A062109 Expansion of ((1-x)/(1-2*x))^4.

1, 4, 14, 44, 129, 360, 968, 2528, 6448, 16128, 39680, 96256, 230656, 546816, 1284096, 2990080, 6909952, 15859712, 36175872, 82051072, 185139200, 415760384, 929562624, 2069889024, 4591714304, 10150215680, 22364028928, 49123688448, 107592286208, 235015241728, 512040632320

Offset: 0

Author

Henry Bottomley, May 30 2001

Keywords

• nonn
• easy

Comments

If X_1,X_2,...,X_n are 2-blocks of a (2n+4)-set X then, for n >= 1, a(n+1) is the number of (n+3)-subsets of X intersecting each X_i, (i=1,2,...,n). - Milan Janjic, Nov 23 2007

If the offset here is set to zero, the binomial transform of A006918. - R. J. Mathar, Jun 29 2009

a(n) is the number of weak compositions of n with exactly 3 parts equal to 0. - Milan Janjic, Jun 27 2010

Binomial transform of A002623. - Carl Najafi, Jan 22 2013

Except for an initial 1, this is the p-INVERT of (1,1,1,1,1,...) for p(S) = (1 - S)^4; see A291000. - Clark Kimberling, Aug 24 2017

Links

• Harry J. Smith, Table of n, a(n) for n = 0..200
• Robert Davis and Greg Simay, Further Combinatorics and Applications of Two-Toned Tilings, arXiv:2001.11089 [math.CO], 2020.
• Nickolas Hein and Jia Huang, Variations of the Catalan numbers from some nonassociative binary operations, arXiv:1807.04623 [math.CO], 2018.
• Milan Janjic, Two Enumerative Functions
• Milan Janjic and B. Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - From _N. J. A. Sloane_, Feb 13 2013
• Milan Janjic and B. 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 (8,-24,32,-16).

Programs

Magma

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(((1-x)/(1-2*x))^4)); // G. C. Greubel, Oct 16 2018

Maple

seq(coeff(series(((1-x)/(1-2*x))^4, x,n+1),x,n),n=0..30); # Muniru A Asiru, Jul 01 2018

Mathematica

CoefficientList[Series[(1 - x)^4/(1 - 2 x)^4, {x, 0, 26}], x] (* Michael De Vlieger, Jul 01 2018 *)
LinearRecurrence[{8,-24,32,-16},{1,4,14,44,129},30] (* Harvey P. Dale, Sep 02 2022 *)

PARI

a(n)=if(n<1,n==0,(n+5)*(n^2+13*n+18)*2^n/96)

Formula

a(n) = (n+5)*(n^2 + 13*n + 18)*2^(n-5)/3, with a(0)=1.

a(n) = A055809(n-5)*2^(n-4).

a(n) = 2*a(n-1) + A058396(n) - A058396(n-1).

a(n) = Sum_{kA058396(n).

a(n) = A062110(4, n).

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