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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).

%I A112339 #10 Jan 09 2021 11:53:12
%S A112339 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,
%T A112339 609,107,7,1,251,2787,6325,4696,1376,168,8,1,506,8840,29484,31947,
%U A112339 14068,2772,244,9,1,1018,27503,131402,199766,124859,36252,5118,345,10
%N 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).
%C A112339 Row sums equal to A085686 second column = A084174 - 1.
%e A112339 Triangle begins:
%e A112339   1;
%e A112339   1,  2;
%e A112339   1,  5,  3;
%e A112339   1, 13, 16,  4;
%e A112339   ...
%p A112339 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)]);
%Y A112339 Cf. A008277, A085686, A112340.
%K A112339 nonn,tabl
%O A112339 2,3
%A A112339 _Mike Zabrocki_, Sep 05 2005