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A237619
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Riordan array (1/(1+x*c(x)), x*c(x)) where c(x) is the g.f. of Catalan numbers (A000108).
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2
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1, -1, 1, 0, 0, 1, -1, 1, 1, 1, -2, 2, 3, 2, 1, -6, 6, 8, 6, 3, 1, -18, 18, 24, 18, 10, 4, 1, -57, 57, 75, 57, 33, 15, 5, 1, -186, 186, 243, 186, 111, 54, 21, 6, 1, -622, 622, 808, 622, 379, 193, 82, 28, 7, 1, -2120, 2120, 2742, 2120, 1312, 690, 311, 118, 36, 8, 1
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OFFSET
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0,11
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LINKS
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FORMULA
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EXAMPLE
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Triangle begins:
1;
-1, 1;
0, 0, 1;
-1, 1, 1, 1;
-2, 2, 3, 2, 1;
-6, 6, 8, 6, 3, 1;
-18, 18, 24, 18, 10, 4, 1;
-57, 57, 75, 57, 33, 15, 5, 1;
Production matrix begins:
-1, 1;
-1, 1, 1;
-1, 1, 1, 1;
-1, 1, 1, 1, 1;
-1, 1, 1, 1, 1, 1;
-1, 1, 1, 1, 1, 1, 1;
-1, 1, 1, 1, 1, 1, 1, 1;
-1, 1, 1, 1, 1, 1, 1, 1, 1;
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MATHEMATICA
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A065602[n_, k_]:= A065602[n, k]= Sum[(k-1+2*j)*Binomial[2*(n-j)-k-1, n-1]/(2*(n - j) -k-1), {j, 0, (n-k)/2}];
T[n_, k_]:= If[k==0, A065602[n, 0], If[n==1 && k==1, 1, A065602[n, k]]];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, May 27 2022 *)
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PROG
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(SageMath)
def A065602(n, k): return sum( (k+2*j-1)*binomial(2*n-2*j-k-1, n-1)/(2*n-2*j-k-1) for j in (0..(n-k)//2) )
if (n<2): return (-1)^(n-k)
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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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