A034007 First differences of A045891.

1, 0, 2, 4, 9, 20, 44, 96, 208, 448, 960, 2048, 4352, 9216, 19456, 40960, 86016, 180224, 376832, 786432, 1638400, 3407872, 7077888, 14680064, 30408704, 62914560, 130023424, 268435456, 553648128, 1140850688, 2348810240

Offset: 0

Author

Wolfdieter Lang

Keywords

• easy
• nonn

Comments

Let M_n be the n X n matrix m_(i,j) = 4 + abs(i-j) then det(M_n) = (-1)^(n+1)*a(n+2). - Benoit Cloitre, May 28 2002

Number of ordered pairs of (possibly empty) ordered partitions, each not beginning with 1. - Christian G. Bower, Jan 23 2004

If X_1, X_2, ..., X_n are 2-blocks of a (2n+4)-set X then, for n>=1, a(n+3) is the number of (n+1)-subsets of X intersecting each X_i, (i=1,2,...,n). - Milan Janjic, Nov 18 2007

Except for an initial 1, this is the p-INVERT of (1,1,1,1,1,...) for p(S) = (1 - S^2)^2; see A291000. - Clark Kimberling, Aug 24 2017

Conjecture 1: For compositions of n+k-1, a(n) is the number of runs of 1 of length k. Example: Among the compositions of 4+2-1 = 5, there are a(4) = 4 runs of two 1's: 3,[1,1]; [1,1],3; 1,2,[1,1] and [1,1],2,1. - Gregory L. Simay, Feb 18 2018

Conjecture 2: More generally, let R(n,m,k) = the number of runs of k m's in all compositions of n. Then R(n,m,k) = A045623(n-m*k) - 2*A045623(n-m*(k+1)) + A045623(n-m*(k+2)). For example, R(7,1,1) = A045623(6) - 2*A045623(5) + A045623(4) = 144 - 2*64 + 28 = 44 = a(7). - Gregory L. Simay, Feb 20 2018

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Frank Ellermann, Illustration of binomial transforms.
• Milan Janjic, Two Enumerative Functions.
• Milan Janjic and Boris Petkovic, A Counting Function, arXiv:1301.4550 [math.CO], 2013.
• Milan Janjic and Boris Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5.
• Index entries for linear recurrences with constant coefficients, signature (4,-4).

Crossrefs

Cf. A045623, A045891, A291000.

Convolution of A034008 with itself.

Columns of A091613 converge to this sequence.

Programs

Magma

[1,0,2] cat [(n+5)*2^(n-4): n in [3..30]]; // G. C. Greubel, Sep 27 2022

Mathematica

Join[{1,0,2,a=4},Table[a=(2*(n+7)*a)/(n+6),{n,2,100}]] (* Vladimir Joseph Stephan Orlovsky, Jan 15 2011 *)

PARI

a(n)=if(n<3,[1,0,2][n+1],(n+5)*2^(n-4)) \\ Charles R Greathouse IV, Jun 01 2011

SageMath

[1,0,2]+[(n+5)*2^(n-4) for n in range(3,30)] # G. C. Greubel, Sep 27 2022

Formula

a(n) = Sum_{k = 0..n-3} (k+4)*binomial(n-3,k) for n >= 3. - N. J. A. Sloane, Jan 30 2008

a(n) = (n+5)*2^(n-4), n >= 3; a(0)=1, a(1)=0, a(2)=2.

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

a(n) = Sum_{k=0..n} (k+1)*C(n-3,n-k). - Peter Luschny, Apr 20 2015

From Amiram Eldar, Jan 13 2021: (Start)

Sum_{n>=2} 1/a(n) = 512*log(2) - 74327/210.

Sum_{n>=2} (-1)^n/a(n) = 14579/70 - 512*log(3/2). (End)

E.g.f.: (1/16)*(11 - 12*x + 2*x^2 + (5+2*x)*exp(2*x)). - G. C. Greubel, Sep 27 2022