A280950 -id:A280950 - cp's OEIS frontend

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

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)).

A280952 Expansion of Product_{k>=0} 1/(1 - x^(5k(k+1)/2+1)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 5, 5, 5, 5, 6, 6, 7, 7, 7, 7, 8, 8, 9, 10, 11, 11, 12, 12, 13, 14, 15, 15, 16, 16, 17, 18, 19, 19, 20, 21, 23, 24, 25, 26, 27, 28, 30, 31, 32, 33, 34, 35, 37, 38, 40, 42, 44, 45, 47, 49, 51, 53, 55, 56, 58, 60, 62, 64, 67, 68, 71, 74, 77, 79, 83, 85, 88, 91, 94, 96, 100

Offset: 0

Ilya Gutkovskiy, Jan 11 2017

nonn

Number of partitions of n into centered pentagonal numbers (A005891).

			a(12) = 3 because we have [6, 6], [6, 1, 1, 1, 1, 1, 1] and [1, 1, 1, 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, Centered Pentagonal Number
Index entries for sequences related to centered polygonal numbers
Index entries for related partition-counting sequences

Cf. A005891, A218379, A279221, A280950, A280951, A280953.

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

nmax = 88; CoefficientList[Series[Product[1/(1 - x^(5 k (k + 1)/2 + 1)), {k, 0, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=0} 1/(1 - x^(5*k*(k+1)/2+1)).

A280953 Expansion of Product_{k>=0} 1/(1 - x^(3k(k+1)+1)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 5, 5, 5, 5, 5, 6, 6, 7, 7, 7, 7, 7, 8, 8, 9, 9, 10, 11, 11, 12, 12, 13, 13, 14, 15, 15, 16, 16, 17, 17, 18, 19, 19, 20, 20, 22, 23, 24, 25, 25, 27, 27, 29, 30, 31, 32, 32, 34, 34, 36, 37, 38, 39, 40, 43, 44, 46, 47, 48, 50, 51, 54, 55, 57, 58, 59

Offset: 0

Ilya Gutkovskiy, Jan 11 2017

nonn

Number of partitions of n into centered hexagonal numbers (A003215).

			a(14) = 3 because we have [7, 7], [7, 1, 1, 1, 1, 1, 1, 1] and [1, 1, 1, 1, 1, 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, Hex Number
Index entries for sequences related to centered polygonal numbers
Index entries for related partition-counting sequences

Cf. A003215, A278949, A279222, A280950, A280951, A280952.

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

nmax = 86; CoefficientList[Series[Product[1/(1 - x^(3 k (k + 1) + 1)), {k, 0, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=0} 1/(1 - x^(3*k*(k+1)+1)).

A281081 Expansion of Product_{k>=0} (1 + x^(3k(k+1)/2+1)).

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Jan 14 2017

nonn

Number of partitions of n into distinct centered triangular numbers (A005448).

			a(46) = 2 because we have [46] and [31, 10, 4, 1].

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

Cf. A005448, A024940, A279278, A280950, A281082, A281083, A281084.

nmax = 105; CoefficientList[Series[Product[1 + x^(3 k (k + 1)/2 + 1), {k, 0, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=0} (1 + x^(3*k*(k+1)/2+1)).

A282502 Expansion of 1/(1 - Sum_{k>=0} x^(3k(k+1)/2+1)).

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 4, 5, 7, 10, 15, 21, 29, 40, 57, 81, 114, 159, 223, 315, 445, 626, 879, 1236, 1741, 2452, 3450, 4852, 6826, 9608, 13524, 19032, 26778, 37680, 53027, 74627, 105017, 147776, 207949, 292636, 411813, 579515, 815499, 1147585, 1614917, 2272566, 3198016, 4500318, 6332952, 8911902, 12541080

Offset: 0

Ilya Gutkovskiy, Feb 16 2017

nonn

Number of compositions (ordered partitions) into centered triangular numbers (A005448).

Conjecture: every number > 1 is the sum of at most 5 centered triangular numbers.

			a(7) = 5 because we have [4, 1, 1, 1], [1, 4, 1, 1], [1, 1, 4, 1], [1, 1, 1, 4] and [1, 1, 1, 1, 1, 1, 1].

Eric Weisstein's World of Mathematics, Centered Triangular Number
Index entries for sequences related to centered polygonal numbers
Index entries for sequences related to compositions

Cf. A005448, A023361, A280950.

nmax = 50; CoefficientList[Series[1/(1 - Sum[x^(3 k (k + 1)/2 + 1), {k, 0, nmax}]), {x, 0, nmax}], x]

G.f.: 1/(1 - Sum_{k>=0} x^(3*k*(k+1)/2+1)).

a(n) ~ c / r^n, where r = 0.71061790420456638132596657780064392952867377958... is the root of the equation r^(5/8)*EllipticTheta(2, 0, r^(3/2)) = 2 and c = 0.478786567198436133936216342628844283927491282611910379922933700360643... . - Vaclav Kotesovec, Feb 17 2017

A332005 Number of compositions (ordered partitions) of n into distinct centered triangular numbers.

Original entry on oeis.org

1, 1, 0, 0, 1, 2, 0, 0, 0, 0, 1, 2, 0, 0, 2, 6, 0, 0, 0, 1, 2, 0, 0, 2, 6, 0, 0, 0, 0, 2, 6, 1, 2, 6, 24, 2, 6, 0, 0, 0, 0, 2, 6, 0, 0, 6, 25, 2, 0, 0, 4, 12, 0, 0, 6, 24, 2, 6, 0, 0, 12, 48, 0, 0, 25, 124, 6, 0, 2, 12, 24, 0, 0, 0, 2, 12, 24, 2, 12, 48, 120, 6, 24, 2, 6, 1, 2, 12

Offset: 0

Ilya Gutkovskiy, Feb 04 2020

nonn

			a(15) = 6 because we have [10, 4, 1], [10, 1, 4], [4, 10, 1], [4, 1, 10], [1, 10, 4] and [1, 4, 10].

Eric Weisstein's World of Mathematics, Centered Triangular Number
Index entries for sequences related to compositions

Cf. A005448, A023361, A280950, A281081, A282502, A331843, A331919, A332006.

A334987 Sum of centered triangular numbers dividing n.

Original entry on oeis.org

1, 1, 1, 5, 1, 1, 1, 5, 1, 11, 1, 5, 1, 1, 1, 5, 1, 1, 20, 15, 1, 1, 1, 5, 1, 1, 1, 5, 1, 11, 32, 5, 1, 1, 1, 5, 1, 20, 1, 15, 1, 1, 1, 5, 1, 47, 1, 5, 1, 11, 1, 5, 1, 1, 1, 5, 20, 1, 1, 15, 1, 32, 1, 69, 1, 1, 1, 5, 1, 11, 1, 5, 1, 1, 1, 24, 1, 1, 1, 15, 1, 1, 1, 5, 86, 1, 1, 5, 1, 11

Offset: 1

Ilya Gutkovskiy, May 18 2020

nonn

Eric Weisstein's World of Mathematics, Centered Triangular Number

Cf. A005448, A185027, A280950, A300409, A334988.

nmax = 90; CoefficientList[Series[Sum[(3 k (k - 1)/2 + 1) x^(3 k (k - 1)/2 + 1)/(1 - x^(3 k (k - 1)/2 + 1)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
nmax = 90; CoefficientList[Series[Log[Product[1/(1 - x^(3 k (k - 1)/2 + 1)), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax] // Rest

isc(n) = my(k=(2*n-2)/3, m); (n==1) || ((denominator(k)==1) && (m=sqrtint(k)) && (m*(m+1)==k));
a(n) = sumdiv(n, d, if (isc(d), d)); \\ Michel Marcus, May 19 2020

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

L.g.f.: log(G(x)), where G(x) is the g.f. for A280950.

cp's OEIS Frontend

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

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A280952 Expansion of Product_{k>=0} 1/(1 - x^(5*k*(k+1)/2+1)).

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A280953 Expansion of Product_{k>=0} 1/(1 - x^(3*k*(k+1)+1)).

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A281081 Expansion of Product_{k>=0} (1 + x^(3*k*(k+1)/2+1)).

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A282502 Expansion of 1/(1 - Sum_{k>=0} x^(3*k*(k+1)/2+1)).

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A332005 Number of compositions (ordered partitions) of n into distinct centered triangular numbers.

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A334987 Sum of centered triangular numbers dividing n.

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

A280952 Expansion of Product_{k>=0} 1/(1 - x^(5k(k+1)/2+1)).

A280953 Expansion of Product_{k>=0} 1/(1 - x^(3k(k+1)+1)).

A281081 Expansion of Product_{k>=0} (1 + x^(3k(k+1)/2+1)).

A282502 Expansion of 1/(1 - Sum_{k>=0} x^(3k(k+1)/2+1)).