A001156 -id:A001156 - cp's OEIS frontend

A278949 Expansion of Product_{k>=1} 1/(1 - x^(k(2k-1))).

1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 8, 9, 9, 11, 11, 11, 12, 13, 13, 15, 15, 15, 16, 17, 17, 19, 20, 20, 23, 24, 24, 26, 27, 27, 30, 31, 31, 33, 34, 35, 38, 40, 40, 44, 45, 46, 49, 51, 51, 56, 57, 58, 61, 63, 64, 69, 72, 73, 78, 80, 81, 86, 89, 90, 96, 98, 99, 105, 108, 110, 116, 120, 121, 130

Offset: 0

Ilya Gutkovskiy, Dec 02 2016

Number of partitions of n into nonzero hexagonal numbers (A000384).

			a(7) = 2 because we have [6, 1] and [1, 1, 1, 1, 1, 1, 1].

Alois P. Heinz, Table of n, a(n) for n = 0..20000
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
Eric Weisstein's World of Mathematics, Hexagonal Number
Index to sequences related to polygonal numbers
Index entries for related partition-counting sequences

Cf. A000384, A001156, A007294, A037444, A218379.

h:= proc(n) option remember; `if`(n<1, 0, (t->
      `if`(t*(2*t-1)>n, t-1, t))(1+h(n-1)))
    end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+(t-> b(n-t, min(i, h(n-t))))(i*(2*i-1))))
    end:
a:= n-> b(n, h(n)):
seq(a(n), n=0..100);  # Alois P. Heinz, Dec 28 2018

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

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

A280951 Expansion of Product_{k>=0} 1/(1 - x^(2k(k+1)+1)).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 4, 4, 5, 5, 5, 6, 6, 7, 7, 7, 8, 8, 10, 11, 11, 12, 12, 14, 15, 15, 16, 16, 18, 19, 19, 21, 22, 24, 26, 26, 28, 29, 31, 33, 33, 35, 36, 39, 42, 43, 45, 47, 50, 53, 54, 56, 58, 61, 65, 66, 69, 72, 76, 81, 83, 86, 89, 93, 98, 100, 103, 107, 112, 118, 121, 125, 130, 136, 142, 146

Offset: 0

Ilya Gutkovskiy, Jan 11 2017

nonn

Number of partitions of n into centered square numbers (A001844).

			a(10) = 3 because we have [5, 5], [5, 1, 1, 1, 1, 1] and [1, 1, 1, 1, 1, 1, 1, 1, 1, 1].

M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
Eric Weisstein's World of Mathematics, Centered Square Number
Index entries for sequences related to centered polygonal numbers
Index entries for related partition-counting sequences

Cf. A001156, A001844, A279220, A280950, A280952, A280953.

nmax = 82; CoefficientList[Series[Product[1/(1 - x^(2 k (k + 1) + 1)), {k, 0, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=0} 1/(1 - x^(2*k*(k+1)+1)).

A292520 Expansion of Product_{k>=1} 1/(1 + x^(k^2)).

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Sep 18 2017

sign

Convolution inverse of A033461.

The difference between the number of partitions of n into an even number of squares and the number of partitions of n into an odd number of squares.

Vaclav Kotesovec, Table of n, a(n) for n = 0..20000
Martin Klazar, What is an answer? — remarks, results and problems on PIO formulas in combinatorial enumeration, part I, arXiv:1808.08449 [math.CO], 2018.
Index entries for related partition-counting sequences

Cf. A001156, A033461, A081362, A243148, A276516, A279225, A279226.

nmax = 100; CoefficientList[Series[Product[1/(1 + x^(k^2)), {k, 1, Floor[Sqrt[nmax]] + 1}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 19 2017 *)

G.f.: Product_{k>=1} 1/(1 + x^(k^2)).
a(n) ~ (-1)^n * exp(3 * Pi^(1/3) * Zeta(3/2)^(2/3) * n^(1/3) / 2^(7/3)) * Zeta(3/2)^(1/3) / (2^(5/3) * sqrt(3) * Pi^(1/3) * n^(5/6)). - Vaclav Kotesovec, Sep 19 2017
a(n) = Sum_{k=0..n} (-1)^k * A243148(n,k). - Alois P. Heinz, Jul 25 2022

A279012 Expansion of Product_{k>=1} 1/(1 - x^(k(5k-3)/2)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 7, 7, 7, 7, 8, 8, 9, 10, 11, 11, 11, 12, 12, 13, 14, 15, 15, 15, 16, 16, 17, 18, 19, 19, 20, 21, 22, 24, 25, 26, 26, 27, 28, 29, 31, 32, 33, 33, 34, 35, 37, 39, 41, 42, 43, 45, 46, 48, 50, 52, 53, 54, 56, 58, 60, 62, 64, 65, 67, 69, 72, 75, 78

Offset: 0

Ilya Gutkovskiy, Dec 03 2016

Number of partitions of n into nonzero heptagonal numbers (A000566).

			a(8) = 2 because we have [7, 1] and [1, 1, 1, 1, 1, 1, 1, 1].

Alois P. Heinz, Table of n, a(n) for n = 0..20000
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
Eric Weisstein's World of Mathematics, Heptagonal Number
Index to sequences related to polygonal numbers
Index entries for related partition-counting sequences

Cf. A000566, A001156, A007294, A037444, A218379, A278949.

h:= proc(n) option remember; `if`(n<1, 0, (t->
      `if`(t*(5*t-3)/2>n, t-1, t))(1+h(n-1)))
    end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+(t-> b(n-t, min(i, h(n-t))))(i*(5*i-3)/2)))
    end:
a:= n-> b(n, h(n)):
seq(a(n), n=0..100);  # Alois P. Heinz, Dec 28 2018

nmax=90; CoefficientList[Series[Product[1/(1 - x^(k (5 k - 3)/2)), {k, 1, nmax}], {x, 0, nmax}], x]

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

A285799 Number of partitions of n into parts with an odd number of distinct prime divisors.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 3, 4, 6, 7, 9, 12, 15, 19, 23, 29, 37, 44, 54, 66, 80, 96, 115, 138, 165, 196, 231, 275, 322, 380, 444, 520, 608, 706, 821, 952, 1102, 1272, 1467, 1688, 1941, 2226, 2549, 2917, 3329, 3798, 4324, 4918, 5587, 6337, 7180, 8125, 9184, 10369, 11695, 13174, 14828, 16671, 18723, 21011, 23551

Offset: 0

Ilya Gutkovskiy, Apr 26 2017

nonn

			a(7) = 4 because we have [7], [5, 2], [4, 3] and [3, 2, 2].

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Index entries for related partition-counting sequences

Cf. A001156 (number of partitions into parts with an odd number of divisors), A030230, A285798.

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

G.f.: Product_{k>=1} 1/(1 - x^A030230(k)).

A291655 Expansion of Product_{k>=1} 1/(1-x^(k^2))^(k^2).

Original entry on oeis.org

1, 1, 1, 1, 5, 5, 5, 5, 15, 24, 24, 24, 44, 80, 80, 80, 131, 221, 266, 266, 386, 566, 746, 746, 990, 1474, 1924, 2089, 2529, 3709, 4609, 5269, 6130, 8576, 11096, 12746, 14937, 19397, 25697, 28997, 34111, 43135, 56365, 65905, 76219, 95770, 120070, 144370, 163661

Offset: 0

Vaclav Kotesovec, Aug 28 2017

nonn

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

Cf. A001156, A033461, A281790, A285047, A291649, A291696, A291720.

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

log(a(n)) ~ 5 * Pi^(1/5) * Zeta(5/2)^(2/5) * n^(3/5) / (2^(6/5) * 3^(3/5)).

A294297 Integers with precisely five partitions into sums of four squares of nonnegative numbers.

Original entry on oeis.org

50, 52, 54, 58, 70, 73, 74, 75, 76, 84, 85, 86, 89, 91, 93, 101, 103, 109, 111, 113, 127, 131, 140, 142, 143, 151, 167, 191, 200, 208, 216, 232, 280, 296, 304, 336, 344, 560, 568, 800, 832, 864, 928, 1120, 1184, 1216, 1344, 1376, 2240, 2272, 3200, 3328, 3456

Offset: 1

Robert Price, Oct 27 2017

nonn

A002635(a(n)) = 5.

Robert Price, Table of n, a(n) for n = 1..81
D. H. Lehmer, On the Partition of Numbers into Squares, The American Mathematical Monthly, Vol. 55, No.8, October 1948, pp. 476-481.
Index entries for sequences related to sums of squares

Cf. A001156, A002635, A006431, A025428, A180149, A245022, A294282.

f[n_] := Length@ PowersRepresentations[n, 4, 2]; Select[ Range@ 3500, f@# == 5 &] (* Robert G. Wilson v, Oct 27 2017 *)

A300974 a(n) = [x^n] Product_{k>=1} 1/(1 - x^(k^2))^n.

Original entry on oeis.org

1, 1, 3, 10, 39, 151, 588, 2304, 9111, 36307, 145553, 586246, 2370264, 9614242, 39105580, 159444160, 651468967, 2666771488, 10934393619, 44899828056, 184616878289, 760010818689, 3132147583744, 12921037206764, 53351800567200, 220478125956426, 911839751015196, 3773836780169050

Offset: 0

Ilya Gutkovskiy, Mar 17 2018

nonn

Number of partitions of n into squares of n kinds.

Cf. A000290, A001156, A008485, A023871, A240944, A279225, A285047, A300975, A301518.

a:= proc(m) option remember; local b; b:= proc(n, i)
      option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      binomial(m+j-1, j)*b(n-i^2*j, i-1), j=0..n/i^2)))
      end: b(n, isqrt(n))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Mar 17 2018

Table[SeriesCoefficient[Product[1/(1 - x^k^2)^n, {k, 1, n}], {x, 0, n}], {n, 0, 27}]

From Vaclav Kotesovec, Mar 23 2018: (Start)
a(n) ~ c * d^n / sqrt(n), where
d = 4.216358447600641565890184638418336163396695730036... and
c = 0.26442245016754864773722176155288663999776... (End)

A339364 Number of partitions of n into an even number of squares.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 1, 1, 2, 1, 3, 1, 3, 3, 3, 3, 4, 4, 6, 4, 7, 6, 7, 7, 8, 9, 11, 9, 13, 12, 14, 14, 16, 16, 20, 17, 23, 22, 25, 25, 28, 29, 33, 31, 37, 38, 41, 42, 45, 48, 54, 51, 61, 60, 67, 67, 72, 76, 84, 81, 93, 93, 102, 104, 110, 117, 125, 125, 139, 140, 153

Offset: 0

Ilya Gutkovskiy, Dec 01 2020

nonn

			a(10) = 3 because we have [9, 1], [4, 4, 1, 1] and [1, 1, 1, 1, 1, 1, 1, 1, 1, 1].

Cf. A000290, A001156, A027187, A292520, A339365, A339366, A339367.

nmax = 70; CoefficientList[Series[(1/2) (Product[1/(1 - x^(k^2)), {k, 1, Floor[nmax^(1/2)] + 1}] + Product[1/(1 + x^(k^2)), {k, 1, Floor[nmax^(1/2)] + 1}]), {x, 0, nmax}], x]

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

a(n) = (A001156(n) + A292520(n)) / 2.

A339365 Number of partitions of n into an odd number of squares.

Original entry on oeis.org

0, 1, 0, 1, 1, 1, 1, 1, 1, 3, 1, 3, 2, 3, 3, 3, 4, 5, 4, 6, 5, 7, 7, 7, 8, 10, 9, 12, 10, 14, 13, 14, 15, 18, 17, 21, 20, 24, 24, 25, 27, 31, 30, 35, 34, 40, 40, 42, 45, 50, 50, 56, 55, 64, 65, 68, 72, 78, 79, 88, 85, 99, 99, 105, 110, 118, 122, 131, 132, 146, 149

Offset: 0

Ilya Gutkovskiy, Dec 01 2020

nonn

			a(9) = 3 because we have [9], [4, 4, 1] and [1, 1, 1, 1, 1, 1, 1, 1, 1].

Cf. A000290, A001156, A027193, A292520, A339364, A339366, A339367.

nmax = 70; CoefficientList[Series[(1/2) (Product[1/(1 - x^(k^2)), {k, 1, Floor[nmax^(1/2)] + 1}] - Product[1/(1 + x^(k^2)), {k, 1, Floor[nmax^(1/2)] + 1}]), {x, 0, nmax}], x]

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

a(n) = (A001156(n) - A292520(n)) / 2.

cp's OEIS Frontend

A278949 Expansion of Product_{k>=1} 1/(1 - x^(k*(2*k-1))).

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A280951 Expansion of Product_{k>=0} 1/(1 - x^(2*k*(k+1)+1)).

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A292520 Expansion of Product_{k>=1} 1/(1 + x^(k^2)).

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A279012 Expansion of Product_{k>=1} 1/(1 - x^(k*(5*k-3)/2)).

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A285799 Number of partitions of n into parts with an odd number of distinct prime divisors.

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A291655 Expansion of Product_{k>=1} 1/(1-x^(k^2))^(k^2).

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A294297 Integers with precisely five partitions into sums of four squares of nonnegative numbers.

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A300974 a(n) = [x^n] Product_{k>=1} 1/(1 - x^(k^2))^n.

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A278949 Expansion of Product_{k>=1} 1/(1 - x^(k(2k-1))).

A280951 Expansion of Product_{k>=0} 1/(1 - x^(2k(k+1)+1)).

A279012 Expansion of Product_{k>=1} 1/(1 - x^(k(5k-3)/2)).