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

%I A371637 #27 Feb 19 2025 11:56:34
%S A371637 1,1,2,1,12,20,1,30,300,488,1,56,1400,13664,22160,1,90,4200,102480,
%T A371637 997200,1616672,1,132,9900,450912,10969200,106700352,172976960,1,182,
%U A371637 20020,1465464,66546480,1618288672,15740903360,25518205568
%N A371637 Triangle read by rows: T(n, k) = (-8)^k*binomial(2*n, 2*k)*Euler(2*k, 1/2).
%F A371637 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
%F A371637 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
%e A371637 Triangle starts:
%e A371637 [0] 1;
%e A371637 [1] 1,   2;
%e A371637 [2] 1,  12,    20;
%e A371637 [3] 1,  30,   300,     488;
%e A371637 [4] 1,  56,  1400,   13664,    22160;
%e A371637 [5] 1,  90,  4200,  102480,   997200,    1616672;
%e A371637 [6] 1, 132,  9900,  450912, 10969200,  106700352,   172976960;
%e A371637 [7] 1, 182, 20020, 1465464, 66546480, 1618288672, 15740903360, 25518205568;
%p A371637 T := (n, k) -> (-8)^k*binomial(2*n, 2*k)*euler(2*k, 1/2):
%p A371637 seq(print(seq(T(n, k), k = 0..n)), n = 0..7);
%t A371637 Table[(-8)^k*Binomial[2*n, 2*k]*EulerE[2*k, 1/2], {n, 0, 10}, {k, 0, n}] (* _Paolo Xausa_, Apr 17 2024 *)
%o A371637 (SageMath)
%o A371637 def DelehamDelta(R, S, dim):
%o A371637     ring = PolynomialRing(ZZ, 'x')
%o A371637     x = ring.gen()
%o A371637     A = [R(k) + x * S(k) for k in range(dim)]
%o A371637     C = [ring(0)] + [ring(1) for i in range(dim)]
%o A371637     for k in range(1, dim + 1):
%o A371637         for n in range(k - 1, 0, -1):
%o A371637             C[n] = C[n-1] + C[n+1] * A[n-1]
%o A371637         yield list(C[1])
%o A371637 def A371637_triangle(dim):
%o A371637     a = lambda n: 1 - n % 2
%o A371637     b = lambda n: 2*(n + 1)^2
%o A371637     for row in DelehamDelta(a, b, dim): print(row)
%o A371637 A371637_triangle(8)  # _Peter Luschny_, Apr 21 2024
%Y A371637 Cf. A001105, A002939 (column 1), A012816 (main diagonal), A371683 (row sums), A371684 (alternating row sums).
%Y A371637 Cf. A081658, A084938, A086645, A000079, A000364.
%K A371637 nonn,tabl
%O A371637 0,3
%A A371637 _Peter Luschny_, Apr 02 2024