A281083 -id:A281083 - cp's OEIS frontend

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

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)).

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)).

A281084 Expansion of Product_{k>=0} (1 + x^(3k(k+1)+1)).

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Jan 14 2017

nonn

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

			a(98) = 2 because we have [91, 7] and [61, 37].

G. C. Greubel, Table of n, a(n) for n = 0..1000
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, A279279, A280953, A281081, A281082, A281083.

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

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

A290942 Number of partitions of n into distinct generalized pentagonal numbers (A001318).

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 2, 2, 1, 1, 0, 2, 2, 2, 3, 1, 2, 2, 2, 3, 2, 4, 3, 3, 3, 2, 5, 4, 5, 4, 2, 3, 3, 6, 6, 5, 5, 4, 5, 7, 8, 8, 7, 6, 6, 6, 8, 9, 9, 9, 7, 8, 9, 9, 11, 10, 11, 11, 10, 12, 10, 14, 15, 14, 14, 11, 13, 13, 17, 17, 14, 15, 14, 17, 20, 19, 20, 20, 20, 21, 20, 21, 21, 25, 26, 23, 22, 21, 24, 27

Offset: 0

Ilya Gutkovskiy, Aug 14 2017

nonn

			a(15) = 3 because we have [15], [12, 2, 1] and [7, 5, 2, 1].

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

Cf. A000009, A001318, A095699, A218379, A218380, A279221, A280952, A281083.

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

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

A322801 Number of compositions (ordered partitions) of n into centered pentagonal numbers (A005891).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 9, 12, 16, 21, 28, 36, 46, 59, 76, 98, 128, 167, 217, 281, 363, 468, 605, 784, 1017, 1320, 1712, 2217, 2869, 3713, 4807, 6227, 8070, 10458, 13549, 17549, 22726, 29430, 38117, 49375, 63962, 82859, 107333, 139026, 180071

Offset: 0

Ilya Gutkovskiy, Dec 26 2018

nonn

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

Cf. A005891, A181324, A280863, A280952, A281083, A282502, A282504, A322802, A322803.

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:
a:= proc(n) option remember; `if`(n=0, 1,
      add(a(n-((i+1)*5*i+2)/2), i=0..h(n)))
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Dec 28 2018

nmax = 50; CoefficientList[Series[1/(1 - Sum[x^(5 k (k + 1)/2 + 1), {k, 0, nmax}]), {x, 0, nmax}], x]

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

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A281084 Expansion of Product_{k>=0} (1 + x^(3*k*(k+1)+1)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A290942 Number of partitions of n into distinct generalized pentagonal numbers (A001318).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A322801 Number of compositions (ordered partitions) of n into centered pentagonal numbers (A005891).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

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

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

A281084 Expansion of Product_{k>=0} (1 + x^(3k(k+1)+1)).