A119308 Triangle for first differences of Catalan numbers.

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

Offset: 0

Author

Paul Barry, May 13 2006

Keywords

• easy
• nonn
• tabl

Comments

Row sums are A000245(n+1). Columns include A000330, A006414, as well as certain Kekulé 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

Examples

			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;
		

Links

• Indranil Ghosh, Rows 0..100, flattened
• Lin Yang and Shengliang Yang, Protected Branches in Ordered Trees, J. Math. Study (2023) Vol. 56, No. 1, 1-17.

Crossrefs

Cf. A000108, A001263.

Programs

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 *)

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

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,k)*(2*C(n+1,k+2)+C(n+1,k+1))/(n+1). - Vladimir Kruchinin, Nov 16 2020