A218115 G.f.: A(x,y) = exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^5 * 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, 17, 1, 1, 98, 98, 1, 1, 354, 2251, 354, 1, 1, 979, 23803, 23803, 979, 1, 1, 2275, 158367, 617036, 158367, 2275, 1, 1, 4676, 773842, 8763293, 8763293, 773842, 4676, 1, 1, 8772, 3031668, 82498785, 241082026, 82498785, 3031668, 8772, 1, 1, 15333

Offset: 0

Author

Paul D. Hanna, Oct 21 2012

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 );

* A181143: exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^3*y^k] * x^n/n );

* A181144: exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^4*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+17*y+y^2)*x^2 + (1+98*y+98*y^2+y^3)*x^3 + (1+354*y+2251*y^2+354*y^3+y^4)*x^4 +...
The logarithm of the g.f. equals the series:
log(A(x,y)) = (1 + y)*x
+ (1 + 2^5*y + y^2)*x^2/2
+ (1 + 3^5*y + 3^5*y^2 + y^3)*x^3/3
+ (1 + 4^5*y + 6^5*y^2 + 4^5*y^3 + y^4)*x^4/4
+ (1 + 5^5*y + 10^5*y^2 + 10^5*y^3 + 5^5*y^4 + y^5)*x^5/5 +...
Triangle begins:
1;
1, 1;
1, 17, 1;
1, 98, 98, 1;
1, 354, 2251, 354, 1;
1, 979, 23803, 23803, 979, 1;
1, 2275, 158367, 617036, 158367, 2275, 1;
1, 4676, 773842, 8763293, 8763293, 773842, 4676, 1;
1, 8772, 3031668, 82498785, 241082026, 82498785, 3031668, 8772, 1;
1, 15333, 10057620, 575963523, 4066874561, 4066874561, 575963523, 10057620, 15333, 1; ...
Note that column 1 forms the sum of fourth powers (A000538).
		

Crossrefs

Cf. A000538 (column 1), A218117 (row sums).

Cf. variants: A001263 (Narayana), A181143, A181144, A218116.

Programs

PARI

{T(n, k)=polcoeff(polcoeff(exp(sum(m=1, n, sum(j=0, m, binomial(m, j)^5*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(""))