A000454 -id:A000454 - cp's OEIS frontend

A346923 Expansion of e.g.f. 1 / (1 - log(1 - x)^4 / 4!).

1, 0, 0, 0, 1, 10, 85, 735, 6839, 69804, 784580, 9680000, 130312336, 1901581968, 29895585356, 503657235900, 9051009737834, 172807817059664, 3493189152511608, 74530548004474584, 1673793045085649146, 39467836062718058100, 974939402596817961050, 25177327470510057799550

Offset: 0

Ilya Gutkovskiy, Aug 07 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..441

Cf. A007840, A346921, A346922, A346924.

Cf. A000454, A346895, A347003, A353119.

nmax = 23; CoefficientList[Series[1/(1 - Log[1 - x]^4/4!), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] Abs[StirlingS1[k, 4]] 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)^4/4!))) \\ Michel Marcus, Aug 07 2021

a(n) = sum(k=0, n\4, (4*k)!*abs(stirling(n, 4*k, 1))/24^k); \\ Seiichi Manyama, May 06 2022

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * |Stirling1(k,4)| * a(n-k).
a(n) ~ n! * 2^(-5/4) * 3^(1/4) / (exp(2^(3/4)*3^(1/4)) * (1 - exp(-2^(3/4)*3^(1/4)))^(n+1)). - Vaclav Kotesovec, Aug 08 2021
a(n) = Sum_{k=0..floor(n/4)} (4*k)! * |Stirling1(n,4*k)|/24^k. - Seiichi Manyama, May 06 2022

A347003 Expansion of e.g.f. exp( log(1 - x)^4 / 4! ).

Original entry on oeis.org

1, 0, 0, 0, 1, 10, 85, 735, 6804, 68544, 754130, 9044750, 117773656, 1656897528, 25061576176, 405667844400, 6997383182854, 128126051451184, 2481884332498848, 50702417505257904, 1089371806098805286, 24555007848629510700, 579348221233739760550, 14278529041496660104450

Offset: 0

Ilya Gutkovskiy, Aug 10 2021

nonn

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

Cf. A000454, A327505, A346923, A347001, A347002, A347004.

nmax = 23; CoefficientList[Series[Exp[Log[1 - x]^4/4!], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] Abs[StirlingS1[k, 4]] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 23}]

a(n) = sum(k=0, n\4, (4*k)!*abs(stirling(n, 4*k, 1))/(24^k*k!)); \\ Seiichi Manyama, May 06 2022

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * |Stirling1(k,4)| * a(n-k).

a(n) = Sum_{k=0..floor(n/4)} (4*k)! * |Stirling1(n,4*k)|/(24^k * k!). - Seiichi Manyama, May 06 2022

A062255 4th level triangle related to Eulerian numbers and binomial transforms (A062254 is third level, 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, 10, 0, 65, 20, 0, 350, 350, 35, 0, 1701, 3696, 1316, 56, 0, 7770, 30660, 24570, 4200, 84, 0, 34105, 220620, 325620, 131020, 12195, 120, 0, 145750, 1447050, 3513345, 2656720, 613140, 33330, 165, 0, 611501, 8901992, 33074448, 41503484, 18444833, 2634192, 87406, 220, 0

Offset: 0

Henry Bottomley, Jun 14 2001

Binomial transform 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)^4.

			Rows start:
 (1),
 (10,0),
 (65,20,0),
 (350,350,35,0), etc.

First column is A000453. Diagonals include A000007 and all but the start of A000292. Row sums are A000454. 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, 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));
A4(n, k) = if ((n<0) || (k<0), 0, (k+4)*A4(n-1, k)+(n-k)*A4(n-1, k-1)+ A3(n, k));
row4(n) = vector(n+1, k, A4(n,k-1)); \\ Michel Marcus, Jan 27 2025

A(n, k) = (k+4)*A(n-1, k)+(n-k)*A(n-1, k-1) + A062254(n, k).

More terms from Michel Marcus, Jan 27 2025

A346946 Expansion of e.g.f. log( 1 + log(1 + x)^4 / 4! ).

Original entry on oeis.org

1, -10, 85, -735, 6734, -66024, 693230, -7774250, 92759821, -1172483598, 15630569591, -218793782025, 3201481037819, -48746860400024, 768683653934928, -12487871805640344, 207761719406853466, -3513910668343842900, 59833161662103132050, -1011244718827893629750

Offset: 4

Ilya Gutkovskiy, Aug 08 2021

sign

Cf. A000454, A003713, A346944, A346945, A346947.

Cf. A346976.

nmax = 23; CoefficientList[Series[Log[1 + Log[1 + x]^4/4!], {x, 0, nmax}], x] Range[0, nmax]! // Drop[#, 4] &
a[n_] := a[n] = StirlingS1[n, 4] - (1/n) Sum[Binomial[n, k] StirlingS1[n - k, 4] k a[k], {k, 1, n - 1}]; Table[a[n], {n, 4, 23}]

a(n) = Stirling1(n,4) - (1/n) * Sum_{k=1..n-1} binomial(n,k) * Stirling1(n-k,4) * k * a(k).

a(n) = Sum_{k=1..floor(n/4)} (-1)^(k-1) * (4*k)! * Stirling1(n,4*k)/(k * 24^k). - Seiichi Manyama, Jan 23 2025

A348064 Coefficient of x^3 in expansion of n!* Sum_{k=0..n} binomial(x,k).

Original entry on oeis.org

1, -2, 25, -75, 1099, -4340, 79064, -382060, 8550916, -48306984, 1303568760, -8346754416, 266955481584, -1894529909376, 70785236377728, -547468189825536, 23610353987137536, -196402650598402560, 9679304091074250240, -85687212859582878720, 4785340778000524477440

Offset: 3

Seiichi Manyama, Sep 26 2021

sign

Column k=3 of A190782.

Cf. A000399, A000454, A054651, A081051, A028340, A347987, A348063, A348065, A348068.

a(n) = n!*polcoef(sum(k=3, n, binomial(x, k)), 3);

N=40; x='x+O('x^N); Vec(serlaplace(log(1+x)^3/(6*(1-x))))

from sympy.abc import x
from sympy import ff, expand
def A348064(n): return sum(ff(n,n-k)*expand(ff(x,k)).coeff(x**3) for k in range(3,n+1)) # Chai Wah Wu, Sep 27 2021

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

A052753 Expansion of e.g.f.: log(1-x)^4.

Original entry on oeis.org

0, 0, 0, 0, 24, 240, 2040, 17640, 162456, 1614816, 17368320, 201828000, 2526193824, 33936357312, 487530074304, 7463742249600, 121367896891776, 2089865973021696, 37999535417459712, 727710096185266176, 14642785817771802624, 308902349883623731200, 6818239581643475251200

Offset: 0

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

Previous name was: A simple grammar.

G. C. Greubel, Table of n, a(n) for n = 0..448
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 709

Column k=4 of A225479.

Cf. A000454, A052517.

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

CoefficientList[Series[(Log[1-x])^4, {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Sep 30 2013 *)

x='x+O('x^30); concat(vector(4), Vec(serlaplace((log(1-x))^4))) \\ G. C. Greubel, Aug 30 2018

a(n) = {4!*stirling(n,4,1)*(-1)^n} \\ Andrew Howroyd, Jul 27 2020

E.g.f.: log(-1/(-1+x))^4.
Recurrence: {a(1)=0, a(0)=0, a(2)=0, (1+4*n+6*n^2+4*n^3+n^4)*a(n+1) + (-4*n^3-15-18*n^2-28*n)*a(n+2) + (6*n^2+24*n+25)*a(n+3) + (-4*n-10)*a(n+4)+a(n+5), a(3)=0, a(4)=24}.
a(n) ~ (n-1)! * 2*log(n)*(2*log(n)^2 + 6*gamma*log(n) - Pi^2 + 6*gamma^2), where gamma is Euler-Mascheroni constant (A001620). - Vaclav Kotesovec, Sep 30 2013
a(n) = 24*A000454(n) = 4!*(-1)^n*Stirling1(n,4). - Andrew Howroyd, Jul 27 2020

New name using e.g.f., Vaclav Kotesovec, Sep 30 2013

A188881 Triangle of coefficients arising from an expansion of Integral( exp(exp(exp(x))), dx).

Original entry on oeis.org

1, 1, 1, 2, 3, 2, 6, 11, 12, 6, 24, 50, 70, 60, 24, 120, 274, 450, 510, 360, 120, 720, 1764, 3248, 4410, 4200, 2520, 720, 5040, 13068, 26264, 40614, 47040, 38640, 20160, 5040, 40320, 109584, 236248, 403704, 538776, 544320, 393120, 181440, 40320

Offset: 1

N. J. A. Sloane, Apr 14 2011

Also the coefficients of the polynomials which are generated by the exponential generating function -log(1 + x*log(1 - t)). The polynomials might be called 'logarithmic polynomials'. Note also A003713, and A263634 for a different use of this term. See the paper of F. Qi for a related, but different family of polynomials. - Peter Luschny, Jul 11 2020
Edgar remarks that these coefficients are related to Stirling numbers of the second kind (cf. A008277).
The first column and the main diagonal are the factorials (A000142). The n-th entry on the first subdiagonal is A001710(n+1). The second column is A000254, the third column is 2*A000399, and the fourth column is 6*A000454. In general, the k-th column is (k-1)!*s(n,k), where s(n,k) is the unsigned Stirling number of the first kind. - Nathaniel Johnston, Apr 15 2011
With offset n=0, k=0 : triangle T(n,k), read by rows,given by T(n,k) = k*T(n-1, k-1) + n*T(n-1, k) with T(0, 0) = 1. - Philippe Deléham, Oct 04 2011

			Triangle begins:
1
1    1
2    3    2
6    11   12   6
24   50   70   60   24
120  274  450  510  360  120
...

Nathaniel Johnston, Table of n, a(n) for n = 1..2500
G. A. Edgar, Transseries for beginners, arXiv:0801.4877 [math.RA], 2008-2009.
F. Qi, On multivariate logarithmic polynomials and their properties, Indagationes Mathematicae (2018).
Wikipedia, Stirling numbers of the first kind

Cf. A277408, A003713, A263634.

S:=proc(n,k)global s:if(n=0 and k=0)then s[0,0]:=1:elif(n=0 or k=0)then s[n,k]:=0:elif(not type(s[n,k],integer))then s[n,k]:=(n-1)*S(n-1,k)+S(n-1,k-1):fi:return s[n,k]:end:
T:=proc(n,k)return (k-1)!*S(n,k);end:
for n from 1 to 6 do for k from 1 to n do print(T(n,k)):od:od: # Nathaniel Johnston, Apr 15 2011
# With offset n = 0, k = 0:
A188881 := (n, k) -> k!*abs(Stirling1(n+1, k+1)):
seq(seq(A188881(n,k), k=0..n), n=0..8); # Peter Luschny, Oct 19 2017
# Alternative:
gf := -log(1 + x*log(1 - t)): ser := series(gf, t, 18):
toeff := n -> n!*expand(coeff(ser, t, n)):
seq(print(seq(coeff(toeff(n), x, k), k=1..n)), n=1..8); # Peter Luschny, Jul 10 2020

Table[(k - 1)! * Sum[StirlingS2[i, k] * (-1)^(n - i) * StirlingS1[n, i], {i, 0, k}], {n, 9}, {k, n}] // Flatten (* Michael De Vlieger, Apr 17 2015 *)

T(n,k):=(k-1)!*sum(stirling2(i,k)*(-1)^(n-i)*stirling1(n,i),i,0,k); /* Vladimir Kruchinin, Apr 17 2015 */

{T(n, k) = if( k<1 || k>n, 0, (n-1)! * polcoeff( (x / (1 - exp(-x * (1 + x * O(x^n)))))^n, n-k))}; /* Michael Somos, May 10 2017 */

{T(n, k) = if( k<1 || n<0, 0, (k-1)! * sum(i=0, k, stirling(i, k, 2) * (-1)^(n-i) * stirling(n, i, 1)))}; /* Michael Somos, May 10 2017 */

T(n, k) = (k-1)!*Sum_{i=0..k}(Stirling2(i,k)*(-1)^(n-i)*Stirling1(n,i)) =
T(n, k) = Sum_{i=0..k}(W(i,k)*(-1)^(n-i)*Stirling1(n,i)), where W(n,k) is the Worpitzky triangle A028246. - Vladimir Kruchinin, Apr 17 2015.
T(n,k) = [x^k] n!*[t^n](-log(1 + x*log(1 - t))). - Peter Luschny, Jul 10 2020
T(n,k) = Sum_{m=0..n-k} abs(Stirling1(n-1,m+k-1))*(k+m-1)!/m!. - Vladimir Kruchinin, Jul 14 2025

a(11)-a(45) from Nathaniel Johnston, Apr 15 2011

A126677 Product_{i=4..n} |Stirling_1(i,4)|.

Original entry on oeis.org

1, 10, 850, 624750, 4228932750, 284539511151000, 205915553429755680000, 1731646846567530390960000000, 182269815381165453023977392960000000, 257732232581979345367168654620866388480000000, 5235508937551175000760375514558779305187938734080000000, 1628187052306629700637438618898904208447222949912051474432000000000

Offset: 4

N. J. A. Sloane, Feb 13 2007

nonn

Partial products of A000454.

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

A346923 Expansion of e.g.f. 1 / (1 - log(1 - x)^4 / 4!).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A347003 Expansion of e.g.f. exp( log(1 - x)^4 / 4! ).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A062255 4th level triangle related to Eulerian numbers and binomial transforms (A062254 is third level, 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).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula

Extensions

A346946 Expansion of e.g.f. log( 1 + log(1 + x)^4 / 4! ).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A348064 Coefficient of x^3 in expansion of n!* Sum_{k=0..n} binomial(x,k).

Views

Author

Keywords

Crossrefs

Programs

PARI

PARI

Python

Formula

A052753 Expansion of e.g.f.: log(1-x)^4.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

Extensions

A188881 Triangle of coefficients arising from an expansion of Integral( exp(exp(exp(x))), dx).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Maxima

PARI

PARI

Formula