A159965 -id:A159965 - cp's OEIS frontend

A108081 a(n) = Sum_{i=0..n} binomial(2*n-i, n+i).

1, 2, 7, 25, 92, 344, 1300, 4950, 18955, 72905, 281403, 1089343, 4227273, 16438345, 64037453, 249855417, 976205516, 3818779616, 14954876080, 58623077586, 230007291334, 903164858092, 3549071519462, 13955918890440, 54912972103772, 216194101316654, 851622127750060

Offset: 0

Ralf Stephan, Jun 03 2005

nonn

A transform of the Fibonacci numbers A000045(n+1) under the mapping g(x)->(1/(c(x)*sqrt(1-4*x)))*g(x*c(x)), c(x) the g.f. of A000108. Hankel transform is the bisection of the Fibonacci numbers F(2*n+2) (A001906(n+1)). - Paul Barry, Sep 28 2007
Diagonal sums of A159965. - Paul Barry, Apr 28 2009
Comment from Li-yao Xia, Oct 22 2015: (Start)
Consider the smallest set X of finite sequences of integer (or words), such that
- 0 belongs to it;
- if a and b are two words in X, let L(a) be the word obtained by reversing a and subtracting 1 from every term, and R(b) be the word obtained by reversing b and adding 1 to every term; then the concatenations L(a).b and a.R(b) belong to X.
Examples of L and R values: L(10,30,20) = 19, 29, 9; R(10,30,20) = 21, 31, 11
List of words of X of lengths 1, 2, 3:
   0
   0,  1
  -1,  0
  -1,  0,  1 = L(0), 0, 1 = -1, 0, R(0)
   0,  2,  1 = 0, R(0, 1)
   1, -1,  0 = L(0), -1, 0
   0,  1,  0 = 0, R(-1, 0)
   0, -1,  0 = L(0, 1), 0
   0,  1,  1 = 0, 1, R(0)
  -1, -2,  0 = L(-1, 0), 0
The number of words of length n for n<=12 is given by a(n+1). Is this always true? (End)

G. C. Greubel and Vincenzo Librandi, Table of n, a(n) for n = 0..1000(terms 0..200 from Vincenzo Librandi)

Cf. A000045, A000108, A001906, A159965.

CoefficientList[Series[(1+Sqrt[1-4*x])/(2*Sqrt[1-4*x]*(x+Sqrt[1-4*x])), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 17 2012 *)
Table[Sum[Binomial[2n-i,n+i],{i,0,n}],{n,0,30}] (* Harvey P. Dale, Oct 20 2013 *)

my(x='x+O('x^66)); Vec((1+sqrt(1-4*x))/(2*sqrt(1-4*x)*(x+sqrt(1-4*x)))) \\ Joerg Arndt, May 15 2013

a(n) = sum(k=0,n, binomial(n+k-1,k)*fibonacci(n-k+1)); \\ G. C. Greubel, Jan 31 2017

G.f.: 1/2*(1-5*x+4*x^2+((1-4*x)*(1-5*x)^2)^(1/2))/(1-4*x)/(1-4*x-x^2). - Vladeta Jovovic, Sep 06 2006
G.f.: (1+sqrt(1-4*x))/(2*sqrt(1-4*x)*(x+sqrt(1-4*x))). - Paul Barry, Sep 28 2007
a(n) = Sum_{k=0..n} C(n+k-1,k)*F(n-k+1). - Paul Barry, Sep 28 2007
Recurrence: n*(n+1)*a(n) = 2*(4*n^2 + 3*n - 6)*a(n-1) - (15*n^2 + 7*n - 48)*a(n-2) - 2*(n+2)*(2*n-3)*a(n-3). - Vaclav Kotesovec, Oct 17 2012
a(n) ~ 2^(2*n+1)/sqrt(Pi*n). - Vaclav Kotesovec, Oct 17 2012
a(n) = [x^n] 1/((1-x-x^2) * (1-x)^n). - Seiichi Manyama, Apr 05 2024

A108080 a(n) = Sum_{i=0..n} binomial(2*n+i,n-i).

Original entry on oeis.org

1, 3, 12, 50, 211, 895, 3805, 16193, 68940, 293526, 1249622, 5318976, 22634700, 96296410, 409573584, 1741574006, 7403616923, 31466106703, 133704121665, 568008916093, 2412570019447, 10245302874071, 43500597657111, 184670002546295, 783850164628721, 3326671128027805, 14116630429874265

Offset: 0

Ralf Stephan, Jun 03 2005

nonn

Apparently a bisection of A026847.

Row sums of A159965. - Paul Barry, Apr 28 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A000108, A026847, A159965, A165201.

CoefficientList[Series[x/(x*Sqrt[1-4*x]-(1-2*x-(1-3*x)*(1-Sqrt[1-4*x])/(2*x))), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 24 2012 *)

x='x+O('x^66); Vec(x/(x*sqrt(1-4*x)-(1-2*x-(1-3*x)*(1-sqrt(1-4*x))/(2*x)))) \\ Joerg Arndt, May 15 2013

From Paul Barry, Apr 28 2009: (Start)
G.f.: x/(x*sqrt(1-4x)-(1-2x-(1-3x)*c(x))), c(x) the g.f. of A000108.
a(n) = Sum_{k=0..n} Sum_{j=0..n} C(n+k,j-k)*C(n,j). (End)
From Paul Barry, Sep 07 2009: (Start)
G.f.: (1/sqrt(1-4x))*(1/(1-xc(x)^3)), c(x) the g.f. of A000108.
a(n) = Sum_{k=0..n} C(2n,n-k)*F(k+1) = Sum_{k=0..n} C(2n,k)*F(n-k+1).
a(n) = Sum_{k=0..n} C(2k,k) * A165201(n-k). (End)
From Vaclav Kotesovec, Oct 24 2012: (Start)
Recurrence: n*(17*n-93)*a(n) = 4*(34*n^2 - 189*n + 98)*a(n-1) - 5*(51*n^2 - 271*n + 252)*a(n-2) - 4*(17*n^2 - 184*n + 406)*a(n-3) + 44*(2*n-7) * a(n-4).
a(n) ~ 1/2*(1+1/sqrt(5))*(sqrt(5)+2)^n. (End)
a(n) = binomial(2*n, n)*hypergeom([1, -n, 1+2*n], [(1+n)/2, 1+n/2], -1/4). - Stefano Spezia, Jun 17 2025

A159971 Riordan array (2c(-x)-1, xc(-x)^3), c(x) the g.f. of A000108.

Original entry on oeis.org

1, -2, 1, 4, -5, 1, -10, 19, -8, 1, 28, -68, 43, -11, 1, -84, 240, -198, 76, -14, 1, 264, -847, 845, -427, 118, -17, 1, -858, 3003, -3458, 2140, -782, 169, -20, 1, 2860, -10712, 13804, -9996, 4503, -1290, 229, -23, 1, -9724, 38454, -54264, 44574

Offset: 0

Paul Barry, Apr 28 2009

Inverse of A159965. Row sums are A159972.

			Triangle begins
1,
-2, 1,
4, -5, 1,
-10, 19, -8, 1,
28, -68, 43, -11, 1,
-84, 240, -198, 76, -14, 1,
264, -847, 845, -427, 118, -17, 1,
-858, 3003, -3458, 2140, -782, 169, -20, 1

cp's OEIS Frontend

A108081 a(n) = Sum_{i=0..n} binomial(2*n-i, n+i).

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A108080 a(n) = Sum_{i=0..n} binomial(2*n+i,n-i).

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A159971 Riordan array (2c(-x)-1, xc(-x)^3), c(x) the g.f. of A000108.

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