A129952 Binomial transform of A124625.

1, 1, 2, 6, 16, 40, 96, 224, 512, 1152, 2560, 5632, 12288, 26624, 57344, 122880, 262144, 557056, 1179648, 2490368, 5242880, 11010048, 23068672, 48234496, 100663296, 209715200, 436207616, 905969664, 1879048192, 3892314112

Offset

0,3

Comments

Essentially the same as A057711: a(n) = A057711(n) for n >= 1.

Number of permutations of length n>=0 avoiding the partially ordered pattern (POP) {1>2, 1>3} of length 4. That is, number of length n permutations having no subsequences of length 4 in which the first element is larger than the second and third elements. - Sergey Kitaev, Dec 08 2020

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, arXiv:1903.08946 [math.CO], 2019.

Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, The Electronic Journal of Combinatorics 26(3) (2019), P3.26.

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

Formula

a(0) = 1, a(1) = 1; for n > 1, a(n) = n*2^(n-2).

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

E.g.f.: (1/2)*(x*exp(2*x) + x + 2). - G. C. Greubel, Jun 08 2016

Mathematica

LinearRecurrence[{4, -4}, {1, 1, 2, 6}, 30] (* G. C. Greubel, Jun 08 2016; corrected by Georg Fischer, Apr 02 2019 *)

Prog

(Magma) m:=15; S:=&cat[ [ 1, 2*i ]: i in [0..m] ]; [ &+[ Binomial(j-1, k-1)*S[k]: k in [1..j] ]: j in [1..2*m] ]; // Klaus Brockhaus, Jun 17 2007
(PARI) {m=29; print1(1, ", ", 1, ", "); for(n=2, m, print1(n*2^(n-2), ", "))} \\ Klaus Brockhaus, Jun 17 2007
(Python)
def A129952(n): return n<<n-2 if n>1 else 1 # Chai Wah Wu, Oct 03 2024

Crossrefs

Cf. A124625, A045623, A057711, A129953 (first differences), A129954 (second differences), A129955 (third differences).

Sequence in context: A078774 A174016 A265725 * A057711 A302239 A264551

Adjacent sequences: A129949 A129950 A129951 * A129953 A129954 A129955

Keyword

nonn,easy

Author

Paul Curtz, Jun 10 2007

Extensions

Edited and extended by Klaus Brockhaus, Jun 17 2007

Status

approved