A112339 Triangle read by rows of numbers b_{n,k}, n >= 2, 1 <= k < n such that (1/(1-q*t))*Product_{n,k} 1/(1 - q^n*t^k)^b_{n,k} = Sum_{i,j>=1} S_{i,j} q^i*t^j where S_{i,j} are entries in the table A008277 (the inverse Euler transformation of the table of Stirling numbers of the second kind).

1, 1, 2, 1, 5, 3, 1, 13, 16, 4, 1, 28, 67, 34, 5, 1, 60, 249, 229, 65, 6, 1, 123, 853, 1265, 609, 107, 7, 1, 251, 2787, 6325, 4696, 1376, 168, 8, 1, 506, 8840, 29484, 31947, 14068, 2772, 244, 9, 1, 1018, 27503, 131402, 199766, 124859, 36252, 5118, 345, 10

Offset: 2

Author

Mike Zabrocki, Sep 05 2005

Keywords

• nonn
• tabl

Comments

Row sums equal to A085686 second column = A084174 - 1.

Examples

			Triangle begins:
  1;
  1,  2;
  1,  5,  3;
  1, 13, 16,  4;
  ...
		

Crossrefs

Cf. A008277, A085686, A112340.

Programs

Maple

EULERitable:=proc(tbl) local ser,out,i,j,tmp; ser:=1+add(add(q^i*t^j*tbl[i][j], j=1..nops(tbl[i])), i=1..nops(tbl)); out:=[]; for i from 1 to nops(tbl) do tmp:=coeff(ser,q,i); ser:=expand(ser*mul(add((-q^i*t^j)^k*choose(abs(coeff(tmp,t,j)),k),k=0..nops(tbl)/i), j = 1..degree(tmp,t))); ser:=subs({seq(q^j=0,j=nops(tbl)+1..degree(ser,q))},ser); out:=[op(out),[seq(abs(coeff(tmp,t,j)), j=1..degree(tmp,t))]]; end do; out; end: EULERitable([seq([seq(combinat[stirling2](n,k),k=1..n)],n=1..11)]);