A270995
Expansion of Product_{k>=1} 1/(1 - A000009(k)*x^k).
Original entry on oeis.org
1, 1, 2, 4, 7, 12, 23, 37, 64, 108, 180, 290, 488, 772, 1251, 2001, 3180, 4982, 7913, 12261, 19162, 29669, 45804, 70187, 108029, 164276, 250267, 379439, 574067, 864044, 1302169, 1949050, 2917900, 4352796, 6481627, 9620256, 14274080, 21090608, 31142909
Offset: 0
a(6)=23: {(6), (5)(1), (51), (4)(2), (42), (4)(1)(1), (41)(1), (3)(3), (3)(2)(1), (3)(21), (32)(1), (31)(2), (21)(3), (321), (3)(1)(1)(1), (31)(1)(1), (2)(2)(2), (2)(2)(1)(1), (21)(2)(1), (21)(21), (2)(1)(1)(1)(1), (21)(1)(1)(1), (1)(1)(1)(1)(1)(1)}.
For compositions instead of partitions we have
A304969, non-strict
A055887.
A072233 counts partitions by sum and length.
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nmax = 50; CoefficientList[Series[Product[1/(1-PartitionsQ[k]*x^k), {k, 1, nmax}], {x, 0, nmax}], x]
A327608
Number of parts in all twice partitions of n where the second partition is strict.
Original entry on oeis.org
0, 1, 3, 8, 17, 34, 74, 134, 254, 470, 842, 1463, 2620, 4416, 7545, 12749, 21244, 34913, 57868, 93583, 151963, 244602, 391206, 620888, 987344, 1550754, 2435087, 3804354, 5920225, 9162852, 14179754, 21785387, 33436490, 51121430, 77935525, 118384318, 179617794
Offset: 0
a(3) = 8 = 1+2+2+3 counting the parts in 3, 21, 2|1, 1|1|1.
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g:= proc(n, i) option remember; `if`(i*(i+1)/2 f+
[0, f[1]])(g(n-i, min(n-i, i-1)))))
end:
b:= proc(n, i) option remember; `if`(n=0, [1, 0],
`if`(i<2, 0, b(n, i-1)) +(h-> (f-> f +[0, f[1]*
h[2]/h[1]])(b(n-i, min(n-i, i))*h[1]))(g(i$2)))
end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..37);
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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]]]]];
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]]}]];
a[n_] := b[n, n][[2]];
a /@ Range[0, 37] (* Jean-François Alcover, Dec 05 2020, after Alois P. Heinz *)
A327590
Number of partitions in all twice partitions of n.
Original entry on oeis.org
0, 1, 4, 10, 29, 63, 164, 339, 797, 1640, 3578, 7139, 15210, 29621, 60381, 117116, 232523, 442388, 863069, 1621560, 3105993, 5785525, 10894394, 20083143, 37434186, 68344449, 125774280, 228088127, 415668548, 747660318, 1351364816, 2413792653, 4327245170
Offset: 0
a(3) = 10 = 1+1+1+2+2+3 counting the partitions in 3, 21, 111, 2|1, 11|1, 1|1|1.
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b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, 0, b(n, i-1)+
(p-> p+[0, p[1]])(combinat[numbpart](i)*b(n-i, min(n-i, i)))))
end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..42);
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b[n_, i_] := b[n, i] = If[n == 0, {1, 0}, If[i<1, {0, 0}, b[n, i-1] + Function[p, p + {0, p[[1]]}][PartitionsP[i] b[n-i, Min[n-i, i]]]]];
a[n_] := b[n, n][[2]];
a /@ Range[0, 42] (* Jean-François Alcover, Dec 16 2020, after Alois P. Heinz *)
Showing 1-3 of 3 results.
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