A109267 -id:A109267 - cp's OEIS frontend

A109262 A Catalan transform of the Fibonacci numbers.

0, 1, 2, 6, 19, 63, 215, 749, 2650, 9490, 34318, 125104, 459152, 1694914, 6287896, 23429158, 87635243, 328917615, 1238303243, 4674847097, 17692789741, 67114622451, 255120892105, 971649360211, 3707176155659, 14167390221873

Offset: 0

Paul Barry, Jun 24 2005

A column of A109267.

Hankel transform is -Fibonacci(2*n). a(n+1) has Hankel transform Fibonacci(2*n+1). - Paul Barry, Nov 22 2007

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Paul Barry, Chebyshev moments and Riordan involutions, arXiv:1912.11845 [math.CO], 2019.
Stoyan Dimitrov, On permutation patterns with constrained gap sizes, arXiv:2002.12322 [math.CO], 2020.
Sergio Falcon, Catalan transform of the K-Fibonacci sequence, Commun. Korean Math. Soc. 28 (2013), No. 4, pp. 827-832.
Guo-Niu Han, Enumeration of Standard Puzzles
Guo-Niu Han, Enumeration of Standard Puzzles [Cached copy]
Merve Taştan and Engin Özkan, Catalan transform of the k-Jacobsthal sequence, Electronic Journal of Mathematical Analysis and Applications (2020) Vol. 8, No. 2, 70-74.

Cf. A000045, A000108, A039599, A081696, A106566, A109267.

[n eq 0 select 0 else (&+[k*Binomial(2*n-k-1,n-1)*Fibonacci(k): k in [0..n]])/n: n in [0..30]]; // G. C. Greubel, May 30 2022

CoefficientList[Series[(1-Sqrt[1-4*x])/(2*(Sqrt[1-4*x]+x)), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 13 2014 *)

[0]+[(1/n)*sum(k*binomial(2*n-k-1, n-1)*fibonacci(k) for k in (1..n)) for n in (1..30)] # G. C. Greubel, May 30 2022

G.f.: x*c(x)/(1 - x*c(x) - x^2*c(x)^2) = (1 - sqrt(1-4*x))/(2*(x + sqrt(1-4*x))) where c(x) is the g.f. of A000108.
a(n) = Sum_{k=0..n} (k/(2*n-k))*binomial(2*n-k, n-k)*Fibonacci(k).
a(n) = Sum_{k=0..n} A106566(n,k)*A000045(k). - Philippe Deléham, Oct 28 2008
a(n) = Sum_{k=0..n} A039599(n,k)*(-1)^(k+1)*A000045(k). - Philippe Deléham, Oct 28 2008
n*a(n) - (7*n-4)*a(n-1) + (7*n-2)*a(n-2) + (19*n-60)*a(n-3) + 2*(2*n-7)*a(n-4) = 0. - R. J. Mathar, Nov 26 2012
Recurrence: n*(5*n-11)*a(n) = 2*(20*n^2 - 59*n + 30)*a(n-1) - 15*(5*n^2 - 19*n + 16)*a(n-2) - 2*(2*n-5)*(5*n-6)*a(n-3). - Vaclav Kotesovec, Feb 13 2014
a(n) ~ 5*4^n/(sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Feb 13 2014
a(n) = (1/(2*sqrt(5)))*Catalan(n-1)*Sum_{j=0..1} ((-1)^j + sqrt(5)) * Hypergeometric2F1([2,1-n], [2*(1-n)], (1+(-1)^j*sqrt(5))/2). - G. C. Greubel, May 30 2022

A109263 A Catalan transform of F(n-1)-0^n.

Original entry on oeis.org

0, 0, 1, 3, 10, 34, 118, 416, 1485, 5355, 19473, 71313, 262735, 973027, 3619955, 13521307, 50684778, 190597594, 718788034, 2717755820, 10300186824, 39121645886, 148884623768, 567643844338, 2167882951110, 8292331115104

Offset: 0

Paul Barry, Jun 24 2005

A column of A109267.

Define a triangle by T(n,0)=A000045(n) and T(n,k)=sum_{r=k-1..n} T(r,k-1). (The column k=1 is A000071, the column k=2 is A001924 etc). Then T(n,n)=a(n+1). - J. M. Bergot, May 22 2013

G.f.: x^2c(x)^2/(1-xc(x)-x^2c(x)^2) where c(x) is the g.f. of A000108; a(n)=sum{k=0..n, (k/(2n-k))binomial(2n-k, n-k)(F(k-1)-0^k)}.

Conjecture: n*(n-3)*a(n) +2*(-4*n^2+15*n-10)*a(n-1) +(15*n^2-69*n+80)*a(n-2) +2*(n-2)*(2*n-5)*a(n-3)=0. - R. J. Mathar, Nov 09 2012

A109264 Riordan array (1-x-x^2,x(1-x)).

Original entry on oeis.org

1, -1, 1, -1, -2, 1, 0, 0, -3, 1, 0, 1, 2, -4, 1, 0, 0, 1, 5, -5, 1, 0, 0, -1, -1, 9, -6, 1, 0, 0, 0, -2, -6, 14, -7, 1, 0, 0, 0, 1, -1, -15, 20, -8, 1, 0, 0, 0, 0, 3, 5, -29, 27, -9, 1, 0, 0, 0, 0, -1, 4, 20, -49, 35, -10, 1, 0, 0, 0, 0, 0, -4, -1, 49, -76, 44, -11, 1, 0, 0, 0, 0, 0, 1, -8, -21, 98, -111, 54, -12, 1

Offset: 0

Paul Barry, Jun 24 2005

Rows sums are A109265. Diagonal sums are A109266. Inverse is A109267.

			Rows begin
1;
-1,1;
-1,-2,1;
0,0,-3,1;
0,1,2,-4,1;
0,0,1,5,-5,1;
0,0,-1,-1,9,-6,1;

A109268 Expansion of 1/((1-xc(x)-x^2c(x)^2)(1-x^2c(x))) with c(x) the g.f. of A000108.

Original entry on oeis.org

1, 1, 4, 11, 36, 119, 407, 1418, 5018, 17971, 64989, 236915, 869518, 3209735, 11907644, 44368636, 165957615, 622880402, 2345005449, 8852862717, 33505203060, 127096279124, 483127286394, 1840030664115, 7020349647422

Offset: 0

Paul Barry, Jun 24 2005

Diagonal sums of A109267.

G.f.: (2-x-x*sqrt(1-4x))(sqrt(1-4x)-x)/(2(1-4x-x^2)(1-x+x^3))

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A109262 A Catalan transform of the Fibonacci numbers.

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A109263 A Catalan transform of F(n-1)-0^n.

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A109264 Riordan array (1-x-x^2,x(1-x)).

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A109268 Expansion of 1/((1-xc(x)-x^2c(x)^2)(1-x^2c(x))) with c(x) the g.f. of A000108.

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