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A181143 G.f.: A(x,y) = exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^3*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. 8
1, 1, 1, 1, 5, 1, 1, 14, 14, 1, 1, 30, 85, 30, 1, 1, 55, 337, 337, 55, 1, 1, 91, 1029, 2230, 1029, 91, 1, 1, 140, 2632, 10549, 10549, 2632, 140, 1, 1, 204, 5922, 39533, 73157, 39533, 5922, 204, 1, 1, 285, 12090, 124805, 384948, 384948, 124805, 12090, 285, 1, 1 (list; table; graph; refs; listen; history; text; internal format)
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 );

* 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 );

* 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..55.

EXAMPLE

G.f.: A(x,y) = 1 + (1+y)*x + (1+5*y+y^2)*x^2 + (1+14*y+14*y^2+y^3)*x^3 + (1+30*y+85*y^2+30*y^3+y^4)*x^4 +...

The logarithm of the g.f. equals the series:

log(A(x,y)) = (1 + y)*x

+ (1 + 2^3*y + y^2)*x^2/2

+ (1 + 3^3*y + 3^3*y^2 + y^3)*x^3/3

+ (1 + 4^3*y + 6^3*y^2 + 4^3*y^3 + y^4)*x^4/4

+ (1 + 5^3*y + 10^3*y^2 + 10^3*y^3 + 5^3*y^4 + y^5)*x^5/5 +...

Triangle begins:

1;

1, 1;

1, 5, 1;

1, 14, 14, 1;

1, 30, 85, 30, 1;

1, 55, 337, 337, 55, 1;

1, 91, 1029, 2230, 1029, 91, 1;

1, 140, 2632, 10549, 10549, 2632, 140, 1;

1, 204, 5922, 39533, 73157, 39533, 5922, 204, 1;

1, 285, 12090, 124805, 384948, 384948, 124805, 12090, 285, 1;

1, 385, 22869, 345389, 1648478, 2748240, 1648478, 345389, 22869, 385, 1;

1, 506, 40678, 861080, 6016297, 15525056, 15525056, 6016297, 861080, 40678, 506, 1; ...

Note that column 1 forms the sum of squares (A000330).

Inverse binomial transform of columns begins:

[1];

[1, 4, 5, 2];

[1, 13, 58, 123, 136, 76, 17];

[1, 29, 278, 1308, 3532, 5867, 6118, 3914, 1407, 218];

[1, 54, 920, 7626, 36916, 114637, 240271, 348354, 350881, 241531, 108551, 28742, 3404]; ...

the g.f. of the rightmost coefficients of which form the g.f. exp( Sum_{n>=1} (3*n)!/(3*n!^3) * x^n/n ), and yield the self-convolution of A229452.

PROG

(PARI) {T(n, k)=polcoeff(polcoeff(exp(sum(m=1, n, sum(j=0, m, binomial(m, j)^3*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. A000330 (column 1), A166990 (row sums), A166896 (antidiagonal sums), A218139.

Cf. variants: A001263 (Narayana), A181144, A218115, A218116.

Sequence in context: A239279 A278880 A111910 * A144438 A157207 A008957

Adjacent sequences:  A181140 A181141 A181142 * A181144 A181145 A181146

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Oct 13 2010

STATUS

approved

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Last modified February 21 22:56 EST 2018. Contains 299427 sequences. (Running on oeis4.)