A049612 a(n) = T(n,3), array T as in A049600.

0, 1, 5, 19, 63, 192, 552, 1520, 4048, 10496, 26624, 66304, 162560, 393216, 940032, 2224128, 5214208, 12124160, 27983872, 64159744, 146210816, 331350016, 747110400, 1676673024, 3746562048, 8338276352, 18488492032

Offset: 0

Author

Clark Kimberling

Keywords

• nonn

Comments

If X_1, X_2, ..., X_n are 2-blocks of a (2n+3)-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 18 2007

References

• Robert Cori, Gabor Hetyei, Genus one partitions, in 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014), 2014, Chicago, United States. Discrete Mathematics and Theoretical Computer Science, DMTCS Proceedings vol. AT, pp. 333-344,

Links

• R. Cori and G. Hetyei, Counting genus one partitions and permutations, arXiv preprint arXiv:1306.4628 [math.CO], 2013.
• R. Cori and G. Hetyei, How to count genus one partitions, FPSAC 2014, Chicago, Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France, 2014, 333-344.
• Milan Janjic, Two Enumerative Functions
• S. Kitaev and J. Remmel, p-Ascent Sequences, arXiv preprint arXiv:1503.00914 [math.CO], 2015.
• Sergey Kitaev and Jeffrey Remmel, A note on p-ascent sequences, preprint, 2016.
• Index entries for linear recurrences with constant coefficients, signature (8,-24,32,-16)

Crossrefs

Cf. A049600.

Row sums of triangle A055252. a(n+1) = A055584(n, 0), n >= 0.

Programs

Mathematica

CoefficientList[Series[x (1-x)^3/(1-2x)^4,{x,0,30}],x] (* or *) Join[ {0},LinearRecurrence[{8,-24,32,-16},{1,5,19,63},30]] (* Harvey P. Dale, Jan 07 2014 *)

Formula

G.f.: x*(1-x)^3 /(1-2*x)^4.

a(n) = Sum_{k=0..floor((n+3)/2)} C(n+3, 2k)*C(k+1, 3). - Paul Barry, May 15 2003

a(n+1) = 2^n*n^3/48 + 5*2^n*n^2/16 + 7*2^n*n/6 + 2^n, n>=1. - Milan Janjic, Nov 18 2007

Binomial transform of the tetrahedral numbers A000292 when omitting the initial 0 in both sequences. - Carl Najafi, Sep 08 2011