A180960 A generalized Narayana triangle for sec(x)^2.

1, 1, 1, 1, 4, 1, 1, 9, 9, 1, 1, 16, 46, 16, 1, 1, 25, 150, 150, 25, 1, 1, 36, 375, 952, 375, 36, 1, 1, 49, 791, 4039, 4039, 791, 49, 1, 1, 64, 1484, 12992, 31078, 12992, 1484, 64, 1, 1, 81, 2556, 34524, 162774, 162774, 34524, 2556, 81, 1, 1, 100, 4125, 79920, 641250, 1484504, 641250, 79920, 4125, 100, 1

Offset

0,5

Links

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

Formula

G.f.: 1/(1 - x - x*y - 1*2*x^2y/(1 - x - x*y - 2*3*x^2y/(1 - x - x*y - 3*4*x^2y/(1 - ...(continued fraction).

E.g.f.: exp((1+y)*x) * sec^2(sqrt(y)*x).

T(n,k) = Sum_{j=0..n} C(n,j)*C(n-j,2(k-j))*A000182(|k-j| + 1).

Example

Triangle begins as:
  1;
  1,   1;
  1,   4,    1;
  1,   9,    9,     1;
  1,  16,   46,    16,      1;
  1,  25,  150,   150,     25,       1;
  1,  36,  375,   952,    375,      36,      1;
  1,  49,  791,  4039,   4039,     791,     49,     1;
  1,  64, 1484, 12992,  31078,   12992,   1484,    64,    1;
  1,  81, 2556, 34524, 162774,  162774,  34524,  2556,   81,   1;
  1, 100, 4125, 79920, 641250, 1484504, 641250, 79920, 4125, 100, 1;

Mathematica

A000182[n_]:= 4^n*(4^n -1)*Abs[BernoulliB[2*n]]/(2*n);
T[n_, k_]:= Sum[Binomial[n, j]*Binomial[n-j, 2*(k-j)]*A000182[Abs[k-j]+1], {j, 0, n}];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Apr 06 2021 *)

Prog

(Magma)
A000182:= func< n | 4^n*(4^n -1)*Abs(Bernoulli(2*n))/(2*n) >;
A180960:= func< n, k | (&+[ Binomial(n, j)*Binomial(n-j, 2*(k-j))*A000182(Abs(k-j) +1): j in [0..n]]) >;
[A180960(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 06 2021
(Sage)
def A000182(n): return 4^n*(4^n -1)*abs(bernoulli(2*n))/(2*n)
def A180960(n, k): return sum( binomial(n, j)*binomial(n-j, 2*(k-j))*A000182(abs(k-j) +1) for j in (0..n))
flatten([[A180960(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 06 2021

Crossrefs

Cf. A000182, A180959.

Sequence in context: A124216 A008459 A259333 * A157192 A154982 A347972

Adjacent sequences: A180957 A180958 A180959 * A180961 A180962 A180963

Keyword

easy,nonn,tabl

Author

Paul Barry, Sep 28 2010

Extensions

Name clarified by G. C. Greubel, Apr 06 2021

Status

approved