A218116 G.f.: A(x,y) = exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^6 * 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, 33, 1, 1, 276, 276, 1, 1, 1300, 12695, 1300, 1, 1, 4425, 221495, 221495, 4425, 1, 1, 12201, 2185350, 11534720, 2185350, 12201, 1, 1, 29008, 14794261, 285715550, 285715550, 14794261, 29008, 1, 1, 61776, 76579851, 4276969276, 15781532964

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 );

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

Links

Table of n, a(n) for n=0..40.

Example

G.f.: A(x,y) = 1 + (1+y)*x + (1+33*y+y^2)*x^2 + (1+276*y+276*y^2+y^3)*x^3 + (1+1300*y+12695*y^2+1300*y^3+y^4)*x^4 +...
The logarithm of the g.f. equals the series:
log(A(x,y)) = (1 + y)*x
+ (1 + 2^6*y + y^2)*x^2/2
+ (1 + 3^6*y + 3^6*y^2 + y^3)*x^3/3
+ (1 + 4^6*y + 6^6*y^2 + 4^6*y^3 + y^4)*x^4/4
+ (1 + 5^6*y + 10^6*y^2 + 10^6*y^3 + 5^6*y^4 + y^5)*x^5/5 +...
Triangle begins:
1;
1, 1;
1, 33, 1;
1, 276, 276, 1;
1, 1300, 12695, 1300, 1;
1, 4425, 221495, 221495, 4425, 1;
1, 12201, 2185350, 11534720, 2185350, 12201, 1;
1, 29008, 14794261, 285715550, 285715550, 14794261, 29008, 1;
1, 61776, 76579851, 4276969276, 15781532964, 4276969276, 76579851, 61776, 1;
1, 120825, 324104715, 44480357175, 478591541712, 478591541712, 44480357175, 324104715, 120825, 1; ...
Note that column 1 forms the sum of fifth powers (A000539).

Prog

(PARI) {T(n, k)=polcoeff(polcoeff(exp(sum(m=1, n, sum(j=0, m, binomial(m, j)^6*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. A000539 (column 1), A218119 (row sums).

Cf. variants: A001263 (Narayana), A181143, A181144, A218115.

Sequence in context: A123238 A372394 A225347 * A279364 A308408 A263169

Adjacent sequences: A218113 A218114 A218115 * A218117 A218118 A218119

Keyword

nonn,tabl

Author

Paul D. Hanna, Oct 21 2012

Status

approved