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%I #19 Dec 05 2020 09:47:20
%S 0,1,3,8,17,34,74,134,254,470,842,1463,2620,4416,7545,12749,21244,
%T 34913,57868,93583,151963,244602,391206,620888,987344,1550754,2435087,
%U 3804354,5920225,9162852,14179754,21785387,33436490,51121430,77935525,118384318,179617794
%N Number of parts in all twice partitions of n where the second partition is strict.
%H Alois P. Heinz, <a href="/A327608/b327608.txt">Table of n, a(n) for n = 0..4000</a>
%e a(3) = 8 = 1+2+2+3 counting the parts in 3, 21, 2|1, 1|1|1.
%p g:= proc(n, i) option remember; `if`(i*(i+1)/2<n, 0,
%p `if`(n=0, [1, 0], g(n, i-1)+ (f-> f+
%p [0, f[1]])(g(n-i, min(n-i, i-1)))))
%p end:
%p b:= proc(n, i) option remember; `if`(n=0, [1, 0],
%p `if`(i<2, 0, b(n, i-1)) +(h-> (f-> f +[0, f[1]*
%p h[2]/h[1]])(b(n-i, min(n-i, i))*h[1]))(g(i$2)))
%p end:
%p a:= n-> b(n$2)[2]:
%p seq(a(n), n=0..37);
%t g[n_, i_] := g[n, i] = If[i(i+1)/2 < n, 0, If[n == 0, {1, 0}, g[n, i - 1] + Function[f, f + {0, f[[1]]}][g[n - i, Min[n - i, i - 1]]]]];
%t b[n_, i_] := b[n, i] = If[n == 0, {1, 0}, If[i < 2, 0, b[n, i - 1]] + Module[{h, f}, h = g[i, i]; f = b[n - i, Min[n - i, i]] h[[1]]; f + {0, f[[1]] h[[2]]/h[[1]]}]];
%t a[n_] := b[n, n][[2]];
%t a /@ Range[0, 37] (* _Jean-François Alcover_, Dec 05 2020, after _Alois P. Heinz_ *)
%Y Cf. A000009, A000041, A270995, A327554, A327594, A327605, A327607.
%K nonn
%O 0,3
%A _Alois P. Heinz_, Sep 18 2019
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