A100335 An inverse Catalan transform of J(2n).

0, 1, 4, 11, 27, 64, 149, 341, 768, 1707, 3755, 8192, 17749, 38229, 81920, 174763, 371371, 786432, 1660245, 3495253, 7340032, 15379115, 32156331, 67108864, 139810133, 290805077, 603979776, 1252698795, 2594876075, 5368709120

Offset

0,3

Comments

The g.f. is obtained from that of A002450 through the mapping g(x) -> g(x*(1-x)). A002450 may be retrieved through the mapping g(x) -> g(x*c(x)), where c(x) is the g.f. of A000108.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (5,-9,8,-4).

Formula

G.f.: x*(1-x)/(1 - 5*x + 9*x^2 - 8*x^3 + 4*x^4).

a(n) = 5*a(n-1) - 9*a(n-2) + 8*a(n-3) - 4*a(n-4).

a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*(-1)^k*(4^(n-k) - 1)/3.

a(n) = (1/3)*((n+1)*2^n - A010892(n)). - Ralf Stephan, May 15 2007

Binomial transform of A042965: (1, 3, 4, 5, 7, 8, 9, 11, 12, 13, ...), also row sums of triangle A133110. - Gary W. Adamson, Sep 12 2007

a(n) = Sum_{k=0..n} A109466(n,k)*A002450(k). - Philippe Deléham, Oct 30 2008

Mathematica

LinearRecurrence[{5, -9, 8, -4}, {0, 1, 4, 11}, 41] (* G. C. Greubel, Jan 24 2023 *)

Prog

(Magma) I:=[0, 1, 4, 11]; [n le 4 select I[n] else 5*Self(n-1) -9*Self(n-2) +8*Self(n-3) -4*Self(n-4): n in [1..41]]; // G. C. Greubel, Jan 24 2023
(SageMath)
def A100335(n): return (1/3)*((n+1)*2^n - chebyshev_U(n, 1/2))
[A100335(n) for n in range(41)] # G. C. Greubel, Jan 24 2023

Crossrefs

Cf. A000108, A001045, A002450, A010892.

Cf. A042965, A100334, A109466, A133110.

Sequence in context: A000253 A276691 A047859 * A340228 A080869 A137229

Adjacent sequences: A100332 A100333 A100334 * A100336 A100337 A100338

Keyword

easy,nonn

Author

Paul Barry, Nov 17 2004

Status

approved