A166231 -id:A166231 - cp's OEIS frontend

A166232 a(n) = A166231(n)/4^n.

1, 1, 3, 20, 272, 7424, 405504, 44302336, 9680453632, 4230542786560, 3697657604210688, 6463791365183504384, 22598414411798807576576, 158015104883301198495481856, 2209780998563745292895322636288

Offset: 0

Paul Barry, Oct 09 2009

a(n+1) is the Hankel transform of A068764.

G. C. Greubel, Table of n, a(n) for n = 0..70

Cf. A068764, A166231.

Table[2^(Binomial[n, 2])*Sum[Binomial[n, 2 k]*2^(-k), {k, 0, Floor[n/2]}], {n, 0, 25}] (* G. C. Greubel, May 07 2016 *)

a(n) = 2^C(n,2)*Sum_{k=0..floor(n/2)} C(n,2k)*2^(-k).

a(n) = A166231(n)/4^n.

A166229 Expansion of (1-2x-sqrt(1-8x+8x^2))/(2x).

Original entry on oeis.org

1, 2, 8, 36, 176, 912, 4928, 27472, 156864, 912832, 5394176, 32282240, 195264000, 1191825920, 7331457024, 45406194944, 282896763904, 1771868302336, 11150040870912, 70461597988864, 446971590516736, 2845144452292608

Offset: 0

Paul Barry, Oct 09 2009

Binomial transform of A166228. Hankel transform is A166231.

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Cf. A006318, A174347.

CoefficientList[Series[(1-2*x-Sqrt[1-8*x+8*x^2])/(2*x), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 20 2012 *)

a(n) = 0^n + Sum_{k = 0..n} C(n-1,k-1)*A006318(k). - Paul Barry, Nov 04 2009
G.f.: 1/(1-2x/(1-x-x/(1-2x/(1-x-x/(1-2x/(1-x-x/(1-... (continued fraction). - Paul Barry, Dec 10 2009
Recurrence: (n+1)*a(n) = 4*(2*n-1)*a(n-1) - 8*(n-2)*a(n-2). - Vaclav Kotesovec, Oct 20 2012
a(n) ~ sqrt(1+sqrt(2))*(4+2*sqrt(2))^n/(sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 20 2012
From Peter Bala, May 01 2024: (Start)
O.g.f.: A(x) = x*S(x/(1 - x)), where S(x) = (1 - x - sqrt(1 - 6*x + x^2))/(2*x) is the o.g.f. for the large Schröder numbers A006318.
a(n) = A174347(n+1) - A174347(n).
The g.f. satisfies x^2*A(x)^2 - x*(1 - 2*x)*A(x) + x*(1 - x) = 0.
A(x) = (1 - x)/(1 - 2*x - x*(1 - x)/(1 - 2*x - x*(1 - x)/(1 - 2*x - ...))). (End)

A166228 Alternating sum of large Schroeder numbers.

Original entry on oeis.org

1, 1, 5, 17, 73, 321, 1485, 7073, 34513, 171585, 866133, 4427313, 22870425, 119208321, 626178717, 3311424321, 17615732385, 94202293633, 506116560293, 2730607756881, 14788011564009, 80361643637953, 438070231780973

Offset: 0

Paul Barry, Oct 09 2009

Hankel transform is A166231. Binomial transform is A166229.

Vincenzo Librandi, Table of n, a(n) for n = 0..300

CoefficientList[Series[(1-x-Sqrt[1-6*x+x^2])/(2*x*(1+x)), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 20 2012 *)

G.f.: (1-x-sqrt(1-6x+x^2))/(2x(1+x));
a(n) = Sum{k=0..n} (-1)^k*A006318(n-k) = Sum_{k=0..n} (-1)^(n-k)*A006318(k).
Conjecture: (n+1)*a(n) +(4-5n)*a(n-1) +(1-5n)*a(n-2) +(n-2)*a(n-3)=0. - R. J. Mathar, Nov 17 2011
a(n) ~ sqrt(48+34*sqrt(2))*(3+2*sqrt(2))^n/(8*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 20 2012

cp's OEIS Frontend

A166232 a(n) = A166231(n)/4^n.

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A166229 Expansion of (1-2x-sqrt(1-8x+8x^2))/(2x).

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A166228 Alternating sum of large Schroeder numbers.

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