A082173 a(0)=1; for n >= 1, a(n) = Sum_{k=0..n} 11^k*N(n,k) where N(n,k) = (1/n)*C(n,k)*C(n,k+1) are the Narayana numbers (A001263).

1, 1, 12, 155, 2124, 30482, 453432, 6936799, 108507180, 1727970542, 27924685416, 456820603086, 7550600079672, 125905525750500, 2115511349837040, 35782547891727495, 608787760350045420, 10411451736723707990

Offset

0,3

Comments

More generally coefficients of (1 + m*x - sqrt(m^2*x^2 -(2*m+4)*x + 1) )/((2*m+2)*x) are given by a(n) = Sum_{k=0..n} (m+1)^k*N(n,k).

The Hankel transform of this sequence is 11^C(n+1,2). - Philippe Deléham, Oct 29 2007

For fixed m > 0, if g.f. = (1 + m*x - sqrt(m^2*x^2 -(2*m+4)*x + 1) )/((2*m+2)*x) then a(n,m) ~ (m + 2 + 2*sqrt(m+1))^(n + 1/2) / (2*sqrt(Pi) * (m+1)^(3/4) * n^(3/2)). - Vaclav Kotesovec, Mar 19 2018

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Formula

G.f.: (1+10*x-sqrt(100*x^2-24*x+1))/(22*x).

a(n) = Sum_{k=0..n} A088617(n, k)*11^k*(-10)^(n-k). - Philippe Deléham, Jan 21 2004

a(n) = (12*(2n-1)*a(n-1) - 100*(n-2)*a(n-2)) / (n+1) for n >= 2, a(0) = a(1) = 1. - Philippe Deléham, Aug 19 2005

From Gary W. Adamson, Jul 08 2011: (Start)

a(n) = upper left term in M^n, M = the production matrix:

1, 1

11, 11, 11

1, 1, 1, 1

11, 11, 11, 11, 11

1, 1, 1, 1, 1, 1

... (End)

a(n) ~ sqrt(22+12*sqrt(11))*(12+2*sqrt(11))^n/(22*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 14 2012

G.f.: 1/(1 - x/(1 - 11*x/(1 - x/(1 - 11*x/(1 - x/(1 - ...)))))), a continued fraction. - Ilya Gutkovskiy, Aug 10 2017

a(n) = hypergeom([1 - n, -n], [2], 11). - Peter Luschny, Mar 19 2018

Maple

A082173_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;
for w from 1 to n do a[w] := a[w-1]+11*add(a[j]*a[w-j-1], j=1..w-1)od;
convert(a, list) end: A082173_list(17); # Peter Luschny, May 19 2011

Mathematica

Table[SeriesCoefficient[(1+10*x-Sqrt[100*x^2-24*x+1])/(22*x), {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 14 2012 *)
a[n_] := Hypergeometric2F1[1 - n, -n, 2, 11];
Table[a[n], {n, 0, 18}] (* Peter Luschny, Mar 19 2018 *)

Prog

(PARI) a(n)=if(n<1, 1, sum(k=0, n, 11^k/n*binomial(n, k)*binomial(n, k+1)))
(Magma) [1] cat [&+[11^k*Binomial(n, k)*Binomial(n, k+1)/n:k in [0..n]]:n in [1..18]]; // Marius A. Burtea, Jan 22 2020
(SageMath)
def A082173_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P( (1+10*x-sqrt(100*x^2-24*x+1))/(22*x) ).list()
A082173_list(30) # G. C. Greubel, Jan 21 2024

Crossrefs

Cf. A001003, A001263, A007564, A059231, A088617.

Sequence in context: A120657 A015612 A085260 * A005723 A097259 A158546

Adjacent sequences: A082170 A082171 A082172 * A082174 A082175 A082176

Keyword

nonn

Author

Benoit Cloitre, May 10 2003

Status

approved