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A081178 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).

%I A081178 #44 Jan 15 2024 01:45:06
%S A081178 1,1,8,71,680,6882,72528,788019,8766248,99362894,1143498224,
%T A081178 13326176998,156950554384,1865210341828,22338852956064,
%U A081178 269355965364459,3267146912972328,39837475762660374,488032452193307568
%N A081178 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 A081178 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 A081178 The Hankel transform of this sequence is 7^C(n+1,2). - _Philippe Deléham_, Oct 29 2007
%C A081178 From _Gary W. Adamson_, Jul 08 2011: (Start)
%C A081178 a(n) = upper left term in M^n, M = the production matrix:
%C A081178   1, 1
%C A081178   7, 7, 7
%C A081178   1, 1, 1, 1
%C A081178   7, 7, 7, 7, 7
%C A081178   1, 1, 1, 1, 1, 1
%C A081178   ...
%C A081178 (End)
%C A081178 Shifts left when INVERT transform applied seven times. - _Benedict W. J. Irwin_, Feb 07 2016
%C A081178 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 A081178 Vincenzo Librandi, <a href="/A081178/b081178.txt">Table of n, a(n) for n = 0..200</a>
%H A081178 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 A081178 G.f.: (1+6*x-sqrt(36*x^2-16*x+1))/(14*x).
%F A081178 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 A081178 a(n) ~ sqrt(14+8*sqrt(7))*(8+2*sqrt(7))^n/(14*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 13 2012
%F A081178 a(n) = hypergeom([1 - n, -n], [2], 7). - _Peter Luschny_, Mar 19 2018
%p A081178 A081178_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;
%p A081178 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 A081178 convert(a, list) end: A081178_list(18); # _Peter Luschny_, May 19 2011
%t A081178 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 A081178 a[n_] := Hypergeometric2F1[1 - n, -n, 2, 7];
%t A081178 Table[a[n], {n, 0, 18}] (* _Peter Luschny_, Mar 19 2018 *)
%o A081178 (PARI) a(n)=if(n<1,1,sum(k=0,n,7^k/n*binomial(n,k)*binomial(n,k+1)))
%o A081178 (Magma)
%o A081178 B:=Binomial;
%o A081178 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 [A081178(n): n in [0..40]]; // _G. C. Greubel_, Jan 15 2024
%o A081178 (SageMath)
%o A081178 def A081178(n):
%o A081178     b=binomial;
%o A081178     if n==0: return 1
%o A081178     else: return (1/n)*sum(7^k*b(n,k)*b(n,k+1) for k in range(n+1))
%o A081178 [A081178(n) for n in range(41)] # _G. C. Greubel_, Jan 15 2024
%Y A081178 Cf. A001003, A001263, A007564, A059231.
%K A081178 nonn
%O A081178 0,3
%A A081178 _Benoit Cloitre_, May 10 2003