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A379378 The n-th term of the n-th forward differences of partition numbers A000041.

%I A379378 #19 Jul 26 2025 16:06:45
%S A379378 1,1,1,2,5,10,22,64,159,328,747,1914,4608,10252,23339,55405,128034,
%T A379378 287855,660549,1541383,3528645,7921187,17870633,40689873,92248847,
%U A379378 207911243,469331387,1059603243,2377923972,5313383490,11889346697,26641635997,59560543885
%N A379378 The n-th term of the n-th forward differences of partition numbers A000041.
%H A379378 Alois P. Heinz, <a href="/A379378/b379378.txt">Table of n, a(n) for n = 0..3209</a>
%p A379378 b:= proc(n, k) option remember; `if`(k=0,
%p A379378       combinat[numbpart](n), b(n+1, k-1)-b(n, k-1))
%p A379378     end:
%p A379378 a:= n-> b(n$2):
%p A379378 seq(a(n), n=0..32);
%t A379378 A379378list[nmax_] := Module[{p = PartitionsP[Range[0, nmax*2]]}, Join[{First[p]}, Table[First[p = Differences[Rest[p]]], nmax]]];
%t A379378 A379378list[50] (* _Paolo Xausa_, Jul 26 2025 *)
%Y A379378 Main diagonal of A175804.
%Y A379378 Cf. A000041, A033815, A281425, A386457 (parity).
%K A379378 nonn
%O A379378 0,4
%A A379378 _Alois P. Heinz_, Dec 21 2024