A342317 T(n, k) = [x^n] 2^n*P(n, x), where P(n, x) = (1 + 4*x)^(n + 1) + (1 - 2^(-2*n-1))*(2 + 4*x)^(n + 1). Triangle read by rows, T(n, k) for 0 <= k <= n+1 if n >= 0 and by convention T(-1, 0) = 0.

0, 2, 6, 9, 44, 60, 35, 234, 564, 504, 135, 1144, 3816, 6112, 4080, 527, 5430, 23000, 51120, 61360, 32736, 2079, 25332, 130500, 368480, 614160, 589632, 262080, 8255, 116466, 709548, 2436840, 5160400, 6880608, 5504576, 2097024

Offset

-1,2

Comments

The polynomials are the numerators of an integral representing the signed Euler numbers A163982 similar to the integral given by Jensen representing the Bernoulli numbers.

References

J. L. W. V. Jensen, Remarques relatives aux réponses de MM. Franel et Kluyver. L'Intermédiaire des mathématiciens, tome II, Gauthier-Villars et Fils, 346-347, 1895.

Links

Table of n, a(n) for n=-1..34.

Formula

A163982(n) = -(2*Pi/(n + 1)) Integral_{x in R}(P(n, i*x)/(exp(-Pi*z) + exp(Pi*z))^2, where R is the real line.

Example

[-1] 0;
[ 0] 2,    6;
[ 1] 9,    44,     60;
[ 2] 35,   234,    564,    504;
[ 3] 135,  1144,   3816,   6112,    4080;
[ 4] 527,  5430,   23000,  51120,   61360,   32736;
[ 5] 2079, 25332,  130500, 368480,  614160,  589632,  262080;
[ 6] 8255, 116466, 709548, 2436840, 5160400, 6880608, 5504576, 2097024;

Maple

CoeffList := p -> op(PolynomialTools:-CoefficientList(p, x)):
p := (n, x) -> (1 + 4*x)^(n + 1) + (1 - 2^(-2*n-1))*(2 + 4*x)^(n + 1);
c := n -> `if`(n = -1, 0, CoeffList(expand(2^n*p(n, x)))):
seq(c(n), n = -1..8);
# The Jensen-type integral:
JInt := n -> (2*Pi/(n + 1))*int(p(n, I*x)/(exp(-Pi*x) + exp(Pi*x))^2,
x = -infinity..infinity): seq(JInt(n), n=0..9);

Mathematica

p[n_, z_] := 2^(2 n)(2 - 4^(-n))(1 + 2 z)^(1 + n) + 2^n (1 + 4 z)^(1 + n);
c[n_] := If[n == -1, {0}, CoefficientList[p[n, z], z]];
Table[c[n], {n, -1, 6}] // Flatten

Crossrefs

Cf. A092431, A248217, A163982.

Sequence in context: A129831 A101713 A240761 * A117541 A244060 A333185

Adjacent sequences: A342314 A342315 A342316 * A342318 A342319 A342320

Keyword

nonn,tabl

Author

Peter Luschny, Mar 20 2021

Status

approved