A270661 a(n) = Sum_{k=0..n}((binomial(2*k,k)*Sum_{i=0..n-k}(binomial(k+1,n-k-i)*binomial(k+i,k)))/(k+1)^2)*(n+1).

1, 5, 14, 45, 164, 662, 2862, 12957, 60590, 290218, 1416216, 7014714, 35172968, 178180968, 910542696, 4688140189, 24296295636, 126640986410, 663465473910, 3491674292814, 18450954171742, 97859178176632, 520755520521982, 2779633126026210

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Formula

G.f.: C((x^2+x)/(1-x))*((x^2-2*x-1)/(x^2-1)*(1+x)/(1-x)), where C(x) is g.f. of Catalan numbers (A000108).

a(n) ~ sqrt(55473 + 4469*sqrt(41)) * (5 + sqrt(41))^n / (sqrt(Pi) * n^(3/2) * 2^(n+7)). - Vaclav Kotesovec, Mar 21 2016

Conjecture: (n+1)*a(n) -4* n*a(n-1) +(-13*n+33)*a(n-2) +2*(8*n-17)*a(n-3) +(19*n-69)*a(n-4) +4*(-4*n+15)*a(n-5) +7*(-n+5)*a(n-6) +2*(2*n-13)*a(n-7)=0. - R. J. Mathar, Jun 07 2016

Conjecture: (n+1)*(n^2-5*n+3)*a(n) -2*n*(3*n^2-16*n+14)*a(n-1) +2*(n+4)*(n-3)*a(n-2) +2*(5*n^3-31*n^2+39*n+9)*a(n-3) +(-n^3+2*n^2+21)*a(n-4) -2*(2*n-9)*(n^2-3*n-1)*a(n-5)=0. - R. J. Mathar, Jun 07 2016

Maple

A270661 := proc(n)
    local a, k ;
    a := 0 ;
    for k from 0 to n do
        a := a+binomial(2*k, k)/(k+1)^2*add(binomial(k+1, n-k-i)*binomial(k+i, k), i=0..n-k) ;
    end do:
    %*(n+1) ;
end proc: # R. J. Mathar, Jun 07 2016

Mathematica

Table[Sum[(Binomial[2 k, k] Sum[Binomial[k + 1, n - k - i] Binomial[k + i, k], {i, 0, n - k}]/(k + 1)^2) (n + 1), {k, 0, n}], {n, 0, 23}] (* Michael De Vlieger, Mar 21 2016 *)

Prog

(Maxima)
C(x):=(1-sqrt(1-4*x))/(2*x);
makelist(coeff(taylor(C((x^2+x)/(1-x))/x*((x^2-2*x-1)/(x^2-1)*(1+x)/(1-x)), x, 0, 10)*x, x, n), n, 0, 10);
a(n):=(sum((binomial(2*k, k)*sum(binomial(k+1, n-k-i)*binomial(k+i, k), i, 0, n-k))/(k+1)^2, k, 0, n))*(n+1);
(PARI) a(n) = sum(k=0, n, (binomial(2*k, k)*sum(i=0, n-k, (binomial(k+1, n-k-i)*binomial(k+i, k)))/(k+1)^2)*(n+1)); \\ Michel Marcus, Mar 21 2016

Crossrefs

Cf. A000108.

Sequence in context: A100059 A270062 A236043 * A222908 A270911 A270154

Adjacent sequences: A270658 A270659 A270660 * A270662 A270663 A270664

Keyword

nonn

Author

Vladimir Kruchinin, Mar 20 2016

Status

approved