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A061261 Limits of diagonals in triangle defined in A061260.

%I A061261 #20 Nov 10 2020 08:24:20
%S A061261 1,2,6,15,37,85,194,423,912,1917,3974,8096,16302,32382,63668,123851,
%T A061261 238756,456190,864821,1627016,3039845,5641884,10406924,19083836,
%U A061261 34802782,63135539,113965033,204739662,366156396,652001918,1156200929,2042173379,3593341512
%N A061261 Limits of diagonals in triangle defined in A061260.
%C A061261 Terms 1, 2, 6, 15, 37, 85, ... are limits of diagonals in the triangle T(n,k) = A061260: 1 2, 1 3, 2, 1, 5, 6, 2, 1, 7, 11, 6, 2, 1, 11, 23, 15, 6, 2, 1, 15, 40, 32, 15, 6, 2, 1, 22, 73, 67, 37, 15, 6, 2, 1, 30, 120, 134, 79, 37, 15, 6, 2, 1, 42, 202, 255, 172, 85, 37, 15, 6, 2, 1, 56, 320, 470, 348, 187, 85, 37, 15, 6, 2, 1
%H A061261 Alois P. Heinz, <a href="/A061261/b061261.txt">Table of n, a(n) for n = 0..5000</a>
%F A061261 G.f.: Product_{k >= 1} (1 - x^k)^( - numbpart(k + 1)), where numbpart(k) = number of partitions of k, cf. A000041. a(n) = 1/n*Sum_{k = 1..n} b(k)*a(n - k), n>0, a(0) = 1, where b(k) = Sum_{d|k} d*numbpart(d + 1).
%F A061261 a(n) = A061260(2n,n). - _Alois P. Heinz_, Oct 21 2018
%p A061261 with(numtheory): with(combinat):
%p A061261 a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
%p A061261       numbpart(d+1), d=divisors(j))*a(n-j), j=1..n)/n)
%p A061261     end:
%p A061261 seq(a(n), n=0..40);  # _Alois P. Heinz_, Apr 14 2017, revised Sep 19 2017
%t A061261 a[n_] := a[n] = If[n==0, 1, Sum[Sum[d PartitionsP[d+1], {d, Divisors[j]}] a[n-j], {j, 1, n}]/n];
%t A061261 a /@ Range[0, 40] (* _Jean-François Alcover_, Nov 10 2020, after _Alois P. Heinz_ *)
%Y A061261 Cf. A061260.
%K A061261 easy,nonn
%O A061261 0,2
%A A061261 _Vladeta Jovovic_, Apr 23 2001