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A236843
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Triangle read by rows related to the Catalan transform of the Fibonacci numbers.
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2
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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, 9, 1, 429, 1001, 572, 429, 154, 54, 10, 1, 1430, 3432, 2002, 1638, 637, 273, 65, 12, 1, 4862, 11934, 7072, 6188, 2548, 1260, 350, 90, 13, 1, 16796, 41990, 25194, 23256, 9996, 5508, 1700, 544, 104, 15, 1
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,4
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COMMENTS
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LINKS
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FORMULA
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G.f. for the column k (with zeros omitted): C(x)^A032766(k+1) where C(x) is g.f. for Catalan numbers (A000108).
Sum_{k=0..n} T(n,k) = A109262(n+1).
Sum_{k=0..n} T(n+k,2k) = A026726(n).
Sum_{k=0..n} T(n+1+k,2k+1} = A026674(n+1).
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
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EXAMPLE
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Triangle begins:
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, 9, 1;
Production matrix is:
1...1
1...2...1
0...1...1...1
0...1...1...2...1
0...0...0...1...1...1
0...0...0...1...1...2...1
0...0...0...0...0...1...1...1
0...0...0...0...0...1...1...2...1
0...0...0...0...0...0...0...1...1...1
0...0...0...0...0...0...0...1...1...2...1
0...0...0...0...0...0...0...0...0...1...1...1
...
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MATHEMATICA
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T[n_, k_]:= (1/4)*(6*k+5-(-1)^k)*(2*n-Floor[k/2])!/((n-k)!*(n+Floor[(k+1)/2]+1)!);
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Jun 13 2022 *)
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PROG
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(Magma)
F:=Factorial;
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)) >;
(SageMath)
F=factorial
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))
(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
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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