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A294033 Triangle read by rows, expansion of exp(x*z)*z*(tanh(z) + sech(z)), T(n, k) for n >= 1 and 0 <= k <= n-1.

%I A294033 #18 Mar 15 2025 02:36:28
%S A294033 1,2,2,-3,6,3,-8,-12,12,4,25,-40,-30,20,5,96,150,-120,-60,30,6,-427,
%T A294033 672,525,-280,-105,42,7,-2176,-3416,2688,1400,-560,-168,56,8,12465,
%U A294033 -19584,-15372,8064,3150,-1008,-252,72,9,79360,124650,-97920,-51240,20160,6300,-1680,-360,90,10,-555731,872960,685575,-359040,-140910,44352,11550,-2640,-495,110,11
%N A294033 Triangle read by rows, expansion of exp(x*z)*z*(tanh(z) + sech(z)), T(n, k) for n >= 1 and 0 <= k <= n-1.
%F A294033 T(n, k) = (k+1)*binomial(n,k+1)*2^(n-k-1)*(Euler(n-k-1, 1/2) + Euler(n-k-1, 1)) for 0 <= k <= n-2.
%F A294033 T(n, k) is the coefficient of x^k of the polynomial p(n) = n*Sum_{k=1..n} binomial(n-1, k-1)*L(k-1)*x^(n-k) and L(n) = (-1)^binomial(n,2)*A000111(n). In particular n divides T(n, k).
%e A294033 Triangle starts:
%e A294033   [1][   1]
%e A294033   [2][   2,   2]
%e A294033   [3][  -3,   6,    3]
%e A294033   [4][  -8, -12,   12,    4]
%e A294033   [5][  25, -40,  -30,   20,    5]
%e A294033   [6][  96, 150, -120,  -60,   30,  6]
%e A294033   [7][-427, 672,  525, -280, -105, 42, 7]
%p A294033 gf := exp(x*z)*z*(tanh(z)+sech(z)):
%p A294033 s := n -> n!*coeff(series(gf,z,n+2),z,n):
%p A294033 C := n -> PolynomialTools:-CoefficientList(s(n),x):
%p A294033 ListTools:-FlattenOnce([seq(C(n), n=1..7)]);
%p A294033 # Alternatively:
%p A294033 T := (n, k) -> `if`(n = k+1, n,
%p A294033 (k+1)*binomial(n,k+1)*2^(n-k-1)*(euler(n-k-1, 1/2)+euler(n-k-1, 1))):
%p A294033 for n from 1 to 7 do seq(T(n,k), k=0..n-1) od;
%t A294033 L[0] := 1; L[n_] := (-1)^Binomial[n, 2] 2 Abs[PolyLog[-n, -I]];
%t A294033 p[n_] := n Sum[Binomial[n - 1, k - 1] L[k - 1] x^(n - k), {k, 0, n}];
%t A294033 Table[CoefficientList[p[n], x], {n, 1, 11}] // Flatten
%Y A294033 T(n, 0) = signed A065619. Row sums of abs(T(n,k)) = A231179.
%Y A294033 Diagonals A000027, A002378, A027480, A162668.
%Y A294033 A003506 (m=1), this seq. (m=2), A294034 (m=3).
%Y A294033 Cf. A000111, A247453.
%K A294033 sign,tabl
%O A294033 1,2
%A A294033 _Peter Luschny_, Oct 24 2017