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

0,5

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 );

* 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 ).

Links

Table of n, a(n) for n=0..48.

Example

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]; ...

Prog

(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(""))

Crossrefs

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

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

Sequence in context: A176490 A370232 A174158 * A142468 A359313 A304321

Adjacent sequences: A181141 A181142 A181143 * A181145 A181146 A181147

Keyword

nonn,tabl

Author

Paul D. Hanna, Oct 13 2010

Status

approved