A281810 Expansion of Sum_{i>=1} x^(i*(i+1)/2) / (1 - Sum_{j>=1} x^(j*(j+1)/2))^2.

1, 2, 4, 8, 14, 25, 45, 77, 131, 224, 377, 629, 1049, 1738, 2863, 4708, 7716, 12598, 20524, 33363, 54102, 87567, 141489, 228216, 367538, 591098, 949372, 1522917, 2440190, 3905747, 6245198, 9976535, 15923083, 25392755, 40462155, 64426278, 102510580, 162997910, 259010672, 411328655, 652842792, 1035591110

Offset: 1

Author

Ilya Gutkovskiy, Jan 30 2017

Keywords

• nonn

Comments

Total number of parts in all compositions (ordered partitions) of n into nonzero triangular numbers (A000217).

Examples

			a(6) = 25 because we have [6], [3, 3], [3, 1, 1, 1], [1, 3, 1, 1], [1, 1, 3, 1], [1, 1, 1, 3], [1, 1, 1, 1, 1, 1] and 1 + 2 + 4 + 4 + 4 + 4 + 6 = 25.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..5238
• Index to sequences related to polygonal numbers
• Index entries for sequences related to compositions

Crossrefs

Cf. A000217, A023361.

Programs

Maple

b:= proc(n) option remember; `if`(n=0, [1, 0], add(
      (p-> p+[0, p[1]])(b(n-j*(j+1)/2)), j=1..isqrt(2*n)))
    end:
a:= n-> b(n)[2]:
seq(a(n), n=1..55);  # Alois P. Heinz, Aug 07 2019

Mathematica

nmax = 42; Rest[CoefficientList[Series[Sum[x^(i (i + 1)/2), {i, 1, nmax}]/(1 - Sum[x^(j (j + 1)/2), {j, 1, nmax}])^2, {x, 0, nmax}], x]]
nmax = 42; Rest[CoefficientList[Series[(2 x^(1/8) EllipticTheta[2, 0, Sqrt[x]] - 4 x^(1/4))/(4 x^(1/8) - EllipticTheta[2, 0, Sqrt[x]])^2, {x, 0, nmax}], x]]

Formula

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