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A117436
Triangle related to exp(x)*sec(2*x).
3
1, 0, 1, 4, 0, 1, 0, 12, 0, 1, 80, 0, 24, 0, 1, 0, 400, 0, 40, 0, 1, 3904, 0, 1200, 0, 60, 0, 1, 0, 27328, 0, 2800, 0, 84, 0, 1, 354560, 0, 109312, 0, 5600, 0, 112, 0, 1, 0, 3191040, 0, 327936, 0, 10080, 0, 144, 0, 1, 51733504, 0, 15955200, 0, 819840, 0, 16800, 0, 180, 0, 1
OFFSET
0,4
COMMENTS
Inverse is A117435.
Conjecture: The d-th diagonal (starting with d=0) is proportional to the sequence of generalized binomial coefficients binomial(-x, d) where x is the column index. - Søren G. Have, Feb 26 2017
FORMULA
Number triangle whose k-th column has e.g.f. (x^k/k!)*sec(2*x).
T(n, 0) = A002436(n).
Sum_{k=0..n} T(n, k) = A117437(n).
T(n, k) = binomial(n,k) * (2*i)^(n-k) * E(n-k), where E(n) are the Euler numbers with E(2*n) = A000364(n) and E(2*n+1) = 0. - G. C. Greubel, Jun 01 2021
EXAMPLE
Triangle begins as:
1;
0, 1;
4, 0, 1;
0, 12, 0, 1;
80, 0, 24, 0, 1;
0, 400, 0, 40, 0, 1;
3904, 0, 1200, 0, 60, 0, 1;
0, 27328, 0, 2800, 0, 84, 0, 1;
354560, 0, 109312, 0, 5600, 0, 112, 0, 1;
0, 3191040, 0, 327936, 0, 10080, 0, 144, 0, 1;
51733504, 0, 15955200, 0, 819840, 0, 16800, 0, 180, 0, 1;
MATHEMATICA
T[n_, k_]:= Binomial[n, k]*(2*I)^(n-k)*EulerE[n-k];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Jun 01 2021 *)
PROG
(Sage) flatten([[binomial(n, k)*(2*i)^(n-k)*euler_number(n-k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 01 2021
CROSSREFS
Cf. A000364, A002436 (1st column), A117435 (inverse), A117437 (row sums).
Sequence in context: A117435 A282252 A268367 * A136448 A166318 A166317
KEYWORD
nonn,tabl
AUTHOR
Paul Barry, Mar 16 2006
STATUS
approved