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%I #13 Apr 09 2021 09:17:43
%S 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,
%T 36,1,1,49,791,4039,4039,791,49,1,1,64,1484,12992,31078,12992,1484,64,
%U 1,1,81,2556,34524,162774,162774,34524,2556,81,1,1,100,4125,79920,641250,1484504,641250,79920,4125,100,1
%N A generalized Narayana triangle for sec(x)^2.
%H G. C. Greubel, <a href="/A180960/b180960.txt">Rows n = 0..50 of the triangle, flattened</a>
%F 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).
%F E.g.f.: exp((1+y)*x) * sec^2(sqrt(y)*x).
%F T(n,k) = Sum_{j=0..n} C(n,j)*C(n-j,2(k-j))*A000182(|k-j| + 1).
%e Triangle begins as:
%e 1;
%e 1, 1;
%e 1, 4, 1;
%e 1, 9, 9, 1;
%e 1, 16, 46, 16, 1;
%e 1, 25, 150, 150, 25, 1;
%e 1, 36, 375, 952, 375, 36, 1;
%e 1, 49, 791, 4039, 4039, 791, 49, 1;
%e 1, 64, 1484, 12992, 31078, 12992, 1484, 64, 1;
%e 1, 81, 2556, 34524, 162774, 162774, 34524, 2556, 81, 1;
%e 1, 100, 4125, 79920, 641250, 1484504, 641250, 79920, 4125, 100, 1;
%t A000182[n_]:= 4^n*(4^n -1)*Abs[BernoulliB[2*n]]/(2*n);
%t T[n_, k_]:= Sum[Binomial[n, j]*Binomial[n-j, 2*(k-j)]*A000182[Abs[k-j]+1], {j,0,n}];
%t Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Apr 06 2021 *)
%o (Magma)
%o A000182:= func< n | 4^n*(4^n -1)*Abs(Bernoulli(2*n))/(2*n) >;
%o A180960:= func< n,k | (&+[ Binomial(n,j)*Binomial(n-j, 2*(k-j))*A000182(Abs(k-j) +1): j in [0..n]]) >;
%o [A180960(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Apr 06 2021
%o (Sage)
%o def A000182(n): return 4^n*(4^n -1)*abs(bernoulli(2*n))/(2*n)
%o 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))
%o flatten([[A180960(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Apr 06 2021
%Y Cf. A000182, A180959.
%K easy,nonn,tabl
%O 0,5
%A _Paul Barry_, Sep 28 2010
%E Name clarified by _G. C. Greubel_, Apr 06 2021
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