A024940 -id:A024940 - cp's OEIS frontend

A357071 Number of partitions of n into at most 3 distinct positive triangular numbers.

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

Offset: 0

Ilya Gutkovskiy, Oct 25 2022

nonn

Cf. A000217, A002636, A024940, A307598, A347731, A357070, A357072.

A357072 Number of partitions of n into at most 4 distinct positive triangular numbers.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 1, 1, 2, 1, 1, 2, 1, 2, 2, 0, 2, 3, 1, 1, 3, 2, 1, 4, 3, 0, 3, 2, 2, 4, 3, 3, 3, 1, 3, 3, 2, 4, 5, 4, 1, 5, 3, 2, 6, 3, 3, 6, 4, 2, 5, 4, 4, 5, 4, 2, 7, 5, 5, 7, 3, 4, 7, 4, 4, 8, 5, 4, 7, 6, 2, 7, 6, 5, 10, 5, 6, 7, 5, 7, 7, 6, 4, 10, 8, 3, 12, 6, 4, 11, 6, 5, 9, 7

Offset: 0

Ilya Gutkovskiy, Oct 25 2022

nonn

Cf. A000217, A024940, A341021, A347586, A347627, A347732, A357070, A357071.

A357352 Number of partitions of n into distinct positive triangular numbers such that the number of parts is a triangular number.

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Sep 25 2022

nonn

			a(56) = 2 because we have [45,10,1] and [21,15,10,6,3,1].

Alois P. Heinz, Table of n, a(n) for n = 0..20000

Cf. A000217, A024940, A045450, A178927, A357354.

b:= proc(n, i, t) option remember; (h-> `if`(n=0,
     `if`(issqr(8*t+1), 1, 0), `if`(n>i*(i+1)*(i+2)/6, 0,
     `if`(h>n, 0, b(n-h, i-1, t+1))+b(n, i-1, t))))(i*(i+1)/2)
    end:
a:= n-> b(n, floor((sqrt(1+8*n)-1)/2), 0):
seq(a(n), n=0..100);  # Alois P. Heinz, Sep 25 2022

b[n_, i_, t_] := b[n, i, t] = With[{h = i(i+1)/2}, If[n == 0, If[IntegerQ@ Sqrt[8t+1], 1, 0], If[n > i(i+1)(i+2)/6, 0, If[h > n, 0, b[n-h, i-1, t+1]] + b[n, i-1, t]]]];
a[n_] := b[n, Floor[(Sqrt[8n+1]-1)/2], 0];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Apr 22 2025, after Alois P. Heinz *)

A099200 Number of partitions of n into distinct odd triangular numbers.

Original entry on oeis.org

1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 2, 2, 0, 2, 2, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1

Offset: 1

Reinhard Zumkeller, Oct 04 2004

nonn

			a(70) = #{55+15, 45+21+3+1} = 2.

Cf. A099199, A024940, A000009, A000217.

A280423 G.f.: Product_{k>=1} (1 + x^(k*(k+1))) / (1 - x^k).

Original entry on oeis.org

1, 1, 3, 4, 7, 10, 17, 23, 36, 49, 71, 96, 136, 180, 248, 326, 437, 569, 752, 967, 1261, 1609, 2069, 2620, 3335, 4189, 5285, 6595, 8249, 10230, 12706, 15661, 19327, 23696, 29063, 35457, 43256, 52519, 63756, 77073, 93126, 112120, 134901, 161781, 193884, 231679

Offset: 0

Vaclav Kotesovec, Jan 02 2017

nonn

Cf. A024940, A000041, A280421.

nmax=60; CoefficientList[Series[Product[(1+x^(k*(k+1)))/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(sqrt(2*n/3)*Pi + 3^(1/4) * (sqrt(2)-1) * Zeta(3/2) * n^(1/4) / 2^(5/4) + 3*(2*sqrt(2)-3) * Zeta(3/2)^2 / (64*Pi)) / (8*sqrt(3)*n).

A281666 Expansion of Sum_{i>=1} x^(i(i+1)/2)/(1 + x^(i(i+1)/2)) * Product_{j>=1} (1 + x^(j*(j+1)/2)).

Original entry on oeis.org

1, 0, 1, 2, 0, 1, 2, 0, 2, 4, 2, 0, 2, 3, 1, 4, 3, 2, 6, 4, 3, 5, 0, 5, 9, 3, 2, 7, 6, 3, 11, 10, 0, 9, 12, 3, 11, 10, 8, 11, 8, 9, 9, 6, 12, 19, 15, 7, 15, 16, 4, 20, 21, 10, 23, 24, 10, 16, 19, 18, 20, 20, 17, 24, 27, 18, 28, 26, 19, 33, 30, 12, 33, 39, 25, 36, 38, 16, 32, 44, 29, 41, 48, 37, 41, 45, 33, 39, 44, 41

Offset: 1

Ilya Gutkovskiy, Jan 26 2017

nonn

Total number of parts in all partitions of n into distinct nonzero triangular numbers (A000217).

			a(10) = 4 because we have [10], [6, 3, 1] and 1 + 3 = 4.

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
Index entries for related partition-counting sequences

Cf. A000217, A015723, A024940, A281542, A281615, A329801.

nmax = 90; Rest[CoefficientList[Series[Sum[x^(i (i + 1)/2)/(1 + x^(i (i + 1)/2)), {i, 1, nmax}] Product[1 + x^(j (j + 1)/2), {j, 1, nmax}], {x, 0, nmax}], x]]

G.f.: Sum_{i>=1} x^(i*(i+1)/2)/(1 + x^(i*(i+1)/2)) * Product_{j>=1} (1 + x^(j*(j+1)/2)).

A298850 Expansion of Product_{k>=1} (1 + x^(k(k+1)/2))^(k(k+1)/2).

Original entry on oeis.org

1, 1, 0, 3, 3, 0, 9, 9, 0, 19, 29, 10, 33, 63, 30, 66, 156, 90, 110, 300, 235, 276, 561, 465, 558, 1083, 1065, 1154, 1877, 1983, 2295, 3834, 3879, 3861, 6858, 7452, 7561, 12613, 13252, 13057, 22161, 25569, 24582, 35985, 44193, 44970, 63495, 79105, 77143, 104046, 134820, 138759, 182511, 222600

Offset: 0

Ilya Gutkovskiy, Mar 09 2018

nonn

Cf. A000217, A024940, A027999, A028377, A291649, A298730.

nmax = 53; CoefficientList[Series[Product[(1 + x^(k (k + 1)/2))^(k (k + 1)/2), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} (1 + x^A000217(k))^A000217(k).

A300417 Expansion of Product_{k>=1} (1 + x^(k*(k+1)/2))^2.

Original entry on oeis.org

1, 2, 1, 2, 4, 2, 3, 6, 3, 4, 10, 8, 5, 10, 11, 8, 14, 16, 11, 18, 22, 18, 23, 22, 22, 34, 31, 26, 39, 40, 33, 50, 56, 36, 53, 74, 51, 62, 86, 68, 77, 98, 86, 88, 102, 106, 120, 130, 120, 136, 157, 134, 157, 194, 155, 182, 241, 194, 196, 256, 237, 236, 288, 282, 273, 324

Offset: 0

Ilya Gutkovskiy, Mar 05 2018

nonn

Number of partitions of n into distinct triangular parts (A000217), with 2 types of each part.

Self-convolution of A024940.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Index entries for related partition-counting sequences

Cf. A000217, A022567, A024940, A279226, A298435.

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

G.f.: Product_{k>=1} (1 + x^A000217(k))^2.

a(n) ~ exp(3*Pi^(1/3) * ((sqrt(2)-1) * Zeta(3/2)/2)^(2/3) * n^(1/3)) * ((sqrt(2)-1) * Zeta(3/2) / (2*Pi))^(1/3) / (4*sqrt(3) * n^(5/6)). - Vaclav Kotesovec, Mar 05 2018

A319256 Expansion of Product_{k>0} (1 + k(k+1)/2 x^(k*(k+1)/2)).

Original entry on oeis.org

1, 1, 0, 3, 3, 0, 6, 6, 0, 18, 28, 10, 0, 30, 30, 15, 75, 60, 45, 225, 180, 111, 111, 0, 333, 483, 150, 126, 604, 478, 378, 1572, 1194, 0, 3498, 3498, 351, 2115, 2044, 1333, 4833, 4620, 2946, 2526, 2100, 8043, 11133, 9810, 5175, 13773, 13638, 810, 31194, 34584, 6630

Offset: 0

Seiichi Manyama, Sep 15 2018

nonn

Sum of products of terms in all partitions of n into distinct triangular numbers (A000217).

			a(16) = 75 because we have [15, 1], [10, 6], 15*1 = 15, 10*6 = 60 and 15 + 60 = 75.

Cf. A000217, A024940, A282866, A319257.

A329466 Expansion of Sum_{k>=1} (-1 + Product_{j>=1} (1 + x^(kj(j + 1)/2))).

Original entry on oeis.org

1, 1, 2, 2, 1, 3, 2, 2, 3, 3, 2, 4, 2, 3, 3, 4, 2, 5, 3, 5, 5, 4, 1, 6, 4, 3, 4, 7, 3, 7, 5, 7, 3, 5, 5, 8, 5, 6, 6, 8, 3, 10, 4, 7, 8, 7, 5, 10, 7, 10, 5, 10, 6, 9, 9, 13, 7, 8, 6, 14, 7, 10, 10, 14, 9, 12, 9, 12, 7, 17, 8, 14, 10, 14, 12, 17, 12, 12, 10, 20

Offset: 1

Ilya Gutkovskiy, Nov 13 2019

nonn

Inverse Moebius transform of A024940.

Cf. A024940, A047966, A047968, A329462, A329465.

nmax = 80; CoefficientList[Series[Sum[-1 + Product[(1 + x^(k j (j + 1)/2)), {j, 1, nmax}], {k, 1, nmax}], {x, 0, nmax}], x] // Rest

G.f.: Sum_{k>=1} A024940(k) * x^k / (1 - x^k).

a(n) = Sum_{d|n} A024940(d).

cp's OEIS Frontend

A357071 Number of partitions of n into at most 3 distinct positive triangular numbers.

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A357072 Number of partitions of n into at most 4 distinct positive triangular numbers.

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A357352 Number of partitions of n into distinct positive triangular numbers such that the number of parts is a triangular number.

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A099200 Number of partitions of n into distinct odd triangular numbers.

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A280423 G.f.: Product_{k>=1} (1 + x^(k*(k+1))) / (1 - x^k).

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A281666 Expansion of Sum_{i>=1} x^(i*(i+1)/2)/(1 + x^(i*(i+1)/2)) * Product_{j>=1} (1 + x^(j*(j+1)/2)).

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A298850 Expansion of Product_{k>=1} (1 + x^(k*(k+1)/2))^(k*(k+1)/2).

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Mathematica

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A300417 Expansion of Product_{k>=1} (1 + x^(k*(k+1)/2))^2.

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Mathematica

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A319256 Expansion of Product_{k>0} (1 + k*(k+1)/2 * x^(k*(k+1)/2)).

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A329466 Expansion of Sum_{k>=1} (-1 + Product_{j>=1} (1 + x^(k*j*(j + 1)/2))).

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Formula

A281666 Expansion of Sum_{i>=1} x^(i(i+1)/2)/(1 + x^(i(i+1)/2)) * Product_{j>=1} (1 + x^(j*(j+1)/2)).

A298850 Expansion of Product_{k>=1} (1 + x^(k(k+1)/2))^(k(k+1)/2).

A319256 Expansion of Product_{k>0} (1 + k(k+1)/2 x^(k*(k+1)/2)).

A329466 Expansion of Sum_{k>=1} (-1 + Product_{j>=1} (1 + x^(kj(j + 1)/2))).