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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. 1
1, 2, 2, -3, 6, 3, -8, -12, 12, 4, 25, -40, -30, 20, 5, 96, 150, -120, -60, 30, 6, -427, 672, 525, -280, -105, 42, 7, -2176, -3416, 2688, 1400, -560, -168, 56, 8, 12465, -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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

Table of n, a(n) for n=1..66.

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

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

Diagonals A000027, A002378, A027480, A162668.

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

Cf. A000111, A247453.

Sequence in context: A209420 A317449 A222310 * A254827 A193862 A258631

Adjacent sequences:  A294030 A294031 A294032 * A294034 A294035 A294036

KEYWORD

sign,tabl

AUTHOR

Peter Luschny, Oct 24 2017

STATUS

approved

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Last modified November 22 02:58 EST 2019. Contains 329383 sequences. (Running on oeis4.)