A181143 G.f.: A(x,y) = exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^3*y^k] * x^n/n ) = Sum_{n>=0,k=0..n} T(n,k)*x^n*y^k, as a triangle of coefficients T(n,k) read by rows.

1, 1, 1, 1, 5, 1, 1, 14, 14, 1, 1, 30, 85, 30, 1, 1, 55, 337, 337, 55, 1, 1, 91, 1029, 2230, 1029, 91, 1, 1, 140, 2632, 10549, 10549, 2632, 140, 1, 1, 204, 5922, 39533, 73157, 39533, 5922, 204, 1, 1, 285, 12090, 124805, 384948, 384948, 124805, 12090, 285, 1, 1

Offset: 0

Author

Paul D. Hanna, Oct 13 2010

Keywords

• nonn
• tabl

Comments

Compare g.f. to that of the following triangle variants:

* Pascal's: exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)*y^k] * x^n/n );

* Narayana: exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^2*y^k] * x^n/n );

* A181144: exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^4*y^k] * x^n/n );

* A218115: exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^5*y^k] * x^n/n );

* A218116: exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^6*y^k] * x^n/n ).

Examples

			G.f.: A(x,y) = 1 + (1+y)*x + (1+5*y+y^2)*x^2 + (1+14*y+14*y^2+y^3)*x^3 + (1+30*y+85*y^2+30*y^3+y^4)*x^4 +...
The logarithm of the g.f. equals the series:
log(A(x,y)) = (1 + y)*x
+ (1 + 2^3*y + y^2)*x^2/2
+ (1 + 3^3*y + 3^3*y^2 + y^3)*x^3/3
+ (1 + 4^3*y + 6^3*y^2 + 4^3*y^3 + y^4)*x^4/4
+ (1 + 5^3*y + 10^3*y^2 + 10^3*y^3 + 5^3*y^4 + y^5)*x^5/5 +...
Triangle begins:
1;
1, 1;
1, 5, 1;
1, 14, 14, 1;
1, 30, 85, 30, 1;
1, 55, 337, 337, 55, 1;
1, 91, 1029, 2230, 1029, 91, 1;
1, 140, 2632, 10549, 10549, 2632, 140, 1;
1, 204, 5922, 39533, 73157, 39533, 5922, 204, 1;
1, 285, 12090, 124805, 384948, 384948, 124805, 12090, 285, 1;
1, 385, 22869, 345389, 1648478, 2748240, 1648478, 345389, 22869, 385, 1;
1, 506, 40678, 861080, 6016297, 15525056, 15525056, 6016297, 861080, 40678, 506, 1; ...
Note that column 1 forms the sum of squares (A000330).
Inverse binomial transform of columns begins:
[1];
[1, 4, 5, 2];
[1, 13, 58, 123, 136, 76, 17];
[1, 29, 278, 1308, 3532, 5867, 6118, 3914, 1407, 218];
[1, 54, 920, 7626, 36916, 114637, 240271, 348354, 350881, 241531, 108551, 28742, 3404]; ...
the g.f. of the rightmost coefficients of which form the g.f. exp( Sum_{n>=1} (3*n)!/(3*n!^3) * x^n/n ), and yield the self-convolution of A229452.
		

Crossrefs

Cf. A000330 (column 1), A166990 (row sums), A166896 (antidiagonal sums), A218139.

Cf. variants: A001263 (Narayana), A181144, A218115, A218116.

Programs

PARI

{T(n,k)=polcoeff(polcoeff(exp(sum(m=1,n,sum(j=0,m,binomial(m,j)^3*y^j)*x^m/m)+O(x^(n+1))),n,x),k,y)}
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))