A037444 -id:A037444 - cp's OEIS frontend

A288126 Number of partitions of n-th triangular number (A000217) into distinct triangular parts.

1, 1, 1, 1, 2, 1, 2, 3, 2, 4, 7, 6, 4, 14, 15, 19, 31, 28, 43, 57, 80, 103, 127, 181, 234, 295, 398, 539, 663, 888, 1178, 1419, 1959, 2519, 3102, 4201, 5282, 6510, 8717, 11162, 13557, 18108, 22965, 28206, 36860, 46350, 58060, 73857, 93541, 117058, 147376, 186158, 232949, 292798, 365639

Offset: 0

Ilya Gutkovskiy, Jun 05 2017

nonn

			a(4) = 2 because 4th triangular number is 10 and we have [10], [6, 3, 1].

Robert Israel, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Triangular Number
Index to sequences related to polygonal numbers
Index entries for related partition-counting sequences

Cf. A000217, A007294, A024940, A030273, A037444, A072964.

N:= 100:
G:= mul(1+x^(k*(k+1)/2),k=1..N):
seq(coeff(G,x,n*(n+1)/2),n=0..N); # Robert Israel, Jun 06 2017

Table[SeriesCoefficient[Product[1 + x^(k (k + 1)/2), {k, 1, n}], {x, 0, n (n + 1)/2}], {n, 0, 54}]

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

a(n) = A024940(A000217(n)).

A092362 Number of partitions of n^2 into squares greater than 1.

Original entry on oeis.org

1, 0, 1, 1, 2, 3, 5, 8, 11, 28, 44, 94, 167, 354, 643, 1314, 2412, 4792, 8981, 17374, 32566, 62008, 115702, 217040, 402396, 745795, 1372266, 2517983, 4595652, 8354350, 15125316, 27265107, 48972467, 87584837, 156119631, 277152178, 490437445, 864534950

Offset: 0

Reinhard Zumkeller, Mar 19 2004

nonn

a(n) = A078134(A000290(n)).

			a(6) = 5: 6^2 = 36 = 16+16+4 = 16+4+4+4+4+4 = 9+9+9+9 = 4+4+4+4+4+4+4+4+4.

Alois P. Heinz, Table of n, a(n) for n = 0..500

Cf. A001156, A037444.

Cf. A093115, A093116.

b:=proc(n, i) option remember; `if`(n=0, 1,
     `if`(i<2, 0, b(n, i-1) +`if`(i^2>n, 0, b(n-i^2, i))))
   end:
a:= n-> b(n^2, n):
seq(a(n), n=0..50);  # Alois P. Heinz, Apr 15 2013

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<2, 0, b[n, i-1] + If[i^2>n, 0, b[n-i^2, i]]]]; a[n_] := b[n^2, n]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Nov 11 2015, after Alois P. Heinz *)

a(n) ~ exp(3*Pi^(1/3) * Zeta(3/2)^(2/3) * n^(2/3) / 2^(4/3)) * Zeta(3/2)^(4/3) / (2^(11/3) * sqrt(3) * Pi^(5/6) * n^(11/3)). - Vaclav Kotesovec, Apr 10 2017

Corrected a(0) and more terms from Alois P. Heinz, Apr 15 2013

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

Original entry on oeis.org

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))).

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)).

A298269 Number of partitions of the n-th tetrahedral number into tetrahedral numbers.

Original entry on oeis.org

1, 1, 2, 4, 11, 29, 94, 304, 1005, 3336, 11398, 38739, 132340, 451086, 1541074, 5242767, 17779666, 60048847, 202124143, 677000711, 2256910444, 7486274436, 24713275977, 81162110629, 265192045408, 862061443031, 2788194736946, 8972104829849, 28726271274133, 91515498561954, 290116750935925

Offset: 0

Ilya Gutkovskiy, Jan 27 2018

nonn

			a(3) = 4 because third tetrahedral number is 10 and we have [10], [4, 4, 1, 1], [4, 1, 1, 1, 1, 1, 1] and [1, 1, 1, 1, 1, 1, 1, 1, 1, 1].

David A. Corneth, Table of n, a(n) for n = 0..500
Eric Weisstein's World of Mathematics, Tetrahedral Number
Index to sequences related to pyramidal numbers
Index entries for related partition-counting sequences

Cf. A000292, A037444, A068980, A072964, A298857.

Table[SeriesCoefficient[Product[1/(1 - x^(k (k + 1) (k + 2)/6)), {k, 1, n}], {x, 0, n (n + 1) (n + 2)/6}], {n, 0, 30}]

a(n) = [x^A000292(n)] Product_{k>=1} 1/(1 - x^A000292(k)).

a(n) = A068980(A000292(n)).

A337762 Number of partitions of the n-th n-gonal number into n-gonal numbers.

Original entry on oeis.org

1, 1, 2, 4, 8, 21, 56, 144, 370, 926, 2275, 5482, 12966, 30124, 68838, 154934, 343756, 752689, 1627701, 3479226, 7355608, 15390682, 31889732, 65465473, 133212912, 268811363, 538119723, 1069051243, 2108416588, 4129355331, 8033439333

Offset: 0

Ilya Gutkovskiy, Sep 19 2020

nonn

			a(3) = 4 because the third triangular number is 6 and we have [6], [3, 3], [3, 1, 1, 1] and [1, 1, 1, 1, 1, 1].

David A. Corneth, Table of n, a(n) for n = 0..278 (first 51 terms from Vaclav Kotesovec)
Eric Weisstein's World of Mathematics, Polygonal Number
Index entries for sequences related to partitions
Index to sequences related to polygonal numbers

Cf. A000041, A037444, A060354, A072964, A337763, A337764.

nmax = 20; Table[SeriesCoefficient[Product[1/(1 - x^(k*((k*(n - 2) - n + 4)/2))), {k, 1, n}], {x, 0, n*(4 - 3*n + n^2)/2}], {n, 0, nmax}] (* Vaclav Kotesovec, Sep 19 2020 *)

a(n) = [x^p(n,n)] Product_{k=1..n} 1 / (1 - x^p(n,k)), where p(n,k) = k * (k * (n - 2) - n + 4) / 2 is the k-th n-gonal number.

A093115 Number of partitions of n^2 into squares not greater than n.

Original entry on oeis.org

1, 1, 1, 1, 5, 7, 10, 13, 17, 108, 159, 228, 317, 430, 572, 748, 5753, 8125, 11266, 15376, 20672, 27430, 35942, 46575, 59717, 523905, 708028, 946875, 1253880, 1645224, 2140099, 2761318, 3535658, 4494602, 5674753, 7118724, 69766770, 90940578, 117756370

Offset: 0

Reinhard Zumkeller, Mar 21 2004

nonn

			n=6: 6^2 = 9*2^2 = 8*2^2+4*1^2 = 7*2^2+8*1^2 = 6*2^2+12*1^2 = 5*2^2+16*1^2 = 4*2^2+20*1^2 = 3*2^2+24*1^2 = 2*2^2+28*1^2 = 1*2^2+32*1^2 = 36*1^2, therefore a(6)=10.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A093116, A092362, A001156, A037444, A078134.
Cf. A072925.
Cf. A072213, A161407. [Reinhard Zumkeller, Jun 10 2009]

b:= proc(n, i) option remember; `if`(n=0, 1,
     `if`(i<1, 0, b(n, i-1) +`if`(i^2>n, 0, b(n-i^2, i))))
    end:
a:= proc(n) local r; r:= isqrt(n);
      b(n^2, r-`if`(r^2>n, 1, 0))
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Apr 15 2013

b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + If[i^2 > n, 0, b[n-i^2, i]]]]; a[n_] := (r = Sqrt[n] // Floor; b[n^2, r - If[r^2 > n, 1, 0]]); Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jul 29 2015, after Alois P. Heinz *)

Coefficient of x^(n^2) in the series expansion of Product_{k=1..floor(sqrt(n))} 1/(1 - x^(k^2)). - Vladeta Jovovic, Mar 24 2004

More terms from Vladeta Jovovic, Mar 24 2004

Corrected a(0) by Alois P. Heinz, Apr 15 2013

A093116 Number of partitions of n^2 into squares not less than n.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 1, 2, 5, 4, 4, 5, 9, 15, 23, 24, 13, 20, 32, 55, 84, 113, 185, 303, 545, 167, 298, 435, 716, 1055, 1701, 2584, 4213, 6471, 10218, 15884, 4856, 7376, 11231, 17221, 26054, 39583, 60109, 91622, 138569, 209951, 318368, 483098, 730183

Offset: 0

Reinhard Zumkeller, Mar 21 2004

nonn

			n=10: 10^2 = 100 = 64+36 = 36+16+16+16+16 = 25+25+25+25, all other partitions of 100 into squares contain parts < 10, therefore a(10) = 4.

Alois P. Heinz, Table of n, a(n) for n = 0..250

Cf. A093115, A092362, A001156, A037444, A078134.

Cf. A072213, A161408. [Reinhard Zumkeller, Jun 10 2009]

b:= proc(n, i) option remember; `if`(n=0, 1,
     `if`(i^2>n, 0, b(n, i+1) +b(n-i^2, i)))
    end:
a:= proc(n) local r; r:= isqrt(n);
      b(n^2, r+`if`(r^2Alois P. Heinz, Apr 15 2013

b[n_, i_] := b[n, i] = If[n==0, 1, If[i^2>n, 0, b[n, i+1] + b[n-i^2, i]]]; a[n_] := With[{r = Sqrt[n]//Floor}, b[n^2, r + If[r^2Jean-François Alcover, Oct 26 2015, after Alois P. Heinz *)

A279041 Expansion of Product_{k>=1} 1/(1 - x^(k(3k-2))).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 10, 10, 11, 11, 11, 12, 12, 12, 14, 14, 15, 15, 15, 16, 16, 16, 18, 18, 19, 19, 19, 21, 21, 22, 24, 25, 26, 26, 26, 28, 28, 29, 31, 32, 33, 33, 33, 35, 35, 36, 39, 40, 42, 42, 43, 45, 46, 47, 50, 51, 53

Offset: 0

Ilya Gutkovskiy, Dec 04 2016

Number of partitions of n into nonzero octagonal numbers (A000567).

			a(9) = 2 because we have [8, 1] and [1, 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, Octagonal Number
Index to sequences related to polygonal numbers
Index entries for related partition-counting sequences

Cf. A000567, A001156, A007294, A037444, A218379, A278949, A279012.

h:= proc(n) option remember; `if`(n<1, 0, (t->
      `if`(t*(3*t-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*(3*i-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 (3 k - 2))), {k, 1, nmax}], {x, 0, nmax}], x]

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

A298935 Number of partitions of n^3 into distinct squares.

Original entry on oeis.org

1, 1, 0, 0, 1, 5, 8, 40, 96, 297, 1269, 3456, 12839, 46691, 153111, 577167, 2054576, 7602937, 29000337, 110645967, 418889453, 1580667760, 6058528796, 23121913246, 89793473393, 350029321425, 1359919742613, 5340642744919, 20948242218543, 82505892314268

Offset: 0

Ilya Gutkovskiy, Jan 29 2018

nonn

			a(5) = 5 because we have [121, 4], [100, 25], [100, 16, 9], [64, 36, 25] and [64, 36, 16, 9].

Cf. A000290, A000578, A030272, A030273, A033461, A037444, A218494, A298330, A298642, A298934.

Table[SeriesCoefficient[Product[1 + x^k^2, {k, 1, Floor[n^(3/2) + 1]}], {x, 0, n^3}], {n, 0, 29}]

a(n) = [x^(n^3)] Product_{k>=1} (1 + x^(k^2)).

a(n) = A033461(A000578(n)).

cp's OEIS Frontend

A288126 Number of partitions of n-th triangular number (A000217) into distinct triangular parts.

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A092362 Number of partitions of n^2 into squares greater than 1.

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

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

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A298269 Number of partitions of the n-th tetrahedral number into tetrahedral numbers.

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A337762 Number of partitions of the n-th n-gonal number into n-gonal numbers.

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A093115 Number of partitions of n^2 into squares not greater than n.

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Mathematica

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

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

A279041 Expansion of Product_{k>=1} 1/(1 - x^(k(3k-2))).