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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 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

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)}.

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. 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 October 17 18:58 EDT 2019. Contains 328127 sequences. (Running on oeis4.)