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A119308
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Triangle for first differences of Catalan numbers.
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4
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1, 2, 1, 3, 5, 1, 4, 14, 9, 1, 5, 30, 40, 14, 1, 6, 55, 125, 90, 20, 1, 7, 91, 315, 385, 175, 27, 1, 8, 140, 686, 1274, 980, 308, 35, 1, 9, 204, 1344, 3528, 4116, 2184, 504, 44, 1, 10, 285, 2430, 8568, 14112, 11340, 4410, 780, 54, 1, 11, 385, 4125
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OFFSET
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0,2
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COMMENTS
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Coefficient array of the second column of the inverse of the Riordan array ((1+r*x)/(1+(r+1)x+r*x^2), x/(1+(r+1)x+r*x^2)). - Paul Barry, Apr 01 2011
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LINKS
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FORMULA
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T(n,k) = sum{j=0..n, C(n,j)*[k<=j]*C(j+1,k+1)*C(k+1,j-k)/(j-k+1)}.
Column k has g.f.: sum{j=0..k, C(k,j)*C(k+1,j)x^j/(j+1)}*x^k/(1-x)^(2(k+1)).
T(n,k) = sum{j=0..n, C(n,j)*if(k<=j, C(j+1,2(j-k))*A000108(j-k),0)}.
G.f.: (((x-1)*sqrt(x^2*y^2+(-2*x^2-2*x)*y+x^2-2*x+1)+(-x^2-x)*y+x^2-2*x+1)/(2*x^3*y^2)). - Vladimir Kruchinin, Nov 15 2020
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EXAMPLE
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Triangle begins:
1;
2, 1;
3, 5, 1;
4, 14, 9, 1;
5, 30, 40, 14, 1;
6, 55, 125, 90, 20, 1;
7, 91, 315, 385, 175, 27, 1;
8, 140, 686, 1274, 980, 308, 35, 1;
9, 204, 1344, 3528, 4116, 2184, 504, 44, 1;
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MATHEMATICA
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a[k_, j_]:=If[k<=j, Binomial[j+1, 2(j-k)]*CatalanNumber[j-k], 0];
Flatten[Table[Sum[Binomial[n, j]*a[k, j], {j, 0, n}], {n, 0, 10}, {k, 0, n}]] (* Indranil Ghosh, Mar 03 2017 *)
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PROG
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(PARI)
catalan(n)=binomial(2*n, n)/(n+1);
a(k, j)=if (k<=j, binomial(j+1, 2*(j-k))*catalan(j-k), 0);
tabl(nn)={for (n=0, nn, for (k=0, n, print1(sum(j=0, n, binomial(n, j)*a(k, j)), ", "); ); print(); ); };
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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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