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 A119308 Triangle for first differences of Catalan numbers. 3
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Row sums are A000245(n+1). Columns include A000330, A006414, as well as certain KekulĂ© numbers (A114242, A108647, ...). Diagonal sums are A188460. 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 LINKS Indranil Ghosh, Rows 0..100, flattened FORMULA 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 T(n,k) = C(n+1,m))*(2*C(n+1,m+2)+C(n+1,m+1))/(n+1). - Vladimir Kruchinin, Nov 16 2020 EXAMPLE 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; MATHEMATICA 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 *) PROG (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(); ); }; tabl(10); \\ Indranil Ghosh, Mar 03 2017 CROSSREFS Cf. A000108, A001263. Sequence in context: A153277 A104029 A208752 * A110197 A124819 A124019 Adjacent sequences:  A119305 A119306 A119307 * A119309 A119310 A119311 KEYWORD easy,nonn,tabl AUTHOR Paul Barry, May 13 2006 STATUS approved

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Last modified September 16 21:28 EDT 2021. Contains 347473 sequences. (Running on oeis4.)