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

Author

Benoit Cloitre, May 10 2003

Keywords

• nonn

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

Crossrefs

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

Programs

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

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 *)

PARI

a(n)=if(n<1,1,sum(k=0,n,11^k/n*binomial(n,k)*binomial(n,k+1)))

SageMath

def A082173_list(prec):
    P. = 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

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