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A371637
Triangle read by rows: T(n, k) = (-8)^k*binomial(2*n, 2*k)*Euler(2*k, 1/2).
4
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
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).
Sequence in context: A164826 A055392 A045873 * A265022 A110060 A061081
KEYWORD
nonn
AUTHOR
Peter Luschny, Apr 02 2024
STATUS
approved