A081178 - OEIS

%I #44 Jan 15 2024 01:45:06

%S 1,1,8,71,680,6882,72528,788019,8766248,99362894,1143498224,

%T 13326176998,156950554384,1865210341828,22338852956064,

%U 269355965364459,3267146912972328,39837475762660374,488032452193307568

%N a(0) = 1; for n>=1, a(n) = Sum_{k=0..n} 7^k*N(n,k), where N(n,k)=(1/n)*C(n,k)*C(n,k+1) are the Narayana numbers (A001263).

%C 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).

%C The Hankel transform of this sequence is 7^C(n+1,2). - _Philippe Deléham_, Oct 29 2007

%C From _Gary W. Adamson_, Jul 08 2011: (Start)

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

%C   1, 1

%C   7, 7, 7

%C   1, 1, 1, 1

%C   7, 7, 7, 7, 7

%C   1, 1, 1, 1, 1, 1

%C   ...

%C (End)

%C Shifts left when INVERT transform applied seven times. - _Benedict W. J. Irwin_, Feb 07 2016

%C G.f.: 1/(1 - x/(1 - 7*x/(1 - x/(1 - 7*x/(1 - x/(1 - ...)))))), a continued fraction. - _Ilya Gutkovskiy_, Apr 21 2017

%H Vincenzo Librandi, <a href="/A081178/b081178.txt">Table of n, a(n) for n = 0..200</a>

%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL9/Barry/barry91.html">On Integer-Sequence-Based Constructions of Generalized Pascal Triangles</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.

%F G.f.: (1+6*x-sqrt(36*x^2-16*x+1))/(14*x).

%F a(n) = (8*(2*n-1)*a(n-1) - 36*(n-2)*a(n-2))/(n+1) for n>=2, a(0) = a(1) = 1. - _Philippe Deléham_, Aug 19 2005

%F a(n) ~ sqrt(14+8*sqrt(7))*(8+2*sqrt(7))^n/(14*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 13 2012

%F a(n) = hypergeom([1 - n, -n], [2], 7). - _Peter Luschny_, Mar 19 2018

%p A081178_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;

%p for w from 1 to n do a[w] := a[w-1]+7*add(a[j]*a[w-j-1],j=1..w-1) od;

%p convert(a, list) end: A081178_list(18); # _Peter Luschny_, May 19 2011

%t Table[SeriesCoefficient[(1+6*x-Sqrt[36*x^2-16*x+1])/(14*x),{x,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Oct 13 2012 *)

%t a[n_] := Hypergeometric2F1[1 - n, -n, 2, 7];

%t Table[a[n], {n, 0, 18}] (* _Peter Luschny_, Mar 19 2018 *)

%o (PARI) a(n)=if(n<1,1,sum(k=0,n,7^k/n*binomial(n,k)*binomial(n,k+1)))

%o (Magma)

%o B:=Binomial;

%o A081178:= func< n | n eq 0 select 1 else (&+[7^k*B(n,k)*B(n,k+1): k in [0..n]])/n >;

%o [A081178(n): n in [0..40]]; // _G. C. Greubel_, Jan 15 2024

%o (SageMath)

%o def A081178(n):

%o     b=binomial;

%o     if n==0: return 1

%o     else: return (1/n)*sum(7^k*b(n,k)*b(n,k+1) for k in range(n+1))

%o [A081178(n) for n in range(41)] # _G. C. Greubel_, Jan 15 2024

%Y Cf. A001003, A001263, A007564, A059231.

%K nonn

%O 0,3

%A _Benoit Cloitre_, May 10 2003