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A046527 A triangle related to A000108 (Catalan) and A000302 (powers of 4).

%I A046527 #9 Jul 29 2024 06:22:40
%S A046527 1,1,1,2,5,1,5,22,9,1,14,93,58,13,1,42,386,325,110,17,1,132,1586,1686,
%T A046527 765,178,21,1,429,6476,8330,4746,1477,262,25,1,1430,26333,39796,27314,
%U A046527 10654,2525,362,29,1,4862,106762,185517,149052,69930,20754,3973,478,33,1
%N A046527 A triangle related to A000108 (Catalan) and A000302 (powers of 4).
%H A046527 G. C. Greubel, <a href="/A046527/b046527.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A046527 T(n, k) = binomial(n, k-1)*( 4^(n-k+1) - binomial(2*n, n)/binomial(2*(k-1), k-1) )/2, for n >= k >= 0, with T(n, 0) = A000108(n).
%F A046527 G.f. for column k: c(x)*(x/(1-4*x))^m, where c(x) = g.f. for Catalan numbers (A000108).
%e A046527 Triangle begins as:
%e A046527      1;
%e A046527      1,      1;
%e A046527      2,      5,      1;
%e A046527      5,     22,      9,      1;
%e A046527     14,     93,     58,     13,     1;
%e A046527     42,    386,    325,    110,    17,     1;
%e A046527    132,   1586,   1686,    765,   178,    21,    1;
%e A046527    429,   6476,   8330,   4746,  1477,   262,   25,   1;
%e A046527   1430,  26333,  39796,  27314, 10654,  2525,  362,  29,  1;
%e A046527   4862, 106762, 185517, 149052, 69930, 20754, 3973, 478, 33,  1;
%t A046527 T[n_, k_]:= If[k==0, CatalanNumber[n], (1/2)*Binomial[n,k-1]*(4^(n-k+ 1) -Binomial[2*n,n]/Binomial[2*(k-1),k-1])];
%t A046527 Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jul 28 2024 *)
%o A046527 (Magma)
%o A046527 A046527:= func< n,k | k eq 0 select Catalan(n) else (1/2)*Binomial(n, k-1)*(4^(n-k+1) - Binomial(2*n, n)/(k*Catalan(k-1))) >;
%o A046527 [A046527(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jul 28 2024
%o A046527 (SageMath)
%o A046527 def A046527(n,k):
%o A046527     if k==0: return catalan_number(n)
%o A046527     else: return (1/2)*binomial(n, k-1)*(4^(n-k+1) - binomial(2*n, n)/binomial(2*(k-1), k-1))
%o A046527 flatten([[A046527(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Jul 28 2024
%Y A046527 Column sequences are: A000108 (k=0), A000346 (k=1), A018218 (k=2), A042941 (k=3), A042985 (k=4), A045505 (k=5), A045622 (k=6).
%Y A046527 Row sums: A046814.
%K A046527 easy,nonn,tabl
%O A046527 0,4
%A A046527 _Wolfdieter Lang_