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

%I A112002 #24 Sep 08 2022 08:45:21
%S A112002 720,13068,118124,723680,3416930,13339535,44990231,135036473,
%T A112002 368411615,928095740,2185031420,4853222764,10246937272,20692933630,
%U A112002 40171771630,75289668850,136717357942,241276443496,414908513800,696829576300
%N A112002 Seventh diagonal of triangle A008275 (Stirling1) and seventh column of |A008276|.
%H A112002 T. D. Noe, <a href="/A112002/b112002.txt">Table of n, a(n) for n = 1..1000</a>
%F A112002 a(n)= Stirling1(n+6, n), n>=1, with Stirling1(n, k)= A008275(n, k).
%F A112002 E.g.f. with offset 6: exp(x)*sum(A112486(6, m)*(x^(6+m))/(6+m)!, m=0..6).
%F A112002 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 A112002 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 A112002 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 A112002 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 A112002 := proc(n) combinat[stirling1](n+6,n) ; end proc: # _R. J. Mathar_, Jun 08 2011
%t A112002 Table[StirlingS1[n+6, n], {n, 1, 20}] (* _Jean-François Alcover_, Mar 05 2014 *)
%o A112002 (Sage) [stirling_number1(n,n-6) for n in range(7, 27)] # _Zerinvary Lajos_, May 16 2009
%o A112002 (Magma) [StirlingFirst(n+6, n): n in [1..20]]; // _Vincenzo Librandi_, Aug 09 2015
%Y A112002 Sixth diagonal A053567; A130534.
%K A112002 nonn,easy
%O A112002 1,1
%A A112002 _Wolfdieter Lang_, Sep 12 2005