A024940 -id:A024940 - cp's OEIS frontend

A193749 Number of partitions of n into distinct parts that are squares or triangular numbers (A005214).

1, 1, 0, 1, 2, 1, 1, 2, 1, 2, 4, 2, 1, 4, 4, 3, 5, 4, 3, 7, 7, 4, 6, 7, 6, 10, 10, 5, 10, 14, 10, 12, 14, 9, 14, 20, 13, 13, 20, 17, 18, 24, 17, 16, 27, 26, 22, 27, 25, 26, 35, 31, 26, 35, 37, 36, 42, 37, 35, 48, 47, 40, 49, 49, 48, 60, 58, 49, 61, 66, 61

Offset: 0

Reinhard Zumkeller, Aug 03 2011

nonn

			a(10) = #{10, 9+1, 6+4, 6+3+1} = 4;
a(11) = #{10+1, 6+4+1} = 2;
a(12) = #{9+3} = 1;
a(13) = #{10+3, 9+4, 9+3+1, 6+4+3} = 4.

Reinhard Zumkeller, Table of n, a(n) for n = 0..250

Cf. A000290, A000217, A033461, A024940, A193748, A000009.

a193749 = p a005214_list where
   p _      0    = 1
   p (k:ks) m
     | m < k     = 0
     | otherwise = p ks (m - k) + p ks m

A225045 Number of partitions of n into distinct non-triangular numbers, cf. A014132.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 3, 5, 5, 5, 7, 7, 10, 10, 13, 13, 16, 18, 21, 25, 27, 32, 33, 41, 44, 53, 57, 65, 73, 81, 93, 102, 118, 128, 145, 159, 181, 200, 224, 246, 275, 304, 337, 375, 413, 460, 503, 559, 614, 679, 749, 821, 907, 991, 1096, 1197, 1319, 1442, 1582, 1733, 1893, 2076, 2265, 2482, 2702, 2956, 3220

Offset: 0

Reinhard Zumkeller, Apr 25 2013

nonn

			a(10) = #{8+2} = 1;
a(11) = #{11, 9+2, 7+4, 5+4+2} = 4;
a(12) = #{12, 8+4, 7+5} = 3;
a(13) = #{13, 11+2, 9+4, 8+5, 7+4+2} = 5.

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)

Cf. A000009, A024940, A225044, A087154.

a225045 = p a014132_list where
   p _      0 = 1
   p (k:ks) m = if m < k then 0 else p ks (m - k) + p ks m

b:= proc(n, i) option remember; `if`(n>i*(i+1)/2, 0,
      `if`(n=0, 1, add(b(n-i*j, i-1), j=0..min(n/i,
      `if`(issqr(8*i+1), 0, 1)))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..80);  # Alois P. Heinz, Apr 01 2014

b[n_, i_] := b[n, i] = If[n > i*(i+1)/2, 0, If[n==0, 1, Sum[b[n-i*j, i-1], {j, 0, Min[n/i, If[IntegerQ[Sqrt[8*i+1]], 0, 1]]}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Jan 15 2016, after Alois P. Heinz *)

N=66; q='q+O('q^N); Vec( prod(n=1,N, 1 + q^n) / prod(n=1,N, 1 + q^(n*(n+1)/2)) ) \\ Joerg Arndt, Apr 01 2014

G.f.: prod(n>=1, 1 + q^n ) / prod(n>=1, 1 + q^(n*(n+1)/2) ). [Joerg Arndt, Apr 01 2014]

a(n) ~ exp(Pi*sqrt(n/3) - 3^(1/4) * Zeta(3/2) * n^(1/4) / (2+sqrt(2)) - 3*(3-2*sqrt(2)) * Zeta(3/2)^2 / (16*Pi)) / (2*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Jan 02 2017

A304876 L.g.f.: log(Product_{k>=1} (1 + x^(k(k+1)/2))) = Sum_{n>=1} a(n)x^n/n.

Original entry on oeis.org

1, -1, 4, -1, 1, 2, 1, -1, 4, 9, 1, -10, 1, -1, 19, -1, 1, 2, 1, -11, 25, -1, 1, -10, 1, -1, 4, 27, 1, -3, 1, -1, 4, -1, 1, 26, 1, -1, 4, -11, 1, -19, 1, -1, 64, -1, 1, -10, 1, 9, 4, -1, 1, 2, 56, -29, 4, -1, 1, -35, 1, -1, 25, -1, 1, 68, 1, -1, 4, 9, 1, -46, 1, -1, 19, -1, 1

Offset: 1

Ilya Gutkovskiy, May 20 2018

sign

			L.g.f.: L(x) = x - x^2/2 + 4*x^3/3 - x^4/4 + x^5/5 + 2*x^6/6 + x^7/7 - x^8/8 + 4*x^9/9 + 9*x^10/10 + ...
exp(L(x)) = 1 + x + x^3 + x^4 + x^6 + x^7 + x^9 + 2*x^10 + x^11 + x^13 + x^14 + x^15 + 2*x^16 + ... + A024940(n)*x^n + ...

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to sums of divisors

Cf. A000217, A010054, A024940, A185027, A300853.

nmax = 77; Rest[CoefficientList[Series[Log[Product[1 + x^(k (k + 1)/2), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]]
nmax = 77; Rest[CoefficientList[Series[Sum[k (k + 1)/2 x^(k (k + 1)/2)/(1 + x^(k (k + 1)/2)), {k, 1, nmax}], {x, 0, nmax}], x]]
Table[DivisorSum[n, (-1)^(n/# + 1) # &, IntegerQ[(8 # + 1)^(1/2)] &], {n, 77}]

A010054(n) = issquare(8*n + 1);
A304876(n) = sumdiv(n,d,(-1)^(1+(n/d)) * A010054(d)*d); \\ Antti Karttunen, Feb 20 2023

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

a(n) = Sum_{d|n} (-1)^(n/d+1)*A010054(d)*d.

A359348 Maximal coefficient of (1 + x) * (1 + x^3) * (1 + x^6) * ... * (1 + x^(n*(n+1)/2)).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 3, 4, 5, 7, 12, 18, 27, 44, 73, 122, 210, 362, 620, 1050, 1857, 3290, 5949, 10665, 19086, 34330, 62252, 113643, 209460, 383888, 706457, 1300198, 2407535, 4468367, 8331820, 15525814, 28987902, 54180854, 101560631, 190708871, 358969426

Offset: 0

Seiichi Manyama, Dec 27 2022

nonn

			(1 + x) * (1 + x^3) * (1 + x^6) * (1 + x^10) = 1 + x + x^3 + x^4 + x^6 + x^7 + x^9 + 2 * x^10 + x^11 + x^13 + x^14 + x^16 + x^17 + x^19 + x^20. So a(4) = 2.

Seiichi Manyama, Table of n, a(n) for n = 0..200

Cf. A000217, A024940, A025591, A158380, A160235.

a(n) = vecmax(Vec(prod(k=1, n, 1+x^(k*(k+1)/2))));

a(n) ~ sqrt(5) * 2^(n + 3/2) / (sqrt(Pi) * n^(5/2)). - Vaclav Kotesovec, Dec 29 2022

A280421 G.f.: Product_{k>=1} (1 + x^(k*(k+1)/2)) / (1 - x^k).

Original entry on oeis.org

1, 2, 3, 6, 10, 15, 24, 36, 52, 76, 109, 152, 211, 290, 393, 530, 709, 938, 1236, 1618, 2102, 2720, 3500, 4477, 5707, 7242, 9146, 11511, 14435, 18030, 22451, 27868, 34476, 42531, 52324, 64186, 78541, 95867, 116721, 141791, 171862, 207844, 250846, 302134

Offset: 0

Vaclav Kotesovec, Jan 02 2017

nonn

Convolution of A024940 and A000041.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A024940, A000041, A280204, A280423.

nmax=60; CoefficientList[Series[Product[(1+x^(k*(k+1)/2))/(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^(3/4) + 3*(2*sqrt(2)-3) * Zeta(3/2)^2 / (32*Pi)) / (8*sqrt(3)*n).

A294623 Number of partitions of n into distinct generalized heptagonal numbers (A085787).

Original entry on oeis.org

1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 0, 2, 3, 1, 0, 3, 3, 1, 2, 2, 1, 1, 3, 3, 3, 2, 1, 2, 3, 4, 3, 2, 2, 3, 3, 3, 5, 3, 1, 3, 4, 3, 4, 5, 2, 3, 5, 4, 3, 4, 5, 4, 4, 3, 5, 5, 3, 5, 7, 5, 3, 6, 6, 6, 6, 5, 5, 6, 6, 5, 8, 7, 5, 5, 6, 7, 8, 8

Offset: 0

Ilya Gutkovskiy, Nov 05 2017

nonn

			a(18) = 2 because we have [18] and [13, 4, 1].

Eric Weisstein's World of Mathematics, Heptagonal Number
Index to sequences related to polygonal numbers
Index entries for related partition-counting sequences

Cf. A024940, A085787, A279280, A290942, A294621, A294624.

nmax = 100; CoefficientList[Series[Product[(1 + x^(k (5 k - 3)/2)) (1 + x^(k (5 k + 3)/2)), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} (1 + x^(k*(5*k-3)/2))*(1 + x^(k*(5*k+3)/2)).

A294624 Number of partitions of n into distinct generalized octagonal numbers (A001082).

Original entry on oeis.org

1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 2, 2, 0, 1, 1, 1, 1, 0, 2, 2, 0, 0, 1, 2, 1, 0, 1, 2, 1, 1, 2, 2, 1, 0, 2, 3, 1, 1, 2, 2, 1, 0, 1, 3, 2, 2, 3, 1, 1, 1, 3, 5, 2, 2, 3, 2, 2, 1, 3, 5, 2, 1, 3, 3, 2, 1, 3, 6, 3, 1, 3, 4, 3, 1, 4, 7, 3, 0, 3, 6, 4, 1, 2, 7, 5, 2, 4, 5, 5, 2

Offset: 0

Ilya Gutkovskiy, Nov 05 2017

nonn

			a(21) = 2 because we have [21] and [16, 5].

Eric Weisstein's World of Mathematics, Octagonal Number
Index to sequences related to polygonal numbers
Index entries for related partition-counting sequences

Cf. A001082, A024940, A279281, A290942, A294622, A294623.

nmax = 100; CoefficientList[Series[Product[(1 + x^(k (3 k - 2))) (1 + x^(k (3 k + 2))), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} (1 + x^(k*(3*k-2)))*(1 + x^(k*(3*k+2))).

A303907 Expansion of Product_{k>=2} (1 + x^(k*(k+1)/2)).

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 2, 0, 0, 2, 1, 0, 1, 2, 0, 1, 3, 0, 0, 3, 0, 2, 2, 1, 2, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 0, 2, 4, 1, 2, 5, 1, 2, 3, 2, 3, 3, 2, 2, 5, 2, 4, 4, 2, 3, 6, 1, 3, 6, 3, 3, 7, 2, 2, 7, 3, 5, 6, 5, 4, 6, 4, 5, 5, 5, 4, 10, 4, 3, 11, 3

Offset: 0

Ilya Gutkovskiy, May 02 2018

nonn

Number of partitions of n into distinct triangular numbers > 1.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Index entries for sequences related to partitions

Cf. A000217, A024940, A025147, A280129, A303906.

nmax = 95; CoefficientList[Series[Product[1 + x^(k (k + 1)/2), {k, 2, nmax}], {x, 0, nmax}], x]

a(n) = Sum_{k=0..n} (-1)^(n-k)*A024940(k).

a(n) ~ exp(3*Pi^(1/3) * ((sqrt(2)-1)*Zeta(3/2))^(2/3) * n^(1/3) / 2^(4/3)) * ((sqrt(2)-1)*Zeta(3/2))^(1/3) / (2^(8/3) * sqrt(3) * Pi^(1/3) * n^(5/6)). - Vaclav Kotesovec, May 04 2018

A348528 Number of partitions of n into two or more distinct triangular numbers.

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Oct 21 2021

nonn

Cf. A000217, A010054, A024940, A033985, A348526, A348529.

A357070 Number of partitions of n into at most 2 distinct positive triangular numbers.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 2, 0, 1, 0, 0, 2, 1, 0, 1, 1, 0, 1, 1, 1, 0, 2, 0, 0, 1, 0, 2, 1, 1, 1, 0, 0, 1, 1, 0, 1, 2, 0, 1, 1, 0, 2, 0, 0, 0, 2, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 2, 1, 0, 1, 1, 0, 1, 1, 0, 0, 2, 0, 1, 1, 0, 3, 0, 1, 1, 0, 0, 1, 1, 0, 0, 2, 1, 1, 2, 0, 0, 1, 0, 1, 1, 1

Offset: 0

Ilya Gutkovskiy, Oct 25 2022

nonn

Cf. A000217, A024940, A052343, A307597, A347730, A357071, A357072.

cp's OEIS Frontend

A193749 Number of partitions of n into distinct parts that are squares or triangular numbers (A005214).

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A225045 Number of partitions of n into distinct non-triangular numbers, cf. A014132.

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A304876 L.g.f.: log(Product_{k>=1} (1 + x^(k*(k+1)/2))) = Sum_{n>=1} a(n)*x^n/n.

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A359348 Maximal coefficient of (1 + x) * (1 + x^3) * (1 + x^6) * ... * (1 + x^(n*(n+1)/2)).

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A280421 G.f.: Product_{k>=1} (1 + x^(k*(k+1)/2)) / (1 - x^k).

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A294623 Number of partitions of n into distinct generalized heptagonal numbers (A085787).

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A294624 Number of partitions of n into distinct generalized octagonal numbers (A001082).

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A303907 Expansion of Product_{k>=2} (1 + x^(k*(k+1)/2)).

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A304876 L.g.f.: log(Product_{k>=1} (1 + x^(k(k+1)/2))) = Sum_{n>=1} a(n)x^n/n.