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

0,5

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 ).

Links

Table of n, a(n) for n=0..46.

Example

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).

Prog

(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(""))

Crossrefs

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

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

Sequence in context: A103637 A229956 A157274 * A144442 A157151 A176794

Adjacent sequences: A218112 A218113 A218114 * A218116 A218117 A218118

Keyword

nonn,tabl

Author

Paul D. Hanna, Oct 21 2012

Status

approved