A196010 a(n) = [x^(n*(n+1)/2)] G(x)^n where G(x) = Sum_{n>=0} x^(n*(n+1)/2).

1, 1, 2, 6, 32, 126, 842, 6594, 50654, 437802, 3962082, 38901699, 398593494, 4291288911, 48518097812, 571756012282, 7011537065184, 89099102516820, 1171925227051470, 15914369767022370, 222668594799098538, 3205203680348068734, 47392228013770511784

Offset

0,3

Comments

Number of ordered ways of writing n-th triangular number as a sum of n triangular numbers (with 0's allowed). - Ilya Gutkovskiy, Jan 27 2018

Links

Paul D. Hanna, Table of n, a(n) for n = 0..100

Example

Let G(x) = 1 + x + x^3 + x^6 + x^10 + x^15 + x^21 + x^28 + x^36 +...
then a(n) = the coefficient of x^(n*(n+1)/2) in G(x)^n.
Coefficients in powers of G(x) begin:
n=0: [(1),0,0,0,0,0,0,0,...];
n=1: [1,(1),0,1,0,0,1,0,0,0,1,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,1,...];
n=2: [1,2,1,(2),2,0,3,2,0,2,2,2,1,2,0,2,4,0,2,0,1,4,2,0,2,2,0,2,2,...];
n=3: [1,3,3,4,6,3,(6),9,3,7,9,6,9,9,6,6,15,9,7,12,3,15,15,6,12,12,...];
n=4: [1,4,6,8,13,12,14,24,18,20,(32),24,31,40,30,32,48,48,38,56,42,...];
n=5: [1,5,10,15,25,31,35,55,60,60,90,90,95,135,125,(126),170,180,...];
n=6: [1,6,15,26,45,66,82,120,156,170,231,276,290,390,435,438,561,630, 651,780,861,(842),...]; ...
the coefficients in parenthesis form the initial terms of this sequence.

Prog

(PARI) {a(n)=local(G=sum(m=0, n, x^(m*(m+1)/2))+x*O(x^(n*(n+1)/2))); polcoeff(G^n, n*(n+1)/2)}

Crossrefs

Cf. A000217, A232108.

Sequence in context: A212762 A108327 A188573 * A121071 A092199 A108485

Adjacent sequences: A196007 A196008 A196009 * A196011 A196012 A196013

Keyword

nonn

Author

Paul D. Hanna, Sep 26 2011

Status

approved