OFFSET
0,2
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, Greg Simay, Further Combinatorics and Applications of Two-Toned Tilings, arXiv:2001.11089 [math.CO], 2020.
Nickolas Hein, Jia Huang, Variations of the Catalan numbers from some nonassociative binary operations, arXiv:1807.04623 [math.CO], 2018.
Milan Janjic, Two Enumerative Functions
M. Janjic and B. Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - From N. J. A. Sloane, Feb 13 2013
M. Janjic, 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).
FORMULA
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 *)
PROG
(PARI) a(n)=if(n<1, n==0, (n+5)*(n^2+13*n+18)*2^n/96)
(PARI) { a=1; for (n=0, 200, if (n, a=(n + 5)*(n^2 + 13*n + 18)*2^n/96); write("b062109.txt", n, " ", a) ) } \\ Harry J. Smith, Aug 01 2009
(Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(((1-x)/(1-2*x))^4)); // G. C. Greubel, Oct 16 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Henry Bottomley, May 30 2001
STATUS
approved