A281084 -id:A281084 - cp's OEIS frontend

A281083 Expansion of Product_{k>=0} (1 + x^(5k(k+1)/2+1)).

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

Offset: 0

Ilya Gutkovskiy, Jan 14 2017

nonn

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

			a(82) = 2 because we have [76, 6] and [51, 31].

Alois P. Heinz, Table of n, a(n) for n = 0..20000 (first 1001 terms from G. C. Greubel)
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, A218380, A280952, A281081, A281082, A281084.

nmax = 105; CoefficientList[Series[Product[1 + x^(5 k (k + 1)/2 + 1), {k, 0, nmax}], {x, 0, nmax}], x]

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

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)).

A322802 Number of compositions (ordered partitions) of n into centered hexagonal numbers (A003215).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 10, 13, 17, 22, 28, 36, 45, 56, 70, 88, 111, 140, 178, 226, 286, 361, 455, 573, 721, 909, 1148, 1451, 1834, 2318, 2928, 3695, 4661, 5880, 7420, 9366, 11826, 14935, 18860, 23812, 30059, 37941, 47888, 60445, 76302, 96327

Offset: 0

Ilya Gutkovskiy, Dec 26 2018

nonn

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

Cf. A003215, A280863, A280953, A281084, A282502, A282504, A322798, A322801, A322803.

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:
a:= proc(n) option remember; `if`(n=0, 1,
      add(a(n-(3*i*(i+1)+1)), i=0..h(n)))
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Dec 28 2018

nmax = 53; CoefficientList[Series[1/(1 - Sum[x^(3 k (k + 1) + 1), {k, 0, nmax}]), {x, 0, nmax}], x]

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

cp's OEIS Frontend

A281083 Expansion of Product_{k>=0} (1 + x^(5*k*(k+1)/2+1)).

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

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

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A322802 Number of compositions (ordered partitions) of n into centered hexagonal numbers (A003215).

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Mathematica

Formula

A281083 Expansion of Product_{k>=0} (1 + x^(5k(k+1)/2+1)).

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)).