A132164 - OEIS

%I #16 Jan 18 2025 09:05:56

%S 1,1,8,78,918,12846,209616,3909228,81859548,1897344828,48135826656,

%T 1325008302696,39292978029768,1247949491330088,42236558731574208,

%U 1516738194700667856,57573649342673292816,2302425590703685075728,96720470167595138898048

%N Row sums of triangle A134141 (S1p(7)).

%H Alois P. Heinz, <a href="/A132164/b132164.txt">Table of n, a(n) for n = 0..409</a>

%F a(n)= sum(A134141(n,m),m=1..n),n>=1.

%F E.g.f.: exp((1-(1-x)^6)/(6*(1-x)^6)). Cf. e.g.f. first column of A134141.

%F From _Seiichi Manyama_, Jan 18 2025: (Start)

%F a(n) = Sum_{k=0..n} |Stirling1(n,k)| * A005012(k).

%F a(n) = (1/exp(1/6)) * (-1)^n * n! * Sum_{k>=0} binomial(-6*k,n)/(6^k * k!). (End)

%p a:= proc(n) option remember; `if`(n=0, 1, add(

%p       binomial(n-1, j-1)*(j+5)!/6!*a(n-j), j=1..n))

%p     end:

%p seq(a(n), n=0..25);  # _Alois P. Heinz_, Aug 01 2017

%t a[n_]:=a[n]=If[n==0, 1, Sum[Binomial[n - 1, j - 1] (j + 5)!/6! a[n - j], {j, n}]]; Table[a[n], {n, 0, 25}] (* _Indranil Ghosh_, Aug 02 2017, after Maple code *)

%Y Cf. A132165 (alternating row sum of A134141), A049428.

%K nonn,easy

%O 0,3

%A _Wolfdieter Lang_, Oct 12 2007

%E a(0)=1 prepended by _Alois P. Heinz_, Aug 01 2017