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

0,2

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

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.

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

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