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A086616 Partial sums of the large Schroeder numbers (A006318).

%I A086616 #49 Sep 02 2024 10:41:38
%S A086616 1,3,9,31,121,515,2321,10879,52465,258563,1296281,6589727,33887465,
%T A086616 175966211,921353249,4858956287,25786112993,137604139011,737922992937,
%U A086616 3974647310111,21493266631001,116642921832963,635074797251889,3467998148181631,18989465797056721,104239408386028035
%N A086616 Partial sums of the large Schroeder numbers (A006318).
%C A086616 Row sums of triangle A086614. - _Paul D. Hanna_, Jul 24 2003
%C A086616 Hankel transform is A136577(n+1). - _Paul Barry_, Jun 03 2009
%H A086616 Vincenzo Librandi, <a href="/A086616/b086616.txt">Table of n, a(n) for n = 0..200</a>
%H A086616 Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Barry3/barry93.html">Continued fractions and transformations of integer sequences</a>, JIS 12 (2009), #09.7.6.
%F A086616 G.f.: A(x) = 1/(1 - x)^2 + x*A(x)^2.
%F A086616 a(1) = 1 and a(n) = n + Sum_{i=1..n-1} a(i)*a(n-i) for n >= 2. - _Benoit Cloitre_, Mar 16 2004
%F A086616 G.f.: (1 - x - sqrt(1 - 6*x + x^2))/(2*x*(1 - x)). Cf. A001003. - _Ralf Stephan_, Mar 23 2004
%F A086616 a(n) = Sum_{k=0..n} C(n+k+1, 2*k+1) * A000108(k). - _Paul Barry_, Jun 03 2009
%F A086616 Recurrence: (n+1)*a(n) = (7*n-2)*a(n-1) - (7*n-5)*a(n-2) + (n-2)*a(n-3). - _Vaclav Kotesovec_, Oct 14 2012
%F A086616 a(n) ~ sqrt(24 + 17*sqrt(2))*(3 + 2*sqrt(2))^n/(4*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 14 2012
%F A086616 A(x) = 1/(1 - x)^2 * c(x/(1-x^2)), where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. - _Peter Bala_, Aug 29 2024
%e A086616 a(1) = 2 + 1 = 3;
%e A086616 a(2) = 3 + 4 + 2 = 9;
%e A086616 a(3) = 4 + 10 + 12 + 5 = 31;
%e A086616 a(4) = 5 + 20 + 42 + 40 + 14 = 121.
%t A086616 Table[SeriesCoefficient[(1-x-Sqrt[1-6*x+x^2])/(2*x*(1-x)),{x,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Oct 14 2012 *)
%o A086616 (Sage) # Generalized algorithm of L. Seidel
%o A086616 def A086616_list(n) :
%o A086616     D = [0]*(n+2); D[1] = 1
%o A086616     b = True; h = 2; R = []
%o A086616     for i in range(2*n) :
%o A086616         if b :
%o A086616             for k in range(h,0,-1) : D[k] += D[k-1]
%o A086616         else :
%o A086616             for k in range(1,h, 1) : D[k] += D[k-1]
%o A086616             R.append(D[h-1]); h += 1;
%o A086616         b = not b
%o A086616     return R
%o A086616 A086616_list(23) # _Peter Luschny_, Jun 02 2012
%o A086616 (PARI) x='x+O('x^66); Vec((1-x-sqrt(1-6*x+x^2))/(2*x*(1-x))) \\ _Joerg Arndt_, May 10 2013
%Y A086616 Cf. A086614 (triangle), A086615 (antidiagonal sums).
%Y A086616 Cf. A006318.
%K A086616 nonn,easy
%O A086616 0,2
%A A086616 _Paul D. Hanna_, Jul 24 2003
%E A086616 Name changed using a comment of _Emeric Deutsch_ from Dec 20 2004. - _Peter Luschny_, Jun 03 2012