A124469 Triangle, read by rows, where row n equals the inverse binomial transform of column n in the rectangular table A124460.

1, 1, 1, 1, 3, 1, 1, 8, 6, 1, 1, 22, 28, 11, 1, 1, 65, 120, 81, 20, 1, 1, 209, 500, 494, 219, 37, 1, 1, 730, 2088, 2733, 1812, 578, 70, 1, 1, 2743, 8884, 14411, 12904, 6299, 1518, 135, 1, 1, 10958, 38803, 74484, 84424, 56590, 21384, 4007, 264, 1, 1, 46057, 174366

Offset: 0

Author

Paul D. Hanna, Nov 03 2006

Keywords

• nonn
• tabl

Comments

In table A124460, the g.f. of row n, R_n(y), simultaneously satisfies: R_n(y) = Sum_{k>=0} y^k * R_k(y)^n for n>=0.

Examples

			Triangle begins:
1;
1, 1;
1, 3, 1;
1, 8, 6, 1;
1, 22, 28, 11, 1;
1, 65, 120, 81, 20, 1;
1, 209, 500, 494, 219, 37, 1;
1, 730, 2088, 2733, 1812, 578, 70, 1;
1, 2743, 8884, 14411, 12904, 6299, 1518, 135, 1;
1, 10958, 38803, 74484, 84424, 56590, 21384, 4007, 264, 1;
1, 46057, 174366, 383391, 526121, 453082, 238853, 72076, 10693, 521, 1;
		

Crossrefs

Cf. A124470 (row sums), A006127 (diagonal T(n+1, n)); A124460 (table).

Programs

PARI

{T(n,k)=local(R=vector(n+2,r,vector(n+2,c,binomial(r+c-2,c-1)))); for(i=0,n,for(r=0,n,R[r+1]=Vec(sum(c=0,n,x^c*Ser(R[c+1])^r+O(x^(n+1)))))); Vec(subst(Ser(vector(n+1,j,R[j][n+1])),x,x/(1+x))/(1+x))[k+1]}

Formula

Secondary diagonal T(n+1,n) = 2^n + n = A006127(n).