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
Examples
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;
Crossrefs
Programs
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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);
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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 *) -
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
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