OFFSET
0,2
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
D. Drake, Bijections from Weighted Dyck Paths to Schröder Paths, J. Int. Seq. 13 (2010) # 10.9.2.
FORMULA
a(n) = Sum_{k=0..n} C(n,k)*C(2*n-k+1,n-k)*(k+1)/(2*n-k+1).
G.f. A(x) satisfies: d/dx[log(1 - 4*x*A(x))] = -4*(1-5*x)/(1-13*x+43*x^2-7*x^3).
O.g.f.: 2*(R+x)/(R*(R+x+1)), where R = sqrt(x^2+6*x+1). [Dan Drake, May 19 2010]
Conjecture: +(2*n+5)*(n+1)*a(n) +4*(-3*n^2-9*n+5)*a(n-1) +(2*n+7)*(n-1)*a(n-2)=0. - R. J. Mathar, Jun 22 2016
EXAMPLE
The dot product of Catalan row 4 and Pascal row 4 equals
a(4) = [14,14,9,4,1]*[1,4,6,4,1] = 141
which is equivalent to obtaining the final term
in these repeated partial sums of Pascal row 4:
1,4, 6, 4, 1
.5,11,15,16
..16,31,47
...47,94
....141
MATHEMATICA
Table[Sum[Binomial[n, j]*Binomial[2*n-j+1, n-j]*(j+1)/(2*n-j+1), {j, 0, n} ], {n, 0, 30}] (* G. C. Greubel, May 12 2019 *)
PROG
(PARI) a(n)=sum(k=0, n, binomial(n, k)*binomial(2*n-k+1, n-k)*(k+1)/(2*n-k+1))
for(n=0, 30, print1(a(n), ", "))
(Magma) [(&+[Binomial(n, j)*Binomial(2*n-j+1, n-j)*(j+1)/(2*n-j+1): j in [0..n]]): n in [0..30]]; // G. C. Greubel, May 12 2019
(Sage) [sum(binomial(n, j)*binomial(2*n-j+1, n-j)*(j+1)/(2*n-j+1) for j in (0..n)) for n in (0..30)] # G. C. Greubel, May 12 2019
(GAP) List([0..30], n-> Sum([0..n], j-> Binomial(n, j)*Binomial(2*n-j+1, n-j)*(j+1)/(2*n-j+1))) # G. C. Greubel, May 12 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 04 2006
STATUS
approved