A339374 -id:A339374 - cp's OEIS frontend

A339373 Number of partitions of n into an even number of triangular numbers.

1, 0, 1, 0, 2, 0, 3, 1, 3, 2, 4, 3, 6, 5, 6, 6, 10, 7, 13, 10, 15, 13, 20, 15, 26, 21, 28, 26, 36, 31, 44, 42, 49, 50, 61, 57, 75, 73, 84, 85, 103, 97, 123, 121, 137, 140, 166, 159, 194, 194, 216, 225, 256, 253, 295, 304, 330, 346, 389, 387, 446, 456, 498, 516, 579, 576

Offset: 0

Ilya Gutkovskiy, Dec 02 2020

nonn

			a(6) = 3 because we have [3, 3], [3, 1, 1, 1] and [1, 1, 1, 1, 1, 1].

Cf. A000217, A007294, A027187, A292519, A339364, A339374, A339375, A339376.

nmax = 65; CoefficientList[Series[(1/2) (Product[1/(1 - x^(k (k + 1)/2)), {k, 1, nmax}] + Product[1/(1 + x^(k (k + 1)/2)), {k, 1, nmax}]), {x, 0, nmax}], x]

G.f.: (1/2) * (Product_{k>=1} 1 / (1 - x^(k*(k + 1)/2)) + Product_{k>=1} 1 / (1 + x^(k*(k + 1)/2))).

a(n) = (A007294(n) + A292519(n)) / 2.

A339375 Number of partitions of n into an even number of distinct triangular numbers.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 2, 0, 1, 0, 1, 1, 1, 0, 1, 2, 0, 1, 0, 2, 0, 3, 1, 0, 2, 1, 1, 1, 3, 1, 2, 0, 2, 2, 0, 2, 3, 3, 1, 2, 2, 2, 2, 2, 1, 4, 4, 1, 3, 2, 3, 2, 3, 1, 5, 4, 2, 4, 2, 4, 4, 3, 2, 6, 4, 3, 4, 5, 2, 3, 6, 5, 6, 5, 4, 5, 5, 4, 5, 6, 4

Offset: 0

Ilya Gutkovskiy, Dec 02 2020

nonn

			a(31) = 3 because we have [28, 3], [21, 10] and [21, 6, 3, 1].

Cf. A000217, A024940, A067661, A292518, A339366, A339373, A339374, A339376.

nmax = 90; CoefficientList[Series[(1/2) (Product[(1 + x^(k (k + 1)/2)), {k, 1, nmax}] + Product[(1 - x^(k (k + 1)/2)), {k, 1, nmax}]), {x, 0, nmax}], x]

G.f.: (1/2) * (Product_{k>=1} (1 + x^(k*(k + 1)/2)) + Product_{k>=1} (1 - x^(k*(k + 1)/2))).

a(n) = (A024940(n) + A292518(n)) / 2.

A339376 Number of partitions of n into an odd number of distinct triangular numbers.

Original entry on oeis.org

0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 1, 0, 1, 0, 2, 0, 1, 1, 0, 1, 1, 1, 0, 3, 0, 1, 1, 2, 0, 1, 2, 1, 3, 0, 2, 1, 2, 1, 1, 2, 2, 3, 1, 1, 3, 2, 0, 4, 3, 2, 3, 2, 2, 2, 3, 2, 4, 2, 3, 2, 3, 4, 4, 4, 1, 5, 4, 2, 3, 5, 3, 6, 4, 2, 6, 4, 3, 5, 6, 5, 5, 5, 5, 5, 4, 5

Offset: 0

Ilya Gutkovskiy, Dec 02 2020

nonn

			a(28) = 3 because we have [28], [21, 6, 1] and [15, 10, 3].

Cf. A000217, A024940, A067659, A292518, A339367, A339373, A339374, A339375.

nmax = 90; CoefficientList[Series[(1/2) (Product[(1 + x^(k (k + 1)/2)), {k, 1, nmax}] - Product[(1 - x^(k (k + 1)/2)), {k, 1, nmax}]), {x, 0, nmax}], x]

G.f.: (1/2) * (Product_{k>=1} (1 + x^(k*(k + 1)/2)) - Product_{k>=1} (1 - x^(k*(k + 1)/2))).

a(n) = (A024940(n) - A292518(n)) / 2.

A339417 Number of compositions (ordered partitions) of n into an odd number of triangular numbers.

Original entry on oeis.org

0, 1, 0, 2, 0, 4, 1, 9, 3, 19, 12, 41, 33, 91, 92, 203, 238, 466, 602, 1080, 1493, 2536, 3661, 6001, 8902, 14278, 21554, 34094, 52013, 81602, 125297, 195582, 301475, 469193, 724881, 1126161, 1742206, 2703888, 4186276, 6493192, 10057553, 15594636, 24161364, 37455851

Offset: 0

Ilya Gutkovskiy, Dec 03 2020

nonn

			a(8) = 3 because we have [6, 1, 1], [1, 6, 1] and [1, 1, 6].

Cf. A000217, A023361, A106507, A166444, A339374, A339416.

b:= proc(n, t) option remember; local r, f, g;
      if n=0 then t else r, f, g:=$0..2; while f<=n
      do r, f, g:= r+b(n-f, 1-t), f+g, g+1 od; r fi
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..50);  # Alois P. Heinz, Dec 03 2020

nmax = 43; CoefficientList[Series[(1/2) (1/(1 - Sum[x^(k (k + 1)/2), {k, 1, nmax}]) - 1/Sum[x^(k (k + 1)/2), {k, 0, nmax}]), {x, 0, nmax}], x]

G.f.: (1/2) * (1 / (1 - Sum_{k>=1} x^(k*(k + 1)/2)) - 1 / Sum_{k>=0} x^(k*(k + 1)/2)).
a(n) = (A023361(n) - A106507(n)) / 2.
a(n) = -Sum_{k=0..n-1} A023361(k) * A106507(n-k).

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A339373 Number of partitions of n into an even number of triangular numbers.

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A339375 Number of partitions of n into an even number of distinct triangular numbers.

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A339376 Number of partitions of n into an odd number of distinct triangular numbers.

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A339417 Number of compositions (ordered partitions) of n into an odd number of triangular numbers.

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