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A106337
Number of ways of writing n as the sum of n triangular numbers.
9
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
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
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Apr 29 2005
EXTENSIONS
a(0) changed to 1 by Alois P. Heinz, Jul 31 2017
STATUS
approved