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A319799 Number of partitions of 2n into exactly n positive triangular numbers. 6
1, 0, 1, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 4, 3, 4, 3, 5, 5, 7, 5, 7, 7, 9, 9, 9, 11, 12, 14, 14, 14, 17, 17, 21, 20, 23, 24, 27, 28, 31, 32, 36, 37, 42, 43, 47, 50, 53, 58, 61, 64, 69, 72, 82, 83, 91, 92, 102, 107, 115, 118, 128, 135, 147, 152, 159, 169, 181 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,10

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..3000

FORMULA

a(n) = [x^(2n) y^n] 1/Product_{j>=1} (1-y*x^A000217(j)).

a(n) = A319797(2n,n).

G.f.: Product_{k>=1} 1 / (1 - x^(k*(k + 3)/2)). - Ilya Gutkovskiy, Jan 31 2021

MAPLE

h:= proc(n) option remember; `if`(n<1, 0,

      `if`(issqr(8*n+1), n, h(n-1)))

    end:

b:= proc(n, i, k) option remember; `if`(n=0, `if`(k=0, 1, 0),

      `if`(i*k<n or k>n, 0, b(n, h(i-1), k)+b(n-i, h(min(n-i, i)), k-1)))

    end:

a:= n-> b(2*n, h(2*n), n):

seq(a(n), n=0..80);

MATHEMATICA

h[n_] := h[n] = If[n < 1, 0, If[IntegerQ@Sqrt[8*n + 1], n, h[n - 1]]];

b[n_, i_, k_] := b[n, i, k] = If[n == 0, If[k == 0, 1, 0], If[i*k < n || k > n, 0, b[n, h[i - 1], k] + b[n - i, h[Min[n - i, i]], k - 1]]];

a[n_] := b[2n, h[2n], n];

a /@ Range[0, 80] (* Jean-Fran├žois Alcover, Mar 12 2021, after Alois P. Heinz *)

CROSSREFS

Cf. A000217, A111178, A319797.

Sequence in context: A319818 A319819 A319820 * A294078 A064133 A295101

Adjacent sequences:  A319796 A319797 A319798 * A319800 A319801 A319802

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Sep 28 2018

STATUS

approved

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Last modified July 26 09:29 EDT 2021. Contains 346294 sequences. (Running on oeis4.)