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 ).
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
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Oct 21 2012
STATUS
approved