A000399 -id:A000399 - cp's OEIS frontend

A346922 Expansion of e.g.f. 1 / (1 + log(1 - x)^3 / 3!).

1, 0, 0, 1, 6, 35, 245, 2044, 19572, 210524, 2513760, 33012276, 472963876, 7340889192, 122703087416, 2197496734224, 41979155247520, 852063971170960, 18312093589455440, 415420659953439840, 9920128280950954080, 248735658391768241280, 6533773435848445617600

Offset: 0

Ilya Gutkovskiy, Aug 07 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..437

Cf. A007840, A346921, A346923, A346924.

Cf. A000399, A346894, A347002, A353118.

nmax = 22; CoefficientList[Series[1/(1 + Log[1 - x]^3/3!), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] Abs[StirlingS1[k, 3]] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 22}]

my(x='x+O('x^25)); Vec(serlaplace(1/(1+log(1-x)^3/3!))) \\ Michel Marcus, Aug 07 2021

a(n) = sum(k=0, n\3, (3*k)!*abs(stirling(n, 3*k, 1))/6^k); \\ Seiichi Manyama, May 06 2022

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * |Stirling1(k,3)| * a(n-k).
a(n) ~ n! * 6^(1/3) / (3 * exp(6^(1/3)) * (1 - exp(-6^(1/3)))^(n+1)). - Vaclav Kotesovec, Aug 08 2021
a(n) = Sum_{k=0..floor(n/3)} (3*k)! * |Stirling1(n,3*k)|/6^k. - Seiichi Manyama, May 06 2022

A347002 Expansion of e.g.f. exp( -log(1 - x)^3 / 3! ).

Original entry on oeis.org

1, 0, 0, 1, 6, 35, 235, 1834, 16352, 163764, 1818030, 22143726, 293476326, 4203311892, 64682865156, 1064154324024, 18636296872320, 346103784493560, 6793394350116600, 140508244952179200, 3054120126193160280, 69596730438090806880, 1659041650323705102840

Offset: 0

Ilya Gutkovskiy, Aug 10 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..447

Cf. A000399, A327504, A346922, A347001, A347003, A347004.

nmax = 22; CoefficientList[Series[Exp[-Log[1 - x]^3/3!], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] Abs[StirlingS1[k, 3]] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 22}]

a(n) = sum(k=0, n\3, (3*k)!*abs(stirling(n, 3*k, 1))/(6^k*k!)); \\ Seiichi Manyama, May 06 2022

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * |Stirling1(k,3)| * a(n-k).

a(n) = Sum_{k=0..floor(n/3)} (3*k)! * |Stirling1(n,3*k)|/(6^k * k!). - Seiichi Manyama, May 06 2022

A243569 Unsigned Stirling numbers of the first kind s(n,8).

Original entry on oeis.org

1, 36, 870, 18150, 357423, 6926634, 135036473, 2681453775, 54631129553, 1146901283528, 24871845297936, 557921681547048, 12953636989943896, 311333643161390640, 7744654310169576800, 199321978221066137360

Offset: 8

Stanislav Sykora, Jun 06 2014

Stanislav Sykora, Table of n, a(n) for n = 8..307
Wikipedia, Stirling numbers of the first kind

Cf. A000254, A000399, A000454, A000482, A001233, A001234, A008275, A243570.

Drop[Table[Abs[StirlingS1[n, 8]], {n, 0, 20}], 8] (* Vaclav Kotesovec, Jun 06 2014 *)

PARI
```
abs(stirling(n,8))
```

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

Original entry on oeis.org

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))
```

A163939 Triangle related to the o.g.f.s. of the right hand columns of A163934 (E(x,m=4,n)).

Original entry on oeis.org

1, 6, 4, 35, 60, 10, 225, 690, 325, 20, 1624, 7588, 6762, 1316, 35, 13132, 85288, 120358, 46928, 4508, 56, 118124, 1004736, 2028660, 1298860, 265365, 13896, 84, 1172700, 12529400, 33896400, 31862400, 11077255, 1313610, 39915, 120

Offset: 1

Johannes W. Meijer, Aug 13 2009

The asymptotic expansions of the higher order exponential integral E(x,m=4,n) lead to triangle A163934, see A163931 for information on the E(x,m,n). The o.g.f.s. of the right hand columns of triangle A163934 have a nice structure Gf(p) = W4(z,p)/(1-z)^(2*p+2) with p = 1 for the first right hand column, p = 2 for the second right hand column, etc.. The coefficients of the W4(z,p) polynomials lead to the triangle given above, n >= 1 and 1 <= m <= n. The row sums of this triangle lead to A000457, see A163936 for more information.

			The first few W4(z,p) polynomials are:
W4(z,p=1) = 1/(1-z)^4
W4(z,p=2) = (6+4*z)/(1-z)^6
W4(z,p=3) = (35+60*z+10*z^2)/(1-z)^8
W4(z,p=4) = (225+690*z+325*z^2+20*z^3)/(1-z)^10

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Row sums equal A000457.
A000399 equals the first left hand column.
A000292 equals the first right hand column.
Cf. A163931 (E(x,m,n)) and A163934.
Cf. A163936 (E(x,m=1,n)), A163937 (E(x,m=2,n)) and A163938 (E(x,m=3,n)).

with(combinat): a := proc(n, m): add((-1)^(n+k+1)*((m-k+2)*(m-k+1)*(m-k)/3!)*binomial(2*n+2, k)*stirling1(m+n-k+1, m-k+2), k=0..m-1) end: seq(seq(a(n, m), m=1..n), n=1..8); # Johannes W. Meijer, revised Nov 27 2012

Table[Sum[(-1)^(n + k + 1)*Binomial[m - k + 2, 3]*Binomial[2*n + 2, k]*StirlingS1[m + n - k + 1, m - k + 2], {k, 0, m - 1}], {n, 1, 50}, {m, 1, n}] // Flatten (* G. C. Greubel, Aug 13 2017 *)

for(n=1,10, for(m=1,n, print1(sum(k=0, m-1, (-1)^(n+k+1)* binomial(m-k+2,3)* binomial(2*n+2,k)*stirling(m+n-k+1,m-k+2,1)), ", "))) \\ G. C. Greubel, Aug 13 2017

a(n,m) = Sum_{k=0..(m-1)} (-1)^(n+k+1)*binomial(m-k+2,3)* binomial(2*n+2,k)*stirling1(m+n-k+1,m-k+2), for 1<= m <=n.

A062254 3rd level triangle related to Eulerian numbers and binomial transforms (A062253 is second level, triangle of Eulerian numbers is first level and triangle with Z(0,0)=1 and Z(n,k)=0 otherwise is 0th level).

Original entry on oeis.org

1, 6, 0, 25, 10, 0, 90, 120, 15, 0, 301, 896, 406, 21, 0, 966, 5376, 5586, 1176, 28, 0, 3025, 28470, 55560, 27910, 3123, 36, 0, 9330, 139320, 456525, 437100, 122520, 7860, 45, 0, 28501, 646492, 3312078, 5339719, 2912833, 494802, 19096, 55, 0

Offset: 0

Henry Bottomley, Jun 14 2001

Binomial transform of n^3*k^n is ((kn)^3 + 3(kn)^2 + (1 - k)(kn))*(k + 1)^(n - 3); of n^4*k^n is ((kn)^4 + 6(kn)^3 + (7 - 4k)(kn)^2 + (1 - 4k + k^2)(kn))*(k + 1)^(n - 4); of n^5*k^n is ((kn)^5 + 10(kn)^4 + (25 - 10k)(kn)^3 + (15 - 30k + 5k^2)(kn)^2 + (1 - 11k + 11k^2 - k^3)(kn))*(k + 1)^(n - 5); of n^6*k^n is ((kn)^6 + 15(kn)^5 + (65 - 20k)(kn)^4 + (90 - 120k + 15k^2)(kn)^3 + (31 - 146k + 91k^2 - 6k^3)(kn)^2 + (1 - 26k + 66k^2 - 26k^3 + k^4)(kn))*(k + 1)^(n - 6). This sequence gives the (unsigned) polynomial coefficients of (kn)^3.

			Rows start:
 (1),
 (6,0),
 (25,10,0),
 (90,120,15,0),
 ...

First column is A000392. Diagonals include A000007 and all but the start of A000217. Row sums are A000399.
Taking all the levels together to create a pyramid, one face would be A010054 as a triangle with a parallel face which is Pascal's triangle (A007318) with two columns removed, another face would be a triangle of Stirling numbers of the second kind (A008277) and a third face would be A000007 as a triangle, (cont.)
(cont.) with a triangle of Eulerian numbers (A008292), A062253, A062254 and A062255 as faces parallel to it. The row sums of this last group would provide a triangle of unsigned Stirling numbers of the first kind (A008275).

E(n, k) = if ((n<0) || (k<0), 0, if ((n==0) && (k==0), 1, (k+1)*E(n-1, k)+(n-k)*E(n-1, k-1)));
A2(n, k) = if ((n<0) || (k<0), 0, (k+2)*A2(n-1, k)+(n-k)*A2(n-1, k-1)+E(n, k));
A3(n, k) = if ((n<0) || (k<0), 0, (k+3)*A3(n-1, k)+(n-k)*A3(n-1, k-1) + A2(n, k));
row3(n) = vector(n+1, k, A3(n,k-1)); \\ Michel Marcus, Jan 27 2025

A(n, k) = (k+3)*A(n-1, k) + (n-k)*A(n-1, k-1) + A062253(n, k).

A341588 E.g.f.: -log(1 + log(1 - x))^3 / 6.

Original entry on oeis.org

1, 12, 130, 1485, 18508, 253400, 3805723, 62437500, 1113510409, 21479997957, 446094038806, 9930796412082, 236037249893092, 5968192832899412, 160007282538148508, 4534905316824903144, 135500246340709682692, 4257646241716404353684, 140366073694357927723936, 4845119946789226304526392

Offset: 3

Ilya Gutkovskiy, Feb 15 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 3..418

Cf. A000399, A000559, A003713, A008275, A039814, A341587.

nmax = 22; CoefficientList[Series[-Log[1 + Log[1 - x]]^3/6, {x, 0, nmax}], x] Range[0, nmax]! // Drop[#, 3] &
Table[Sum[Abs[StirlingS1[n, k] StirlingS1[k, 3]], {k, 3, n}], {n, 3, 22}]

a(n) = Sum_{k=3..n} |Stirling1(n, k) * Stirling1(k, 3)|.

a(n) ~ (n-1)! * log(n)^2 / (2 * (1 - exp(-1))^n) * (1 + (2*gamma - 2*log(exp(1) - 1)) / log(n) + (gamma^2 - Pi^2/6 - 2*log(exp(1) - 1)*gamma + log(exp(1)-1)^2) / log(n)^2). - Vaclav Kotesovec, Jun 04 2022

A052748 Expansion of e.g.f.: -(log(1-x))^3.

Original entry on oeis.org

0, 0, 0, 6, 36, 210, 1350, 9744, 78792, 708744, 7036200, 76521456, 905507856, 11589357312, 159580302336, 2352940786944, 36994905688320, 617953469022720, 10929614667747840, 204073497562936320, 4011658382046919680, 82822558521844224000, 1791791417179298304000

Offset: 0

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

Original name: A simple grammar.

Andrew Howroyd, Table of n, a(n) for n = 0..200
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 704

Column k=3 of A225479.

Cf. A000399, A052517.

spec := [S,{B=Cycle(Z),S=Prod(B,B,B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
with(combinat):seq(stirling1(j, 3)*3!*(-1)^(j+1), j=0..50); # Leonid Bedratyuk, Aug 07 2012

a(n) = {3!*stirling(n,3,1)*(-1)^(n+1)} \\ Andrew Howroyd, Jul 27 2020

E.g.f.: log(1/(1-x))^3.
Recurrence: {a(1)=0, a(0)=0, a(2)=0, a(3)=6, (-1 - 3*n - 3*n^2 - n^3)*a(n+1) + (9*n + 7 + 3*n^2)*a(n+2) + (-6 - 3*n)*a(n+3) + a(n+4)}.
a(n) = stirling1(n, 3)*3!*(-1)^(n+1). - Leonid Bedratyuk, Aug 07 2012
a(n) = 6*A000399(n). - Andrew Howroyd, Jul 27 2020

Name changed and terms a(20) and beyond from Andrew Howroyd, Jul 27 2020

A127054 Rectangular table, read by antidiagonals, defined by the following rule: start with all 1's in row zero; from then on, row n+1 equals the partial sums of row n excluding terms in columns k = m*(m+1)/2 (m>=1).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 10, 9, 4, 1, 1, 34, 33, 15, 5, 1, 1, 154, 153, 65, 23, 6, 1, 1, 874, 873, 339, 119, 32, 7, 1, 1, 5914, 5913, 2103, 719, 186, 42, 8, 1, 1, 46234, 46233, 15171, 5039, 1230, 267, 54, 9, 1, 1, 409114, 409113, 124755, 40319, 9258, 1891, 380

Offset: 0

Paul D. Hanna, Jan 04 2007

Variant of table A125781. Generated by a method similar to Moessner's factorial triangle (A125714).

			Rows are partial sums excluding terms in columns k = {1,3,6,10,...}:
row 2 = partial sums of [1, 3, 5,6, 8,9,10, 12,13,14,15, ...];
row 3 = partial sums of [1, 9, 23,32, 54,67,81, 113,131,150,170, ...];
row 4 = partial sums of [1, 33, 119,186, 380,511,661, 1045,1283,...].
The terms that are excluded in the partial sums are shown enclosed in
parenthesis in the table below. Rows of this table begin:
1,(1), 1, (1), 1, 1, (1), 1, 1, 1, (1), 1, 1, 1, 1, (1), 1, ...;
1,(2), 3, (4), 5, 6, (7), 8, 9, 10, (11), 12, 13, 14, 15, (16), ...;
1,(4), 9, (15), 23, 32, (42), 54, 67, 81, (96), 113, 131, 150, ...;
1,(10), 33, (65), 119, 186, (267), 380, 511, 661, (831), 1045, ...;
1,(34), 153, (339), 719, 1230, (1891), 2936, 4219, 5765, (7600), ...;
1,(154), 873, (2103), 5039, 9258, (15023), 25148, 38203, 54625, ..;
1,(874), 5913, (15171), 40319, 78522, (133147), 238124, 379339, ...;
1,(5914), 46233, (124755), 362879, 742218, (1305847), 2477468, ...;
1,(46234), 409113, (1151331), 3628799, 7742058, (14059423), ...;
1,(409114), 4037913, (11779971), 39916799, 88369098, (164977399),...;
Columns include:
k=1: A003422 (Left factorials: !n = Sum k!, k=0..n-1);
k=2: A007489 (Sum of k!, k=1..n);
k=3: A097422 (Sum{k=1 to n} H(k) k!, where H(k) = sum{j=1 to k} 1/j);
k=4: A033312 (n! - 1);
k=5: Partial sums of A001705;
k=6: partial sums of A000399 (Stirling numbers of first kind s(n,3)).

Cf. variants: A125781, A125714; antidiagonal sums: A127055; diagonal: A127056; columns: A003422, A007489, A097422, A033312.

{T(n,k)=local(A=0,b=2,c=0,d=0);if(n==0,A=1, until(d>k,if(c==b*(b-1)/2,b+=1,A+=T(n-1,c);d+=1);c+=1));A}

A348063 Coefficient of x^2 in expansion of n!* Sum_{k=0..n} binomial(x,k).

Original entry on oeis.org

1, 0, 11, 5, 304, 364, 15980, 34236, 1368936, 4429656, 173699712, 771653376, 30605906304, 175622947200, 7149130156800, 50800930272000, 2137822335475200, 18241636315507200, 796397873127782400, 7971407298921830400, 361615771356450508800, 4168685961862906982400, 196587429737202833817600

Offset: 2

Seiichi Manyama, Sep 26 2021

nonn

Column k=2 of A190782.

Cf. A000399, A028339, A054651, A347987, A348064, A348065, A348068.

a(n) = n!*polcoef(sum(k=2, n, binomial(x, k)), 2);

a(n) = if(n<2, 0, a(n-1)+(n-1)^2*a(n-2)+(-1)^n*(n-2)!);

N=40; x='x+O('x^N); Vec(serlaplace(log(1+x)^2/(2*(1-x))))

from sympy.abc import x
from sympy import ff, expand
def A348063(n): return sum(ff(n,n-k)*expand(ff(x,k)).coeff(x**2) for k in range(2,n+1)) # Chai Wah Wu, Sep 27 2021

a(n) = a(n-1) + (n-1)^2 * a(n-2) + (-1)^n * (n-2)!.
E.g.f.: (log(1 + x))^2/(2 * (1 - x)).
a(n) ~ n! * log(2)^2 / 2 * (1 + (-1)^n*log(n)/(log(2)^2*n)). - Vaclav Kotesovec, Sep 27 2021

cp's OEIS Frontend

A346922 Expansion of e.g.f. 1 / (1 + log(1 - x)^3 / 3!).

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A347002 Expansion of e.g.f. exp( -log(1 - x)^3 / 3! ).

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A243569 Unsigned Stirling numbers of the first kind s(n,8).

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

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A163939 Triangle related to the o.g.f.s. of the right hand columns of A163934 (E(x,m=4,n)).

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A062254 3rd level triangle related to Eulerian numbers and binomial transforms (A062253 is second level, triangle of Eulerian numbers is first level and triangle with Z(0,0)=1 and Z(n,k)=0 otherwise is 0th level).

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A341588 E.g.f.: -log(1 + log(1 - x))^3 / 6.

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A052748 Expansion of e.g.f.: -(log(1-x))^3.

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