A294033 - OEIS

%I #15 Oct 31 2017 17:58:23

%S 1,2,2,-3,6,3,-8,-12,12,4,25,-40,-30,20,5,96,150,-120,-60,30,6,-427,

%T 672,525,-280,-105,42,7,-2176,-3416,2688,1400,-560,-168,56,8,12465,

%U -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 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 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 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 Triangle starts:

%e [1][   1]

%e [2][   2,   2]

%e [3][  -3,   6,    3]

%e [4][  -8, -12,   12,    4]

%e [5][  25, -40,  -30,   20,    5]

%e [6][  96, 150, -120,  -60,   30,  6]

%e [7][-427, 672,  525, -280, -105, 42, 7]

%p gf := exp(x*z)*z*(tanh(z)+sech(z)):

%p s := n -> n!*coeff(series(gf,z,n+2),z,n):

%p C := n -> PolynomialTools:-CoefficientList(s(n),x):

%p ListTools:-FlattenOnce([seq(C(n), n=1..7)]);

%p # Alternatively:

%p T := (n, k) -> `if`(n = k+1, n,

%p (k+1)*binomial(n,k+1)*2^(n-k-1)*(euler(n-k-1, 1/2)+euler(n-k-1, 1))):

%p for n from 1 to 7 do seq(T(n,k), k=0..n-1) od;

%t L[0] := 1; L[n_] := (-1)^Binomial[n, 2] 2 Abs[PolyLog[-n, -I]];

%t p[n_] := n Sum[Binomial[n - 1, k - 1] L[k - 1] x^(n - k), {k, 0, n}];

%t Table[CoefficientList[p[n], x], {n, 1, 11}] // Flatten

%Y T(n, 0) = signed A065619. Row sums of abs(T(n,k)) = A231179.

%Y Diagonals A000027, A002378, A027480, A162668.

%Y A003506 (m=1), this seq. (m=2), A294034 (m=3).

%Y Cf. A000111, A247453.

%K sign,tabl

%O 1,2

%A _Peter Luschny_, Oct 24 2017