums), A179865 (subdiagonal), A243569 (k=8), A243570 (k=9).

Triangle A130534 with 1st column A000142.

T[n_,0]:=n!; T[n_,k_]:=(n-1)!/(k-1)!SeriesCoefficient[1/(1-x)Log[1/(1-x)]^(k-1),{x,0,n-1}]; Table[T[n,k],{n,0,9},{k,0,n}]//Flatten

T(n,0) = n!; T(n,k) = (n-1)!/(k-1)! * [x^(n-1)] log(1/(1-x))^(k-1)/(1-x).
T(n,1) = (n-1)! for n > 0.
T(n,2) = A000254(n-1) for n > 1.

cp's OEIS Frontend

A243570 Unsigned Stirling numbers of the first kind s(n,9).

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A346924 Expansion of e.g.f. 1 / (1 + log(1 - x)^5 / 5!).

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A347004 Expansion of e.g.f. exp( -log(1 - x)^5 / 5! ).

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A348065 Coefficient of x^4 in expansion of n!* Sum_{k=0..n} binomial(x,k).

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A348068 Coefficient of x^5 in expansion of n!* Sum_{k=0..n} binomial(x,k).

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A346947 Expansion of e.g.f. log( 1 + log(1 + x)^5 / 5! ).

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Mathematica

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A052767 Expansion of e.g.f.: -(log(1-x))^5.

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A126678 Product_{i=5..n} |Stirling_1(i,5)|.

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A372973 Triangle read by rows: the exponential almost-Riordan array ( 1/(1-x) | 1/(1-x), log(1/(1-x)) ).

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