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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.
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%I #16 Mar 22 2021 03:21:45

%S 0,2,6,9,44,60,35,234,564,504,135,1144,3816,6112,4080,527,5430,23000,

%T 51120,61360,32736,2079,25332,130500,368480,614160,589632,262080,8255,

%U 116466,709548,2436840,5160400,6880608,5504576,2097024

%N 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.

%C 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.

%D 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.

%F 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.

%e [-1] 0;

%e [ 0] 2, 6;

%e [ 1] 9, 44, 60;

%e [ 2] 35, 234, 564, 504;

%e [ 3] 135, 1144, 3816, 6112, 4080;

%e [ 4] 527, 5430, 23000, 51120, 61360, 32736;

%e [ 5] 2079, 25332, 130500, 368480, 614160, 589632, 262080;

%e [ 6] 8255, 116466, 709548, 2436840, 5160400, 6880608, 5504576, 2097024;

%p CoeffList := p -> op(PolynomialTools:-CoefficientList(p, x)):

%p p := (n, x) -> (1 + 4*x)^(n + 1) + (1 - 2^(-2*n-1))*(2 + 4*x)^(n + 1);

%p c := n -> `if`(n = -1, 0, CoeffList(expand(2^n*p(n, x)))):

%p seq(c(n), n = -1..8);

%p # The Jensen-type integral:

%p JInt := n -> (2*Pi/(n + 1))*int(p(n, I*x)/(exp(-Pi*x) + exp(Pi*x))^2,

%p x = -infinity..infinity): seq(JInt(n), n=0..9);

%t p[n_, z_] := 2^(2 n)(2 - 4^(-n))(1 + 2 z)^(1 + n) + 2^n (1 + 4 z)^(1 + n);

%t c[n_] := If[n == -1, {0}, CoefficientList[p[n, z], z]];

%t Table[c[n], {n, -1, 6}] // Flatten

%Y Cf. A092431, A248217, A163982.

%K nonn,tabl

%O -1,2

%A _Peter Luschny_, Mar 20 2021