A100248 Row sums of the slanted Catalan convolution table A100247.

1, 2, 10, 79, 777, 8606, 102512, 1282129, 16605538, 220781427, 2995985345, 41325515589, 577713950666, 8166924383923, 116550061698966, 1676836298476274, 24295472856858786, 354190017808427947, 5191706917095917442, 76469028773023897070, 1131207622704483680933, 16799374652884761512521

Offset

0,2

Links

Table of n, a(n) for n=0..21.

Formula

a(n) = Sum_{k=0..2n} C(n+2*k-[k/2], k)*(n-[k/2])/(n+2*k-[k/2]).

G.f. A(x) satisfies: A(x^2) = ((1+x)/(2 - x*(1-sqrt(1 - 4*x))) - (1-x)/(2 + x*(1-sqrt(1 + 4*x))))/x.

a(n) ~ 5 * 2^(4*n + 1/2) / (49*sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Oct 09 2020

Maple

A100248 := proc(n)
    if n = 0 then
        1;
    else
        add(binomial(n+2*k-floor(k/2), k) * (n-floor(k/2)) / (n+2*k-floor(k/2)), k=0..2*n) ;
    fi;
end proc: # R. J. Mathar, May 06 2016

Mathematica

CoefficientList[Series[(1 + Sqrt[x])/(2*Sqrt[x] + (-1 + Sqrt[1 - 4*Sqrt[x]])*x) + (1 - Sqrt[x])/(-2*Sqrt[x] + (-1 + Sqrt[1 + 4*Sqrt[x]])*x), {x, 0, 25}], x] (* Vaclav Kotesovec, Oct 09 2020 *)

Prog

(PARI) {a(n)=sum(k=0, 2*n, polcoeff(((1-sqrt(1-4*z+z^2*O(z^k)))/(2*z))^(n-k\2), k, z))}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=if(n==0, 1, sum(k=0, 2*n, binomial(n+2*k-(k\2), k)*(n-(k\2))/(n+2*k-(k\2))))}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A100247.

Sequence in context: A081363 A279908 A245903 * A368951 A108486 A152168

Adjacent sequences: A100245 A100246 A100247 * A100249 A100250 A100251

Keyword

nonn

Author

Paul D. Hanna, Nov 09 2004

Status

approved