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
KEYWORD
AUTHOR
Paul Barry, Sep 28 2010
EXTENSIONS
Name clarified by G. C. Greubel, Apr 06 2021
STATUS
approved