A326724 Triangle with Euler (secant) numbers, read by rows, T(n, k) for 0 <= k <= n.

1, -1, 1, 5, -10, 5, -61, 183, -183, 61, 1385, -5540, 8310, -5540, 1385, -50521, 252605, -505210, 505210, -252605, 50521, 2702765, -16216590, 40541475, -54055300, 40541475, -16216590, 2702765, -199360981, 1395526867, -4186580601, 6977634335, -6977634335, 4186580601, -1395526867, 199360981

Offset

0,4

Links

Table of n, a(n) for n=0..35.

Formula

T(n, k) = (2*n)! [x^k] [y^(2*n)] sec(y*sqrt(x - 1)).

Sum_{k=0..n} (-1)^(n-k)*T(n, k) = |A012816(n+1)|.

Example

Triangle starts:
[0]       1;
[1]      -1,         1;
[2]       5,       -10,        5;
[3]     -61,       183,     -183,        61;
[4]    1385,     -5540,     8310,     -5540,     1385;
[5]  -50521,    252605,  -505210,    505210,  -252605,     50521;
[6] 2702765, -16216590, 40541475, -54055300, 40541475, -16216590, 2702765;

Mathematica

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

Crossrefs

Cf. A000364, A012816, A326722.

Sequence in context: A365479 A374537 A360560 * A103697 A185341 A067843

Adjacent sequences: A326721 A326722 A326723 * A326725 A326726 A326727

Keyword

sign,tabl

Author

Peter Luschny, Aug 06 2019

Status

approved