A326723 Triangle read by rows: T(n, k) = (-1)^(n - k) * binomial(n, k) * A000182(n).

0, -1, 1, 2, -4, 2, -16, 48, -48, 16, 272, -1088, 1632, -1088, 272, -7936, 39680, -79360, 79360, -39680, 7936, 353792, -2122752, 5306880, -7075840, 5306880, -2122752, 353792, -22368256, 156577792, -469733376, 782888960, -782888960, 469733376, -156577792, 22368256

Offset: 0

Author

Peter Luschny, Aug 08 2019

Keywords

• sign
• tabl

Examples

			Triangle starts:
[0]      0;
[1]     -1,        1;
[2]      2,       -4,       2;
[3]    -16,       48,     -48,       16;
[4]    272,    -1088,    1632,    -1088,     272;
[5]  -7936,    39680,  -79360,    79360,  -39680,     7936;
[6] 353792, -2122752, 5306880, -7075840, 5306880, -2122752, 353792;
		

Crossrefs

Cf. A000182, A261042, A326722.

Programs

Maple

T := (n, k) -> (-1)^(n - k)*binomial(n, k)*A000182(n):
seq(seq(T(n, k), k = 0..n), n = 0..6);  # Peter Luschny, Apr 23 2024

Mathematica

gf := Sqrt[x - 1] Tan[y Sqrt[x - 1]];
ser := Series[gf, {y, 0, 26}];
cy[n_] := n! Coefficient[ser, y, n];
row[n_] := If[n == 0, 0, CoefficientList[cy[2 n - 1], x]];
Table[row[n], {n, 0, 7}] // Flatten

Formula

T(n, k) = (2*n-1)! [x^k] [y^(2*n-1)] sqrt(x - 1)*tan(y*sqrt(x - 1)) for n > 0.

Sum_{k=0..n} (-1)^(n-k)*T(n, k) = 2*A261042(n-1) for n > 0.

Extensions

Offset set to 0, T(0,0) = 0 and new name by Peter Luschny, Apr 23 2024