A280951 -id:A280951 - cp's OEIS frontend

A280950 Expansion of Product_{k>=0} 1/(1 - x^(3k(k+1)/2+1)).

1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 9, 11, 11, 12, 13, 15, 15, 16, 17, 19, 20, 22, 24, 26, 27, 29, 31, 33, 34, 37, 40, 43, 45, 48, 51, 54, 56, 60, 63, 67, 70, 76, 80, 84, 87, 93, 97, 102, 106, 113, 118, 125, 130, 138, 143, 151, 157, 166, 172, 181, 189, 200, 207, 217, 225, 237, 245, 257, 267, 280

Offset: 0

Ilya Gutkovskiy, Jan 11 2017

nonn

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

			a(8) = 3 because we have [4, 4], [4, 1, 1, 1, 1] and [1, 1, 1, 1, 1, 1, 1, 1].

Robert Israel, Table of n, a(n) for n = 0..10000
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 Triangular Number
Index entries for sequences related to centered polygonal numbers
Index entries for related partition-counting sequences

Cf. A005448, A007294, A068980, A280951, A280952, A280953.

N:= 100:
kmax:= floor((sqrt(24*N-15)-3)/6):
S:= series(mul(1/(1-x^(3*k*(k+1)/2+1)),k=0..kmax),x,N+1):
seq(coeff(S,x,j),j=0..N); # Robert Israel, Jan 25 2017

nmax = 78; CoefficientList[Series[Product[1/(1 - x^(3 k (k + 1)/2 + 1)), {k, 0, nmax}], {x, 0, nmax}], x]

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

A281082 Expansion of Product_{k>=0} (1 + x^(2k(k+1)+1)).

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Jan 14 2017

nonn

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

			a(66) = 2 because we have [61, 5] and [41, 25].

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. A001844, A033461, A280951, A281081, A281083, A281084.

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

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

A282504 Expansion of 1/(1 - Sum_{k>=0} x^(2k(k+1)+1)).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 8, 11, 15, 21, 28, 37, 49, 65, 88, 119, 160, 214, 285, 381, 511, 687, 923, 1237, 1656, 2217, 2971, 3985, 5345, 7166, 9603, 12867, 17244, 23115, 30989, 41543, 55684, 74634, 100032, 134081, 179729, 240919, 322935, 432858, 580191, 777680, 1042407, 1397262, 1872911, 2510457

Offset: 0

Ilya Gutkovskiy, Feb 16 2017

nonn

Number of compositions (ordered partitions) into centered square numbers (A001844).
Conjecture: every number > 1 is the sum of at most 6 centered square numbers.
Extended conjecture: every number > 1 is the sum of at most k+2 centered k-gonal numbers.

			a(8) = 5 because we have [5, 1, 1, 1], [1, 5, 1, 1], [1, 1, 5, 1], [1, 1, 1, 5] and [1, 1, 1, 1, 1, 1, 1, 1].

Indranil Ghosh, Table of n, a(n) for n = 0..200
Eric Weisstein's World of Mathematics, Centered Square Number
Index entries for sequences related to centered polygonal numbers
Index entries for sequences related to compositions

Cf. A001844, A006456, A280951.

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

Vec(1/(1 - sum(k=0, 54, x^(2*k*(k + 1) + 1))) + O(x^54)) \\ Indranil Ghosh, Mar 15 2017

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

a(n) ~ c / r^n, where r = 0.746043978237212782246711857485153004976647... is the root of the equation sqrt(r) * EllipticTheta(2, 0, r^2) = 2 and c = 0.453173429667590077751072798128748901015122665... . - Vaclav Kotesovec, Feb 17 2017

A286934 Number of partitions of n into centered square primes (A027862).

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, May 16 2017

nonn

			a(41) = 2 because we have [41] and [13, 13, 5, 5, 5].

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

Cf. A000607, A001844, A027862, A280951, A282970.

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

G.f.: Product_{k>=1} 1/(1 - x^A027862(k)).

A332006 Number of compositions (ordered partitions) of n into distinct centered square numbers.

Original entry on oeis.org

1, 1, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 2, 6, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 2, 6, 0, 0, 0, 0, 0, 0, 2, 6, 0, 1, 2, 6, 24, 0, 2, 6, 0, 0, 0, 0, 0, 0, 2, 6, 0, 0, 0, 6, 24, 1, 2, 0, 0, 0, 4, 12, 0, 0, 0, 6, 24, 0, 2, 6, 0, 0, 0, 12, 48, 0, 0, 0, 24, 121, 4, 6

Offset: 0

Ilya Gutkovskiy, Feb 04 2020

nonn

			a(19) = 6 because we have [13, 5, 1], [13, 1, 5], [5, 13, 1], [5, 1, 13], [1, 13, 5] and [1, 5, 13].

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

Cf. A001844, A006456, A280951, A281082, A282504, A331844, A331984, A332005.

cp's OEIS Frontend

A280950 Expansion of Product_{k>=0} 1/(1 - x^(3*k*(k+1)/2+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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A281082 Expansion of Product_{k>=0} (1 + x^(2*k*(k+1)+1)).

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A282504 Expansion of 1/(1 - Sum_{k>=0} x^(2*k*(k+1)+1)).

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A286934 Number of partitions of n into centered square primes (A027862).

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Formula

A332006 Number of compositions (ordered partitions) of n into distinct centered square numbers.

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A280950 Expansion of Product_{k>=0} 1/(1 - x^(3k(k+1)/2+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)).

A281082 Expansion of Product_{k>=0} (1 + x^(2k(k+1)+1)).

A282504 Expansion of 1/(1 - Sum_{k>=0} x^(2k(k+1)+1)).