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A096078 Triangle read by rows: T(n,k) = (k+1)*T(n-1,k) + (n-k+1)*T(n,k-1).

%I A096078 #14 Jun 05 2025 20:38:22
%S A096078 1,1,1,1,4,4,1,11,34,34,1,26,180,496,496,1,57,768,4288,11056,11056,1,
%T A096078 120,2904,28768,141584,349504,349504,1,247,10194,166042,1372088,
%U A096078 6213288,14873104,14873104,1,502,34096,868744,11204160,82096368,350400832
%N A096078 Triangle read by rows: T(n,k) = (k+1)*T(n-1,k) + (n-k+1)*T(n,k-1).
%H A096078 Paul Barry, <a href="https://arxiv.org/abs/1803.06408">Three Études on a sequence transformation pipeline</a>, arXiv:1803.06408 [math.CO], 2018.
%F A096078 T(n-1, 1) given by Eulerian numbers, 2^n - n - 1 (A000295). T(n-1, n-1) given by 2^n*(2^{2n} - 1)*|B_{2n}|/n, B_n = Bernoulli numbers (A002105).
%e A096078 Table begins:
%e A096078   1
%e A096078   1 1
%e A096078   1 4 4
%e A096078   1 11 34 34
%e A096078   1 26 180 496 496
%e A096078   1 57 768 4288 11056 11056
%t A096078 T[n_, 0] := 1; T[n_, 1] := 2^(n+1) - n - 2; T[n_, n_] := 2^(n+1)*(2^(2n+2) - 1)*Abs[ BernoulliB[2n + 2]]/ (n + 1); T[n_, k_] := (k + 1)T[n - 1, k] + (n - k + 1)T[n, k - 1]; Flatten[ Table[ T[n, k], {n, 0, 8}, {k, 0, n}]] (* _Robert G. Wilson v_, Jul 23 2004 *)
%Y A096078 Cf. A000295, A002105.
%K A096078 easy,nonn,tabl
%O A096078 0,5
%A A096078 _Paul Boddington_, Jul 22 2004
%E A096078 Edited and extended by _Robert G. Wilson v_, Jul 23 2004