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A108947 Triangle: T(n,k) is the partition function G(n-k,k).

%I A108947 #18 Dec 12 2021 22:32:02
%S A108947 1,0,1,0,1,1,0,1,1,1,0,1,2,1,1,0,1,4,2,1,1,0,1,10,5,2,1,1,0,1,26,14,5,
%T A108947 2,1,1,0,1,76,46,15,5,2,1,1,0,1,232,166,51,15,5,2,1,1,0,1,764,652,196,
%U A108947 52,15,5,2,1,1,0,1,2620,2780,827,202,52,15,5,2,1,1
%N A108947 Triangle: T(n,k) is the partition function G(n-k,k).
%C A108947 See entries for A001680 and A001681 for appropriate references.
%H A108947 Alois P. Heinz, <a href="/A108947/b108947.txt">Rows n = 0..140, flattened</a>
%F A108947 E.g.f. for sequence G(0, k), G(1, k), ... is exp(x + (1/2)*x^2 + ... + (1/k!)*x^k).
%p A108947 G:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A108947       add(G(n-i*j, i-1)*n!/i!^j/(n-i*j)!/j!, j=0..n/i)))
%p A108947     end:
%p A108947 T:= (n, k)-> G(n-k, k):
%p A108947 seq(seq(T(n, k), k=0..n), n=0..12);  # _Alois P. Heinz_, Sep 15 2013
%t A108947 G[n_, i_] := G[n, i] = If[n == 0, 1, If[i<1, 0, Sum[G[n-i*j, i-1]*n!/i!^j/(n-i*j)! /j!, {j, 0, n/i}]]]; T[n_, k_] := G[n-k, k]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* _Jean-François Alcover_, Feb 24 2015, after _Alois P. Heinz_ *)
%Y A108947 Cf. A000110. First differences of a sequence G(k, 0), G(k, 1), ... give a row of A080510 (e.g., 0, 1, 10, 14, 15, 15, ... gives 1, 9, 4, 1).
%K A108947 easy,nonn,tabl
%O A108947 0,13
%A A108947 _Paul Boddington_, Jul 21 2005
%E A108947 One term corrected by _Alois P. Heinz_, Sep 15 2013