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A327605 Number of parts in all twice partitions of n where both partitions are strict.

%I A327605 #32 Jun 03 2020 12:30:12
%S A327605 0,1,1,5,8,15,28,49,86,156,259,412,679,1086,1753,2826,4400,6751,10703,
%T A327605 16250,24757,38047,57459,85861,129329,192660,286177,424358,624510,
%U A327605 915105,1347787,1961152,2847145,4144089,5988205,8638077,12439833,17837767,25536016
%N A327605 Number of parts in all twice partitions of n where both partitions are strict.
%H A327605 Alois P. Heinz, <a href="/A327605/b327605.txt">Table of n, a(n) for n = 0..5000</a>
%e A327605 a(3) = 5 = 1+2+2 counting the parts in 3, 21, 2|1.
%p A327605 g:= proc(n, i) option remember; `if`(i*(i+1)/2<n, 0,
%p A327605       `if`(n=0, [1, 0], g(n, i-1) +(f-> f+
%p A327605        [0, f[1]])(g(n-i, min(n-i, i-1)))))
%p A327605     end:
%p A327605 b:= proc(n, i) option remember; `if`(i*(i+1)/2<n, 0,
%p A327605       `if`(n=0, [1, 0], b(n, i-1) +(h-> (f-> f+[0, f[1]*
%p A327605        h[2]/h[1]])(b(n-i, min(n-i, i-1))*h[1]))(g(i$2))))
%p A327605     end:
%p A327605 a:= n-> b(n$2)[2]:
%p A327605 seq(a(n), n=0..42);
%t A327605 b[n_, i_, k_] := b[n, i, k] = With[{}, If[n == 0, Return@{1, 0}]; If[k == 0, Return@{1, 1}]; If[i (i + 1)/2 < n, Return@{0, 0}]; b[n, i - 1, k] + Function[h, Function[f, f + {0, f[[1]] h[[2]]/h[[1]]}][h[[1]] b[n - i, Min[n - i, i - 1], k]]][b[i, i, k - 1]]];
%t A327605 a[n_] := b[n, n, 2][[2]];
%t A327605 a /@ Range[0, 42] (* _Jean-François Alcover_, Jun 03 2020, after _Alois P. Heinz_ in A327622 *)
%Y A327605 Cf. A000009, A279785, A327553, A327594, A327607, A327608, A327795.
%Y A327605 Column k=2 of A327622.
%K A327605 nonn
%O A327605 0,4
%A A327605 _Alois P. Heinz_, Sep 18 2019