A181144 G.f.: A(x,y) = exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^4*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, 9, 1, 1, 36, 36, 1, 1, 100, 419, 100, 1, 1, 225, 2699, 2699, 225, 1, 1, 441, 12138, 35052, 12138, 441, 1, 1, 784, 42865, 286206, 286206, 42865, 784, 1, 1, 1296, 127191, 1696820, 3932898, 1696820, 127191, 1296, 1, 1, 2025, 330903, 7958563

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 );

* A181143: exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^3*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+9*y+y^2)*x^2 + (1+36*y+36*y^2+y^3)*x^3 + (1+100*y+419*y^2+100*y^3+y^4)*x^4 +...
The logarithm of the g.f. equals the series:
log(A(x,y)) = (1 + y)*x
+ (1 + 2^4*y + y^2)*x^2/2
+ (1 + 3^4*y + 3^4*y^2 + y^3)*x^3/3
+ (1 + 4^4*y + 6^4*y^2 + 4^4*y^3 + y^4)*x^4/4
+ (1 + 5^4*y + 10^4*y^2 + 10^4*y^3 + 5^4*y^4 + y^5)*x^5/5 +...
Triangle begins:
1;
1, 1;
1, 9, 1;
1, 36, 36, 1;
1, 100, 419, 100, 1;
1, 225, 2699, 2699, 225, 1;
1, 441, 12138, 35052, 12138, 441, 1;
1, 784, 42865, 286206, 286206, 42865, 784, 1;
1, 1296, 127191, 1696820, 3932898, 1696820, 127191, 1296, 1;
1, 2025, 330903, 7958563, 36955542, 36955542, 7958563, 330903, 2025, 1;
1, 3025, 776688, 31205941, 261852055, 525079969, 261852055, 31205941, 776688, 3025, 1; ...
Note that column 1 forms the sum of cubes (A000537), and forms the squares of the triangular numbers.
Inverse binomial transform of columns begins:
[1];
[1, 8, 19, 18, 6];
[1, 35, 348, 1549, 3713, 5154, 4161, 1818, 333];
[1, 99, 2500, 27254, 161793, 589819, 1409579, 2282850, 2529900, 1893972, 917349, 259854, 32726]; ...
		

Crossrefs

Cf. A000537 (column 1), A166992 (row sums), A166898 (antidiagonal sums), A218140.

Cf. variants: A001263 (Narayana), A181143, A218115, A218116.

Programs

PARI

{T(n,k)=polcoeff(polcoeff(exp(sum(m=1,n,sum(j=0,m,binomial(m,j)^4*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(""))