A106337 Number of ways of writing n as the sum of n triangular numbers.

1, 1, 1, 4, 13, 31, 82, 253, 757, 2173, 6341, 18888, 56266, 167324, 499773, 1499059, 4503557, 13546893, 40824379, 123233868, 372472353, 1127080252, 3414310032, 10353722919, 31425764410, 95463814056, 290222666436, 882954212908, 2688037654049, 8188468874808

Offset

0,4

Comments

Number of compositions of n into n triangular numbers with 0's allowed. a(3) = 4: [1,1,1], [0,0,3], [0,3,0], [3,0,0]. - Alois P. Heinz, Jul 31 2017

The radius of convergence is equal to A106335. - Vaclav Kotesovec, Nov 15 2017

Links

Alois P. Heinz, Table of n, a(n) for n = 0..2000

Formula

Log.g.f.: Sum_{n>=1} a(n)/n*x^n = log(G106336(x)), where G106336(x) is the g.f. of A106336 and satisfies: Sum_{n>=0} (x*G106336(x))^(n*(n+1)/2) = G106336(x).

a(n) = [x^n] Product_{j=1..n} (1+x^j-x^(2*j)-x^(3*j))^n. - Alois P. Heinz, Aug 01 2017

Example

G106336(x) = exp(x + 1/2*x^2 + 4/3*x^3 + 13/4*x^4 + 31/5*x^5 +...).
G106336(x) = 1 + x + x^2 + 2*x^3 + 5*x^4 + 11*x^5 +...+ A106336(n)*x^n +...
G106336(x) = 1 + x*G106336(x) + (x*G106336(x))^3 + (x*G106336(x))^6 +...

Maple

b:= proc(n) option remember; expand(`if`(n=0, 1,
      add(`if`(issqr(8*j+1), x*b(n-j), 0), j=1..n)))
    end:
a:= n-> (p-> add(coeff(p, x, i)*binomial(n, i), i=0..n))(b(n)):
seq(a(n), n=0..50);  # Alois P. Heinz, Jul 31 2017

Mathematica

QP = QPochhammer; a[0] = 1; a[n_] := SeriesCoefficient[(QP[-1, x]*QP[x^2]/2 )^n, {x, 0, n}]; Table[a[n], {n, 0, 27}] (* Jean-François Alcover, Jun 04 2017 *)

Prog

(PARI) {a(n)=local(X); if(n<1, 1, X=x+x*O(x^n); polcoeff(eta(X^2)^(2*n)/eta(X)^n, n))}

Crossrefs

Cf. A000217, A007294, A023361, A106333, A106334, A106335, A106336, A296045.

Main diagonal of A286180.

Sequence in context: A097122 A116411 A333047 * A027998 A367010 A026567

Adjacent sequences: A106334 A106335 A106336 * A106338 A106339 A106340

Keyword

nonn

Author

Paul D. Hanna, Apr 29 2005

Extensions

a(0) changed to 1 by Alois P. Heinz, Jul 31 2017

Status

approved