A133110 -id:A133110 - cp's OEIS frontend

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

Paul Barry, Nov 17 2004

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.

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).

Cf. A000108, A001045, A002450, A010892.

Cf. A042965, A100334, A109466, A133110.

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

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

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

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

A133109 Triangle read by rows, A042965 on the diagonal, 0 elsewhere.

Original entry on oeis.org

1, 0, 3, 0, 0, 4, 0, 0, 0, 5, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 11, 0, 0, 0, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 15, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 17

Offset: 1

Gary W. Adamson, Sep 12 2007

			First few rows of the triangle:
  1;
  0, 3;
  0, 0, 4;
  0, 0, 0, 5;
  0, 0, 0, 0, 7;
  ...

Cf. A042965, A133110 (binomial transform).

row(n) = vector(n, k, if (k==n, (4*n+1)\3)); \\ Michel Marcus, Mar 07 2022

A042965: (1, 3, 4, 5, 7, 8, 9, 11, 12, 13, ...) as the diagonal of an infinite lower triangular matrix and the rest zeros.

More terms from Michel Marcus, Mar 07 2022

cp's OEIS Frontend

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

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