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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.
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%I #9 May 24 2014 18:37:11

%S 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,

%T 1029,91,1,1,140,2632,10549,10549,2632,140,1,1,204,5922,39533,73157,

%U 39533,5922,204,1,1,285,12090,124805,384948,384948,124805,12090,285,1,1

%N 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.

%C Compare g.f. to that of the following triangle variants:

%C * Pascal's: exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)*y^k] * x^n/n );

%C * Narayana: exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^2*y^k] * x^n/n );

%C * A181144: exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^4*y^k] * x^n/n );

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

%C * A218116: exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^6*y^k] * x^n/n ).

%e 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 +...

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

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

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

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

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

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

%e Triangle begins:

%e 1;

%e 1, 1;

%e 1, 5, 1;

%e 1, 14, 14, 1;

%e 1, 30, 85, 30, 1;

%e 1, 55, 337, 337, 55, 1;

%e 1, 91, 1029, 2230, 1029, 91, 1;

%e 1, 140, 2632, 10549, 10549, 2632, 140, 1;

%e 1, 204, 5922, 39533, 73157, 39533, 5922, 204, 1;

%e 1, 285, 12090, 124805, 384948, 384948, 124805, 12090, 285, 1;

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

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

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

%e Inverse binomial transform of columns begins:

%e [1];

%e [1, 4, 5, 2];

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

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

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

%e 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.

%o (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)}

%o for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))

%Y Cf. A000330 (column 1), A166990 (row sums), A166896 (antidiagonal sums), A218139.

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

%K nonn,tabl

%O 0,5

%A _Paul D. Hanna_, Oct 13 2010