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A090285
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Triangle T(n,k), 0<=k<=n, read by rows, defined by: T(n,k)=0 if k>n, T(n,0) = A000108(n); T(n+1,k)= Sum_{j=0..n} T(n-j,k-1)*binomial(2j+1,j+1).
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8
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1, 1, 1, 2, 4, 1, 5, 15, 7, 1, 14, 56, 37, 10, 1, 42, 210, 176, 68, 13, 1, 132, 792, 794, 392, 108, 16, 1, 429, 3003, 3473, 2063, 731, 157, 19, 1, 1430, 11440, 14893, 10254, 4395, 1220, 215, 22, 1, 4862, 43758, 63004, 49024, 24465, 8249, 1886, 282, 25, 1
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OFFSET
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0,4
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COMMENTS
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The matrix inverse starts
1;
-1, 1;
2, -4, 1;
-4, 13, -7, 1;
8, -38, 33, -10, 1;
-16, 104, -129, 62, -13, 1;
32, -272, 450, -304, 100, -16, 1;
-64, 688, -1452, 1289, -590, 147, -19, 1;
128, -1696, 4424, -4942, 2945, -1014, 203, -22, 1;
Riordan array (c(x), x*c(x)^2/(1-x*c(x)^2)) where c(x) is the g.f. for the Catalan numbers (A000108). - Philippe Deléham, Jun 02 2013
The matrix inverse is the Riordan array ((1+x)/(1+2*x), x*(1+x)/(1+2*x)^2). - Philippe Deléham, Jan 26 2014
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LINKS
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FORMULA
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T(n, 2) = 2^(2n-1) - binomial(2n+1, n) + binomial(2n-1, n-1) = A006419(n).
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MAPLE
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if k < 0 or k > n then
0 ;
elif k = 0 then
else
add(procname(n-1-j, k-1)*binomial(2*j+1, j+1), j=0..n-1) ;
end if;
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MATHEMATICA
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T[n_, k_] := T[n, k] = If[k == 0, CatalanNumber@ n, Sum[T[(n - 1) - j, k - 1] Binomial[2 j + 1, j + 1], {j, 0, n - 1}]]; Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Michael De Vlieger, Jun 26 2017 *)
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CROSSREFS
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See also A001700 for binomial(2n+1,n+1).
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KEYWORD
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AUTHOR
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STATUS
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approved
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