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A236843 Triangle read by rows related to the Catalan transform of the Fibonacci numbers.

%I A236843 #41 Nov 05 2024 23:40:42
%S A236843 1,1,1,2,3,1,5,9,4,1,14,28,14,6,1,42,90,48,27,7,1,132,297,165,110,35,
%T A236843 9,1,429,1001,572,429,154,54,10,1,1430,3432,2002,1638,637,273,65,12,1,
%U A236843 4862,11934,7072,6188,2548,1260,350,90,13,1,16796,41990,25194,23256,9996,5508,1700,544,104,15,1
%N A236843 Triangle read by rows related to the Catalan transform of the Fibonacci numbers.
%C A236843 Row sums are A109262(n+1).
%H A236843 Andrew Howroyd, <a href="/A236843/b236843.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50)
%F A236843 G.f. for the column k (with zeros omitted): C(x)^A032766(k+1) where C(x) is g.f. for Catalan numbers (A000108).
%F A236843 Sum_{k=0..n} T(n,k) = A109262(n+1).
%F A236843 Sum_{k=0..n} T(n+k,2k) = A026726(n).
%F A236843 Sum_{k=0..n} T(n+1+k,2k+1) = A026674(n+1).
%F A236843 T(n, k) = (1/4)*(6*k + 5 - (-1)^k)*(2*n - floor(k/2))!/((n-k)!*(n + floor((k+1)/2) + 1)!). - _G. C. Greubel_, Jun 13 2022
%e A236843 Triangle begins:
%e A236843     1;
%e A236843     1,   1;
%e A236843     2,   3,   1;
%e A236843     5,   9,   4,   1;
%e A236843    14,  28,  14,   6,  1;
%e A236843    42,  90,  48,  27,  7, 1;
%e A236843   132, 297, 165, 110, 35, 9, 1;
%e A236843 Production matrix is:
%e A236843   1...1
%e A236843   1...2...1
%e A236843   0...1...1...1
%e A236843   0...1...1...2...1
%e A236843   0...0...0...1...1...1
%e A236843   0...0...0...1...1...2...1
%e A236843   0...0...0...0...0...1...1...1
%e A236843   0...0...0...0...0...1...1...2...1
%e A236843   0...0...0...0...0...0...0...1...1...1
%e A236843   0...0...0...0...0...0...0...1...1...2...1
%e A236843   0...0...0...0...0...0...0...0...0...1...1...1
%e A236843   ...
%t A236843 T[n_, k_]:= (1/4)*(6*k+5-(-1)^k)*(2*n-Floor[k/2])!/((n-k)!*(n+Floor[(k+1)/2]+1)!);
%t A236843 Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jun 13 2022 *)
%o A236843 (Magma)
%o A236843 F:=Factorial;
%o A236843 A236843:= func< n,k | (1/4)*(6*k+5-(-1)^k)*F(2*n-Floor(k/2))/(F(n-k)*F(n+Floor((k+1)/2)+1)) >;
%o A236843 [A236843(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jun 13 2022
%o A236843 (SageMath)
%o A236843 F=factorial
%o A236843 def A236843(n,k): return (1/2)*(3*k+2+(k%2))*F(2*n-(k//2))/(F(n-k)*F(n+((k+1)//2)+1))
%o A236843 flatten([[A236843(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jun 13 2022
%o A236843 (PARI) T(n, k) = (1/4)*(6*k + 5 - (-1)^k)*(2*n - (k\2))!/((n-k)!*(n + (k+1)\2 + 1)!) \\ _Andrew Howroyd_, Jan 04 2023
%Y A236843 Columns: A000108 (k=0), A000245 (k=1), A002057 (k=2), A003517 (k=3), A000588 (k=4), A001392 (k=5), A003519 (k=6), A090749 (k=7), A000590 (k=8).
%Y A236843 Cf. A026674, A026726, A032766, A109262.
%Y A236843 Cf. A000045, A039599, A106566.
%K A236843 nonn,tabl
%O A236843 0,4
%A A236843 _Philippe Deléham_, Feb 01 2014