A124539 Triangle, read by rows, where row n equals the inverse binomial transform of column n in the rectangular table A124530.

1, 1, 0, 1, 1, 0, 1, 4, 1, 0, 1, 15, 8, 1, 0, 1, 61, 51, 14, 1, 0, 1, 273, 311, 138, 24, 1, 0, 1, 1331, 1901, 1191, 349, 42, 1, 0, 1, 6977, 11838, 9693, 4100, 868, 76, 1, 0, 1, 38872, 75556, 76950, 43257, 13459, 2163, 142, 1, 0, 1, 228089, 495146, 606275, 430517

Offset: 0

Author

Paul D. Hanna, Nov 05 2006

Keywords

• nonn
• tabl

Comments

In table A124530, the g.f. of row n, R_n(y), simultaneously satisfies: R_n(y) = Sum_{k>=0} y^k*R_k(y)^(n*k) for n>=0.

Examples

			Triangle begins:
1;
1, 0;
1, 1, 0;
1, 4, 1, 0;
1, 15, 8, 1, 0;
1, 61, 51, 14, 1, 0;
1, 273, 311, 138, 24, 1, 0;
1, 1331, 1901, 1191, 349, 42, 1, 0;
1, 6977, 11838, 9693, 4100, 868, 76, 1, 0;
1, 38872, 75556, 76950, 43257, 13459, 2163, 142, 1, 0;
1, 228089, 495146, 606275, 430517, 180000, 43274, 5442, 272, 1, 0; ...
		

Crossrefs

Cf. A124530 (table).

Programs

PARI

T(n,k)=local(m=max(n,k),R);R=vector(m+1,r,vector(m+1,c,if(r==1 || c<=2,1,r^(c-2)))); for(i=0,m, for(r=0,m, R[r+1]=Vec(sum(c=0,m, x^c*Ser(R[c+1])^(r*c)+O(x^(m+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 + 2n.