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Seventh diagonal of triangle A008275 (Stirling1) and seventh column of |A008276|.
4

%I #24 Sep 08 2022 08:45:21

%S 720,13068,118124,723680,3416930,13339535,44990231,135036473,

%T 368411615,928095740,2185031420,4853222764,10246937272,20692933630,

%U 40171771630,75289668850,136717357942,241276443496,414908513800,696829576300

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

%H T. D. Noe, <a href="/A112002/b112002.txt">Table of n, a(n) for n = 1..1000</a>

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

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

%F 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.

%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.

%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.

%F 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

%p A112002 := proc(n) combinat[stirling1](n+6,n) ; end proc: # _R. J. Mathar_, Jun 08 2011

%t Table[StirlingS1[n+6, n], {n, 1, 20}] (* _Jean-François Alcover_, Mar 05 2014 *)

%o (Sage) [stirling_number1(n,n-6) for n in range(7, 27)] # _Zerinvary Lajos_, May 16 2009

%o (Magma) [StirlingFirst(n+6, n): n in [1..20]]; // _Vincenzo Librandi_, Aug 09 2015

%Y Sixth diagonal A053567; A130534.

%K nonn,easy

%O 1,1

%A _Wolfdieter Lang_, Sep 12 2005