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A119308 Triangle for first differences of Catalan numbers.

%I A119308 #40 Sep 13 2024 03:31:34
%S A119308 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,
%T A119308 27,1,8,140,686,1274,980,308,35,1,9,204,1344,3528,4116,2184,504,44,1,
%U A119308 10,285,2430,8568,14112,11340,4410,780,54,1,11,385,4125
%N A119308 Triangle for first differences of Catalan numbers.
%C A119308 Row sums are A000245(n+1). Columns include A000330, A006414, as well as certain Kekulé numbers (A114242, A108647, ...).
%C A119308 Diagonal sums are A188460.
%C A119308 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
%H A119308 Indranil Ghosh, <a href="/A119308/b119308.txt">Rows 0..100, flattened</a>
%H A119308 Lin Yang and Shengliang Yang, <a href="https://doi.org/10.4208/jms.v56n1.23.01">Protected Branches in Ordered Trees</a>, J. Math. Study (2023) Vol. 56, No. 1, 1-17.
%F A119308 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).
%F A119308 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)).
%F A119308 T(n,k) = Sum_{j=0..n} C(n,j)*if(k<=j, C(j+1,2(j-k))*A000108(j-k), 0).
%F A119308 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
%F A119308 T(n,k) = C(n+1,k)*(2*C(n+1,k+2)+C(n+1,k+1))/(n+1). - _Vladimir Kruchinin_, Nov 16 2020
%e A119308 Triangle begins:
%e A119308 1;
%e A119308 2,   1;
%e A119308 3,   5,    1;
%e A119308 4,  14,    9,    1;
%e A119308 5,  30,   40,   14,    1;
%e A119308 6,  55,  125,   90,   20,    1;
%e A119308 7,  91,  315,  385,  175,   27,   1;
%e A119308 8, 140,  686, 1274,  980,  308,  35,  1;
%e A119308 9, 204, 1344, 3528, 4116, 2184, 504, 44, 1;
%t A119308 a[k_,j_]:=If[k<=j,Binomial[j+1,2(j-k)]*CatalanNumber[j-k],0];
%t A119308 Flatten[Table[Sum[Binomial[n,j]*a[k,j],{j,0,n}],{n,0,10},{k,0,n}]] (* _Indranil Ghosh_, Mar 03 2017 *)
%o A119308 (PARI)
%o A119308 catalan(n)=binomial(2*n,n)/(n+1);
%o A119308 a(k,j)=if (k<=j,binomial(j+1,2*(j-k))*catalan(j-k),0);
%o A119308 tabl(nn)={for (n=0, nn, for (k=0, n, print1(sum(j=0, n, binomial(n,j)*a(k,j)),", "););print(););};
%o A119308 tabl(10); \\ _Indranil Ghosh_, Mar 03 2017
%Y A119308 Cf. A000108, A001263.
%K A119308 easy,nonn,tabl
%O A119308 0,2
%A A119308 _Paul Barry_, May 13 2006