A261711 Triangle read by rows: T(n,k) is the number of words over alphabet {0,1,2,3} having exactly k occurrences of the string 01, where n>=0 and k>=0.

1, 4, 15, 1, 56, 8, 209, 46, 1, 780, 232, 12, 2911, 1091, 93, 1, 10864, 4912, 592, 16, 40545, 21468, 3366, 156, 1, 151316, 91824, 17784, 1200, 20, 564719, 386373, 89238, 8010, 235, 1, 2107560, 1604984, 430992, 48624, 2120, 24, 7865521, 6598282, 2021103, 275724, 16255, 330, 1

Offset: 0

Author

Ran Pan, Aug 29 2015

Keywords

• easy
• nonn
• tabf

Examples

			1
4
15 1
56 8
209 46 1
780 232 12
2911 1091 93 1
10864 4912 592 16
40545 21468 3366 156 1
151316 91824 17784 8010 20
		

Links

• Alois P. Heinz, Rows n = 0..200, flattened
• Rigoberto Flórez, Leandro Junes, and José L. Ramírez, Further Results on Paths in an n-Dimensional Cubic Lattice, Journal of Integer Sequences, Vol. 21 (2018), Article 18.1.2.
• Ran Pan, Problem 5, Project P.

Crossrefs

Column k=0 is A001353(n+1). The triangle is shifted from A207823.

Programs

Maple

b:= proc(n, t) option remember; expand(`if`(n=0, 1,
      add(b(n-1, i)*`if`(t=1 and i=2, x, 1), i=1..4)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0)):
seq(T(n), n=0..12);  # Alois P. Heinz, Aug 29 2015

Mathematica

CoefficientList[#, y]& /@ CoefficientList[1/(1-4x-(y-1)x^2) + O[x]^13, x] // Flatten (* Jean-François Alcover, Jan 10 2023 *)

Formula

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