A229362 -id:A229362 - cp's OEIS frontend

A007279 Number of partitions of n into partition numbers.

1, 1, 2, 3, 4, 6, 8, 11, 14, 18, 23, 29, 36, 44, 54, 66, 79, 95, 113, 133, 157, 184, 216, 250, 290, 335, 385, 442, 505, 576, 656, 743, 842, 951, 1070, 1204, 1351, 1514, 1691, 1887, 2102, 2336, 2595, 2875, 3184, 3519, 3883, 4282, 4713, 5181, 5690, 6241, 6839, 7482

Offset: 0

N. J. A. Sloane, Mira Bernstein

nonn

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Igor Pak, Complexity problems in enumerative combinatorics, arXiv:1803.06636 [math.CO], 2018.

Cf. A000041.

Cf. A086209, A229362.

with(combinat): gf := 1/product((1-q^numbpart(k)), k=1..20): s := series(gf, q, 200): for i from 0 to 199 do printf(`%d,`,coeff(s, q, i)) od: # James Sellers, Feb 08 2002

CoefficientList[ Series[1/Product[1 - x^PartitionsP[i], {i, 1, 15}], {x, 0, 50}], x]

seq(n)={my(t=1); while(numbpart(t+1)<=n, t++); Vec(1/prod(k=1, t, 1-x^numbpart(k) + O(x*x^n)))} \\ Andrew Howroyd, Jun 22 2018

G.f.: 1/Product_{k>=1} (1-q^A000041(k)). - Michel Marcus, Jun 20 2018

More terms from James Sellers, Feb 08 2002

a(0)=1 prepended by Alois P. Heinz, Jul 02 2017

A293806 a(0) = a(1) = 1; a(n) = [x^n] Product_{k=1..n-1} 1/(1 - x^a(k)).

Original entry on oeis.org

1, 1, 1, 4, 6, 8, 11, 14, 19, 24, 30, 37, 47, 57, 70, 84, 102, 121, 144, 170, 202, 235, 275, 319, 372, 429, 495, 567, 652, 742, 848, 963, 1095, 1237, 1399, 1574, 1775, 1990, 2235, 2499, 2795, 3114, 3473, 3859, 4292, 4755, 5271, 5827, 6444, 7107, 7840, 8625, 9493, 10422, 11444, 12541

Offset: 0

Ilya Gutkovskiy, Oct 16 2017

nonn

a(n) = number of partitions of n into preceding terms starting from a(1), a(2), a(3), ... (for n > 1).

			a(4) = 6 because we have [4], [1a, 1a, 1a, 1a], [1a, 1a, 1a, 1b], [1a, 1a, 1b, 1b],  [1a, 1b, 1b, 1b] and [1b, 1b, 1b, 1b].
G.f. = -x - 2*x^2 + 1/((1 - x)*(1 - x)*(1 - x^4)*(1 - x^6)*(1 - x^8)*(1 - x^11)*(1 - x^14)*(1 - x^19)*...) = 1 + x + x^2 + 4*x^3 + 6*x^4 + 8*x^5 + 11*x^6 + 14*x^7 + 19*x^8 + ...

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Index entries for sequences related to partitions

Cf. A000041, A007279, A151945, A229362.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+`if`(a(i)>n, 0, b(n-a(i), i))))
    end:
a:= n-> `if`(n<2, 1, b(n, n-1)):
seq(a(n), n=0..60);  # Alois P. Heinz, Oct 16 2017

a[n_] := a[n] = SeriesCoefficient[Product[1/(1 - x^a[k]), {k, 1, n - 1}], {x, 0, n}]; a[0] = a[1] = 1; Table[a[n], {n, 0, 55}]

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies A(x) = -x - 2*x^2 + Product_{n>=1} 1/(1 - x^a(n)).

A296387 a(0) = a(1) = 1; a(n) = [x^n] Product_{k=1..n-1} (1 + x^a(k))/(1 - x^a(k)).

Original entry on oeis.org

1, 1, 2, 6, 8, 10, 14, 18, 26, 34, 46, 58, 74, 90, 114, 138, 174, 210, 260, 310, 378, 446, 536, 626, 748, 870, 1034, 1198, 1410, 1622, 1892, 2162, 2510, 2858, 3306, 3754, 4316, 4878, 5576, 6274, 7144, 8014, 9096, 10178, 11508, 12838, 14458, 16078, 18048, 20018, 22410, 24802, 27690, 30578, 34040

Offset: 0

Ilya Gutkovskiy, Dec 11 2017

nonn

Index entries for sequences related to partitions

Cf. A015128, A151945, A229362, A293806.

a[n_] := a[n] = SeriesCoefficient[Product[(1 + x^a[k])/(1 - x^a[k]), {k, 1, n - 1}], {x, 0, n}]; a[0] = a[1] = 1; Table[a[n], {n, 0, 54}]

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies A(x) = -x - 2*x^2 + Product_{n>=1} (1 + x^a(n))/(1 - x^a(n)).

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A007279 Number of partitions of n into partition numbers.

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A293806 a(0) = a(1) = 1; a(n) = [x^n] Product_{k=1..n-1} 1/(1 - x^a(k)).

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A296387 a(0) = a(1) = 1; a(n) = [x^n] Product_{k=1..n-1} (1 + x^a(k))/(1 - x^a(k)).

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