cp's OEIS Frontend

A351429 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. 1/(1 + f^k(x)), where f(x) = exp(x) - 1.

%I A351429 #20 May 12 2023 12:40:58
%S A351429 1,1,-1,1,-1,2,1,-1,1,-6,1,-1,0,-1,24,1,-1,-1,1,1,-120,1,-1,-2,0,1,-1,
%T A351429 720,1,-1,-3,-4,6,-2,1,-5040,1,-1,-4,-11,-2,32,-9,-1,40320,1,-1,-5,
%U A351429 -21,-41,76,115,-9,1,-362880,1,-1,-6,-34,-129,-75,953,172,50,-1,3628800
%N A351429 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. 1/(1 + f^k(x)), where f(x) = exp(x) - 1.
%F A351429 T(n,k) = Sum_{j=0..n} Stirling2(n,j) * T(j,k-1), k>1, T(n,0) = (-1)^n * n!.
%e A351429 Square array begins:
%e A351429      1,  1,  1,   1,   1,    1,     1, ...
%e A351429     -1, -1, -1,  -1,  -1,   -1,    -1, ...
%e A351429      2,  1,  0,  -1,  -2,   -3,    -4, ...
%e A351429     -6, -1,  1,   0,  -4,  -11,   -21, ...
%e A351429     24,  1,  1,   6,  -2,  -41,  -129, ...
%e A351429   -120, -1, -2,  32,  76,  -75,  -806, ...
%e A351429    720,  1, -9, 115, 953, 1540, -3334, ...
%p A351429 A:= (n, k)-> n!*(g->coeff(series(1/(1+(g@@k)(x)), x, n+1), x, n))(x->exp(x)-1):
%p A351429 seq(seq(A(n, d-n), n=0..d), d=0..10);  # _Alois P. Heinz_, Feb 11 2022
%t A351429 T[n_, 0] := (-1)^n*n!; T[n_, k_] := T[n, k] = Sum[StirlingS2[n, j]*T[j, k - 1], {j, 0, n}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Amiram Eldar_, Feb 11 2022 *)
%o A351429 (PARI) T(n, k) = if(k==0, (-1)^n*n!, sum(j=0, n, stirling(n, j, 2)*T(j, k-1)));
%Y A351429 Columns k=0..5 give A133942, A033999, A000587, A130410, A351427, A351428.
%Y A351429 Main diagonal gives A351433.
%Y A351429 Cf. A111933, A351420.
%K A351429 sign,tabl
%O A351429 0,6
%A A351429 _Seiichi Manyama_, Feb 11 2022