A001233 -id:A001233 - cp's OEIS frontend

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

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)).

A052779 Expansion of e.g.f.: (log(1-x))^6.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 720, 15120, 231840, 3265920, 45556560, 649479600, 9604465200, 148370508000, 2402005525920, 40797624067200, 726963917097600, 13580328282393600, 265689107448756480, 5437099866285377280, 116229410301685651200, 2591985252922277184000, 60218914823672258142720

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 736

Column k=6 of A225479.

Cf. A001233, A052517.

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

a(n) = {6!*stirling(n,6,1)*(-1)^n} \\ Andrew Howroyd, Jul 27 2020

E.g.f.: log(-1/(-1+x))^6.
Recurrence: {a(1)=0, a(0)=0, a(2)=0, a(4)=0, a(3)=0, a(5)=0, a(6)=720, (1+15*n^2+6*n+6*n^5+15*n^4+20*n^3+n^6)*a(n+1) + (-63-186*n-225*n^2-6*n^5-45*n^4-140*n^3)*a(n+2) + (540*n+120*n^3+375*n^2+15*n^4+301)*a(n+3) + (-390*n-20*n^3-350-150*n^2)*a(n+4) + (140+15*n^2+90*n)*a(n+5) + (-21-6*n)*a(n+6) + a(n+7)}.
a(n) = 720*A001233(n) = 6!*(-1)^n*Stirling1(n,6). - Andrew Howroyd, Jul 27 2020

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

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.

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

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A052779 Expansion of e.g.f.: (log(1-x))^6.

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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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