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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). 7
1, 1, 12, 155, 2124, 30482, 453432, 6936799, 108507180, 1727970542, 27924685416, 456820603086, 7550600079672, 125905525750500, 2115511349837040, 35782547891727495, 608787760350045420, 10411451736723707990 (list; graph; refs; listen; history; text; internal format)
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

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

...

- Gary W. Adamson, Jul 08 2011

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)))

CROSSREFS

Cf. A001003, A007564, A059231.

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

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Last modified November 18 19:59 EST 2019. Contains 329288 sequences. (Running on oeis4.)