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A327607 Number of parts in all twice partitions of n where the first partition is strict.

%I A327607 #20 Dec 05 2020 03:31:31
%S A327607 0,1,3,11,21,58,128,276,516,1169,2227,4324,8335,15574,29116,55048,
%T A327607 97698,176291,323277,563453,1005089,1770789,3076868,5293907,9184885,
%U A327607 15668638,26751095,45517048,76882920,128738414,217219751,360525590,599158211,995474365
%N A327607 Number of parts in all twice partitions of n where the first partition is strict.
%H A327607 Alois P. Heinz, <a href="/A327607/b327607.txt">Table of n, a(n) for n = 0..4000</a>
%e A327607 a(3) = 11 = 1+2+3+2+3 counting the parts in 3, 21, 111, 2|1, 11|1.
%p A327607 g:= proc(n) option remember; (p-> [p(n), add(p(n-j)*
%p A327607       numtheory[tau](j), j=1..n)])(combinat[numbpart])
%p A327607     end:
%p A327607 b:= proc(n, i) option remember; `if`(i*(i+1)/2<n, 0,
%p A327607       `if`(n=0, [1, 0], b(n, i-1) +(h-> (f-> f+[0, f[1]*
%p A327607        h[2]/h[1]])(b(n-i, min(n-i, i-1))*h[1]))(g(i))))
%p A327607     end:
%p A327607 a:= n-> b(n$2)[2]:
%p A327607 seq(a(n), n=0..37);
%t A327607 g[n_] := g[n] = {PartitionsP[n], Sum[PartitionsP[n - j] DivisorSigma[0, j], {j, 1, n}]};
%t A327607 b[n_, i_] := b[n, i] = If[i(i+1)/2 < n, 0, If[n == 0, {1, 0}, Module[{h, f}, h = g[i]; f = b[n - i, Min[n - i, i - 1]] h[[1]]; b[n, i - 1] + f + {0, f[[1]] h[[2]] / h[[1]]}]]];
%t A327607 a[n_] := b[n, n][[2]];
%t A327607 a /@ Range[0, 37] (* _Jean-François Alcover_, Dec 05 2020, after _Alois P. Heinz_ *)
%Y A327607 Cf. A000009, A000041, A271619, A327552, A327594, A327605, A327608.
%K A327607 nonn
%O A327607 0,3
%A A327607 _Alois P. Heinz_, Sep 18 2019