A006478 -id:A006478 - cp's OEIS frontend

A113684 Expansion of x(1-x^2-x^3)/((1-x)(1-x-x^2))^2.

0, 1, 4, 11, 25, 52, 102, 193, 356, 645, 1153, 2040, 3580, 6241, 10820, 18671, 32089, 54956, 93826, 159745, 271300, 459721, 777409, 1312176, 2211000, 3719617, 6248452, 10482323, 17562841, 29391460, 49132638, 82048705, 136884260

Offset: 0

Paul Barry, Nov 05 2005

Index entries for linear recurrences with constant coefficients, signature (4,-4,-2,4,0,-1)

Cf. A045925, A006478, A023610.

a(n)=4a(n-1)-4a(n-2)-2a(n-3)+4a(n-4)-a(n-6); a(n)=sum{k=0..n, (n-k)*C(n-k, k+1)}; a(n)=n*(F(n+2)-1)-(1+((n-5)*F(n-1)+(3n-8)*F(n))/5).

A334658 Triangular array read by rows. T(n,k) is the number of length n words on alphabet {0,1} with k maximal runs of 0's having length 2 or more, n>=0, 0<=k<=nearest integer to n/3.

Original entry on oeis.org

1, 2, 3, 1, 5, 3, 8, 8, 13, 18, 1, 21, 38, 5, 34, 76, 18, 55, 147, 53, 1, 89, 277, 139, 7, 144, 512, 336, 32, 233, 932, 766, 116, 1, 377, 1676, 1670, 364, 9, 610, 2984, 3516, 1032, 50, 987, 5269, 7198, 2714, 215, 1, 1597, 9239, 14402, 6734, 785, 11

Offset: 0

Geoffrey Critzer, Jul 25 2020

			1,
2,
3,  1,
5,  3,
8,  8,
13, 18,  1,
21, 38,  5,
34, 76,  18,
55, 147, 53, 1
T(6,2) = 5 because we have: 000100, 001000, 001001, 001100, 100100.

Sergi Elizalde, Johnny Rivera Jr., and Yan Zhuang, Counting pattern-avoiding permutations by big descents, arXiv:2408.15111 [math.CO], 2024. See p. 6.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 304.

Cf. A000045 (column k=0) A006478 (column k=1).

Row sums give A000079.

nn = 15; c[z_, u_] := ((1 - z^r)/(1 - z) + u z^r/(1 - z))*1/(1 - z ((1 - z^r)/(1 - z) + u z^r/(1 - z))) /. r -> 2; Map[Select[#, # > 0 &] &, CoefficientList[Series[c[z, u], {z, 0, nn}], {z, u}]] // Grid

O.g.f.: ((u x^2)/(1 - x) + (1 - x^2)/(1 - x))/(1 - x ((u x^2)/(1 - x) + (1 - x^2)/(1 - x))).

Generally, the o.g.f. for such words having maximal runs of length at least r is: ((u x^r)/(1 - x) + (1 - x^r)/(1 - x))/(1 - x ((u x^r)/(1 - x) + (1 - x^r)/(1 - x))).

cp's OEIS Frontend

A113684 Expansion of x(1-x^2-x^3)/((1-x)(1-x-x^2))^2.

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