OFFSET
1,2
FORMULA
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.
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).
EXAMPLE
Triangle starts:
[1][ 1]
[2][ 2, 2]
[3][ -3, 6, 3]
[4][ -8, -12, 12, 4]
[5][ 25, -40, -30, 20, 5]
[6][ 96, 150, -120, -60, 30, 6]
[7][-427, 672, 525, -280, -105, 42, 7]
MAPLE
gf := exp(x*z)*z*(tanh(z)+sech(z)):
s := n -> n!*coeff(series(gf, z, n+2), z, n):
C := n -> PolynomialTools:-CoefficientList(s(n), x):
ListTools:-FlattenOnce([seq(C(n), n=1..7)]);
# Alternatively:
T := (n, k) -> `if`(n = k+1, n,
(k+1)*binomial(n, k+1)*2^(n-k-1)*(euler(n-k-1, 1/2)+euler(n-k-1, 1))):
for n from 1 to 7 do seq(T(n, k), k=0..n-1) od;
MATHEMATICA
L[0] := 1; L[n_] := (-1)^Binomial[n, 2] 2 Abs[PolyLog[-n, -I]];
p[n_] := n Sum[Binomial[n - 1, k - 1] L[k - 1] x^(n - k), {k, 0, n}];
Table[CoefficientList[p[n], x], {n, 1, 11}] // Flatten
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Peter Luschny, Oct 24 2017
STATUS
approved