A112002 Seventh diagonal of triangle A008275 (Stirling1) and seventh column of |A008276|.

720, 13068, 118124, 723680, 3416930, 13339535, 44990231, 135036473, 368411615, 928095740, 2185031420, 4853222764, 10246937272, 20692933630, 40171771630, 75289668850, 136717357942, 241276443496, 414908513800, 696829576300

Offset

1,1

Links

T. D. Noe, Table of n, a(n) for n = 1..1000

Formula

a(n)= Stirling1(n+6, n), n>=1, with Stirling1(n, k)= A008275(n, k).

E.g.f. with offset 6: exp(x)*sum(A112486(6, m)*(x^(6+m))/(6+m)!, m=0..6).

a(n)= (f(n+5, 6)/12!)*sum(A112486(6, m)*f(12, 6-m)*f(n-1, m), m=0..min(6, n-1)), with the falling factorials f(n, k):=n*(n-1)*...*(n-(k-1)). From the e.g.f.

a(n)=(binomial(n+6, 7)/r(8, 5))*sum(A112007(5, m)*r(n+7, 5-m)*f(n-1, m), m=0..5), with rising factorials r(n, k):=n*(n+1)*...*(n+(k-1)) and falling factorials f(n, m). From the g.f.

G.f.: x*(720+3708*x+4400*x^2+1452*x^3+114*x^4+x^5)/(1-x)^13. See row k=5 of triangles A112007 or A008517 for the coefficients.

Explicit formula: a(n) = n(n + 1)(n + 2)(n + 3)(n + 4)(n + 5)(n + 6)(63n^5 + 1575n^4 + 15435n^3 + 73801n^2 + 171150n + 152696)/2903040. - Vaclav Kotesovec, Jan 30 2010

Maple

A112002 := proc(n) combinat[stirling1](n+6, n) ; end proc: # R. J. Mathar, Jun 08 2011

Mathematica

Table[StirlingS1[n+6, n], {n, 1, 20}] (* Jean-François Alcover, Mar 05 2014 *)

Prog

(Sage) [stirling_number1(n, n-6) for n in range(7, 27)] # Zerinvary Lajos, May 16 2009
(Magma) [StirlingFirst(n+6, n): n in [1..20]]; // Vincenzo Librandi, Aug 09 2015

Crossrefs

Sixth diagonal A053567; A130534.

Sequence in context: A213876 A052785 A052783 * A004033 A056271 A000920

Adjacent sequences: A111999 A112000 A112001 * A112003 A112004 A112005

Keyword

nonn,easy

Author

Wolfdieter Lang, Sep 12 2005

Status

approved