A278949 -id:A278949 - cp's OEIS frontend

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

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)).

A322798 Number of compositions (ordered partitions) of n into hexagonal numbers (A000384).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 9, 12, 16, 22, 29, 37, 47, 60, 77, 101, 133, 174, 226, 292, 376, 486, 632, 823, 1072, 1394, 1808, 2342, 3036, 3939, 5116, 6648, 8636, 11211, 14548, 18875, 24493, 31795, 41283, 53604, 69594, 90338, 117251, 152184, 197540

Offset: 0

Ilya Gutkovskiy, Dec 26 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..8828
Eric Weisstein's World of Mathematics, Hexagonal Number
Index entries for sequences related to compositions

Cf. A000384, A006456, A023361, A181324, A278949, A279279, A322799, A322800.

h:= proc(n) option remember; `if`(n<1, 0, (t->
      `if`(t*(2*t-1)>n, t-1, t))(1+h(n-1)))
    end:
a:= proc(n) option remember; `if`(n=0, 1,
      add(a(n-i*(2*i-1)), i=1..h(n)))
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Dec 28 2018

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

G.f.: 1/(1 - Sum_{k>=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)).

A279279 Expansion of Product_{k>=1} (1 + x^(k(2k-1))).

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Dec 09 2016

nonn

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

			a(67) = 2 because we have [66, 1] and [45, 15, 6, 1].

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

Cf. A000384, A024940, A033461, A218380, A278949, A279280, A279281.

nmax = 120; CoefficientList[Series[Product[1 + x^(k (2 k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]

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

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))).

A332014 Number of compositions (ordered partitions) of n into distinct hexagonal numbers.

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 2, 6, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 2, 6, 0, 0, 0, 0, 0, 0, 0, 2, 6, 1, 2, 0, 0, 6, 24, 2, 6, 0, 0, 0, 0, 0, 0, 0, 2, 6, 0, 0, 0, 0, 7, 26, 0, 0, 0, 0, 2, 8, 6, 0, 0, 0, 0, 6, 24

Offset: 0

Ilya Gutkovskiy, Feb 04 2020

nonn

			a(22) = 6 because we have [15, 6, 1], [15, 1, 6], [6, 15, 1], [6, 1, 15], [1, 15, 6] and [1, 6, 15].

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

Cf. A000384, A278949, A279279, A322798, A331843, A331844, A332007, A332015, A332016.

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A322798 Number of compositions (ordered partitions) of n into hexagonal numbers (A000384).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A279279 Expansion of Product_{k>=1} (1 + x^(k*(2*k-1))).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A332014 Number of compositions (ordered partitions) of n into distinct hexagonal numbers.

Views

Author

Keywords

Examples

Links

Crossrefs

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

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

A279279 Expansion of Product_{k>=1} (1 + x^(k(2k-1))).

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