A180959 Generalized Narayana triangle for secant.

1, 1, 1, 1, 3, 1, 1, 6, 6, 1, 1, 10, 23, 10, 1, 1, 15, 65, 65, 15, 1, 1, 21, 150, 321, 150, 21, 1, 1, 28, 301, 1197, 1197, 301, 28, 1, 1, 36, 546, 3584, 7531, 3584, 546, 36, 1, 1, 45, 918, 9114, 35523, 35523, 9114, 918, 45, 1, 1, 55, 1455, 20490, 132045, 276433, 132045, 20490, 1455, 55, 1

Offset

0,5

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Formula

G.f.: 1/(1 -x -x*y -x^2*y/(1 -x -x*y -4*x^2*y/(1 -x -x*y -9*x^2*y/(1 - ... (continued fraction).

E.g.f.: exp((1+y)*x) * sec(sqrt(y)*x).

T(n,k) = Sum_{j=0..n} C(n,j)*C(n-j,2(k-j))*E_(k-j), E_n = A000364(n).

Example

Triangle begins
  1;
  1,  1;
  1,  3,    1;
  1,  6,    6,     1;
  1, 10,   23,    10,      1;
  1, 15,   65,    65,     15,      1;
  1, 21,  150,   321,    150,     21,      1;
  1, 28,  301,  1197,   1197,    301,     28,     1;
  1, 36,  546,  3584,   7531,   3584,    546,    36,    1;
  1, 45,  918,  9114,  35523,  35523,   9114,   918,   45,  1;
  1, 55, 1455, 20490, 132045, 276433, 132045, 20490, 1455, 55, 1;

Mathematica

T[n_, k_]:= Sum[Binomial[n, j]*Binomial[n-j, 2*(k-j)]*(-1)^(k-j)*EulerE[2*Abs[k-j]], {j, 0, n}];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Apr 06 2021 *)

Prog

(Magma)
A000364:= func< n | (4^(n+1)/(2*n+2))*( Evaluate(BernoulliPolynomial(n+1), 3/4) - Evaluate(BernoulliPolynomial(n+1), 1/4) ) >;
A180959:= func< n, k | (&+[ Binomial(n, j)*Binomial(n-j, 2*(k-j))*Abs(A000364(2*Abs(k-j))): j in [0..n]]) >;
[A180959(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 06 2021
(Sage)
def A180959(n, k): return sum( binomial(n, j)*binomial(n-j, 2*(k-j))*abs(euler_number(2*abs(k-j))) for j in (0..n))
flatten([[A180959(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 06 2021

Crossrefs

Cf. A000364.

Sequence in context: A088925 A100862 A098568 * A131235 A202812 A157243

Adjacent sequences: A180956 A180957 A180958 * A180960 A180961 A180962

Keyword

easy,nonn,tabl

Author

Paul Barry, Sep 28 2010

Status

approved