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A382100 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of 1/(2 - B_k(x)), where B_k(x) = 1 + x*B_k(x)^k.

%I A382100 #24 Mar 16 2025 12:20:09
%S A382100 1,1,1,1,1,1,1,1,2,1,1,1,3,4,1,1,1,4,10,8,1,1,1,5,19,35,16,1,1,1,6,31,
%T A382100 98,126,32,1,1,1,7,46,213,531,462,64,1,1,1,8,64,396,1556,2974,1716,
%U A382100 128,1,1,1,9,85,663,3651,11843,17060,6435,256,1
%N A382100 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of 1/(2 - B_k(x)), where B_k(x) = 1 + x*B_k(x)^k.
%F A382100 A(n,k) = ( binomial(k*n,n) - (k-2) * Sum_{j=0..n-1} binomial(k*n,j) )/2 for n > 0.
%F A382100 G.f. of column k: 1/( 1 - Series_Reversion( x/(1+x)^k ) ).
%e A382100 Square array begins:
%e A382100   1,  1,   1,    1,     1,     1,     1, ...
%e A382100   1,  1,   1,    1,     1,     1,     1, ...
%e A382100   1,  2,   3,    4,     5,     6,     7, ...
%e A382100   1,  4,  10,   19,    31,    46,    64, ...
%e A382100   1,  8,  35,   98,   213,   396,   663, ...
%e A382100   1, 16, 126,  531,  1556,  3651,  7391, ...
%e A382100   1, 32, 462, 2974, 11843, 35232, 86488, ...
%o A382100 (PARI) a(n, k) = polcoef(1/(2-sum(j=0, n, binomial(k*j+1, j)/(k*j+1)*x^j+x*O(x^n))), n);
%o A382100 (PARI) a(n, k) = if(n==0, 1, (binomial(k*n, n)-(k-2)*sum(j=0, n-1, binomial(k*n, j)))/2);
%Y A382100 Columns k=0..5 give A000012, A011782, A088218, A047099 (for n > 0), A107026(n)/2 (for n > 0), A304979(n)/2 (for n > 0).
%Y A382100 Cf. A107027, A107030, A355262, A382101.
%K A382100 nonn,tabl
%O A382100 0,9
%A A382100 _Seiichi Manyama_, Mar 15 2025