A223659 Number of unimodal maps [1..n]->[0..3].

1, 4, 16, 50, 130, 296, 610, 1163, 2083, 3544, 5776, 9076, 13820, 20476, 29618, 41941, 58277, 79612, 107104, 142102, 186166, 241088, 308914, 391967, 492871, 614576, 760384, 933976, 1139440, 1381300, 1664546, 1994665, 2377673, 2820148, 3329264

Offset: 0

Author

R. H. Hardin, Mar 25 2013

Keywords

• nonn

Comments

Column 1 of A223663.

Apparently also column 4 of A071920. - R. J. Mathar, May 17 2014

Examples

			Some solutions for n=3:
  2  2  0  1  1  3  1  0  3  1  2  1  2  1  0  2
  2  2  1  3  3  3  3  2  2  2  2  3  0  1  1  1
  2  0  2  2  0  1  3  3  1  0  3  1  0  1  1  0
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000 (terms n = 1..210 from R. H. Hardin)
• Kyu-Hwan Lee and Se-jin Oh, Catalan triangle numbers and binomial coefficients, arXiv:1601.06685 [math.CO], 2016.

Crossrefs

Cf. A071920, A223663, A223718.

Formula

Empirical: a(n) = (1/720)*n^6 + (1/48)*n^5 + (23/144)*n^4 + (9/16)*n^3 + (241/180)*n^2 + (11/12)*n + 1 = 1 + n*(n+1)*(n^4 + 14*n^3 + 101*n^2 + 304*n + 660)/720.

Empirical g.f.: 1-x*(x^2-2*x+2)*(x^4-4*x^3+6*x^2-4*x+2) / (x-1)^7. - R. J. Mathar, May 14 2014

Extensions

a(0)=1 prepended by Alois P. Heinz, Feb 11 2024