A371637 Triangle read by rows: T(n, k) = (-8)^k*binomial(2*n, 2*k)*Euler(2*k, 1/2).

1, 1, 2, 1, 12, 20, 1, 30, 300, 488, 1, 56, 1400, 13664, 22160, 1, 90, 4200, 102480, 997200, 1616672, 1, 132, 9900, 450912, 10969200, 106700352, 172976960, 1, 182, 20020, 1465464, 66546480, 1618288672, 15740903360, 25518205568

Offset

0,3

Links

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

Formula

Triangle T(n, k), 0 <= k <=n, read by rows, given by [1, 0, 1, 0, 1, 0, 1, 0, 1, ...] DELTA [2, 8, 18, 32, 50, 72, 98, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Apr 21 2024

T(n, k) = binomial(2*n, 2*k) * 2^k * abs(Euler(2*k)) = A086645(n, k) * A000079(k) * A000364(k). - Philippe Deléham, Apr 23 2024

Example

Triangle starts:
[0] 1;
[1] 1,   2;
[2] 1,  12,    20;
[3] 1,  30,   300,     488;
[4] 1,  56,  1400,   13664,    22160;
[5] 1,  90,  4200,  102480,   997200,    1616672;
[6] 1, 132,  9900,  450912, 10969200,  106700352,   172976960;
[7] 1, 182, 20020, 1465464, 66546480, 1618288672, 15740903360, 25518205568;

Maple

T := (n, k) -> (-8)^k*binomial(2*n, 2*k)*euler(2*k, 1/2):
seq(print(seq(T(n, k), k = 0..n)), n = 0..7);

Mathematica

Table[(-8)^k*Binomial[2*n, 2*k]*EulerE[2*k, 1/2], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Apr 17 2024 *)

Prog

(SageMath)
def DelehamDelta(R, S, dim):
    ring = PolynomialRing(ZZ, 'x')
    x = ring.gen()
    A = [R(k) + x * S(k) for k in range(dim)]
    C = [ring(0)] + [ring(1) for i in range(dim)]
    for k in range(1, dim + 1):
        for n in range(k - 1, 0, -1):
            C[n] = C[n-1] + C[n+1] * A[n-1]
        yield list(C[1])
def A371637_triangle(dim):
    a = lambda n: 1 - n % 2
    b = lambda n: 2*(n + 1)^2
    for row in DelehamDelta(a, b, dim): print(row)
A371637_triangle(8)  # Peter Luschny, Apr 21 2024

Crossrefs

Cf. A001105, A002939 (column 1), A012816 (main diagonal), A371683 (row sums), A371684 (alternating row sums).

Cf. A081658, A084938, A086645, A000079, A000364.

Sequence in context: A164826 A055392 A045873 * A265022 A110060 A061081

Adjacent sequences: A371634 A371635 A371636 * A371638 A371639 A371640

Keyword

nonn

Author

Peter Luschny, Apr 02 2024

Status

approved