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URL: https://oeis.org/A073778

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A073778
a(n) = Sum_{k=0..n} T(k)*T(n-k), where T is A000073; convolution of A000073 with itself.
11
0, 0, 0, 0, 1, 2, 5, 12, 26, 56, 118, 244, 499, 1010, 2027, 4040, 8004, 15776, 30956, 60504, 117845, 228818, 443057, 855732, 1649022, 3171128, 6086626, 11662252, 22309543, 42614178, 81286743, 154856528, 294660040, 560052736, 1063367384, 2017030256
OFFSET
0,6
COMMENTS
Number of binary sequences of length n+1 that have exactly one subsequence 000. Example: a(4)=5 because we have 00010,00011,01000,10001 and 11000. Column 1 of A118390. - Emeric Deutsch, Apr 27 2006
Let (b(n)) be the p-INVERT of (1,1,1,0,0,0,0,0,0,...) using p(S) = 1 - S^2; then b(n) = a(n+3) for n >= 0. See A292324. - Clark Kimberling, Sep 15 2017
a(n) is the number of binary strings of length n that contain exactly 4 ones in runs of ones of length >=3. - Félix Balado, Feb 26 2026
LINKS
Michael De Vlieger, Table of n, a(n) for n = 0..3770, terms n = 2..1002 from Vincenzo Librandi.
Daniel Birmajer, Juan B. Gil, and Michael D. Weiner, Linear recurrence sequences and their convolutions via Bell polynomials, arXiv:1405.7727 [math.CO], 2014; see also, J. Int. Seq. 18 (2015) # 15.1.2.
Michael Dairyko, Lara Pudwell, Samantha Tyner and Casey Wynn, Non-contiguous pattern avoidance in binary trees, Electron. J. Combin. 19 (2012), no. 3, Paper 22, 21 pp. MR2967227. - From N. J. A. Sloane, Feb 01 2013
Omar Khadir, László Németh, and László Szalay, Tiling of dominoes with ranked colors, Results in Math. (2024) Vol. 79, Art. No. 253. See p. 2.
Emrah Kilic and Helmut Prodinger, A Note on the Conjecture of Ramirez and Sirvent, J. Int. Seq. 17 (2014) # 14.5.8.
László Németh and László Szalay, Explicit solution of system of two higher-order recurrences, arXiv:2408.12196 [math.NT], 2024. See p. 10.
José L. Ramirez and Víctor F. Sirvent, Incomplete Tribonacci Numbers and Polynomials, Journal of Integer Sequences, Vol. 17, 2014, #14.4.2.
FORMULA
G.f.: x^4/(1 - x - x^2 - x^3)^2.
a(n) = Sum_{k=0..n-4} (k+1)*Sum_{j=0..k} binomial(j,n-3*k+2*j-4)*binomial(k,j). - Vladimir Kruchinin, Dec 14 2011
(n-2)*a(n) - (n-1)*a(n-1) - n*a(n-2) - (n+1)*a(n-3) = 0, n > 2. - Michael D. Weiner, Nov 18 2014
G.f.: (1 - x*y)/x/(1 - x (y+1) + x^2*(y-1) + x^3*(y-1) - x^4*y*(y^2-1)); a(n) is the coefficient of x^n*y^4 in this bivariate g.f. - Félix Balado, Feb 26 2026
MAPLE
A073778:=proc(n) coeftayl(x^4/(1-x-x^2-x^3)^2, x=0, n); end proc: seq(A073778(n), n=0..40); # Wesley Ivan Hurt, Nov 17 2014
MATHEMATICA
CoefficientList[Series[x^4/(1-x-x^2-x^3)^2, {x, 0, 40}], x]
PROG
(SageMath) [( x^4/(1-x-x^2-x^3)^2 ).series(x, n+1).list()[n] for n in (0..40)] # Zerinvary Lajos, Jun 02 2009; modified by G. C. Greubel, Dec 15 2021
(Maxima)
a(n):= sum((k+1)*sum(binomial(j, n-3*k+2*j-4)*binomial(k, j), j, 0, k), k, 0, n-4);
makelist(a(n), n, 0, 30); /* Vladimir Kruchinin, Dec 14 2011 */
(PARI) T(n) = ([0, 1, 0; 0, 0, 1; 1, 1, 1]^n)[1, 3]; \\ A000073
a(n) = sum(k=0, n, T(k)*T(n-k)); \\ Michel Marcus, Oct 20 2021
CROSSREFS
Sequence in context: A116726 A228078 A125180 * A381858 A033490 A221950
KEYWORD
easy,nonn
AUTHOR
Mario Catalani (mario.catalani(AT)unito.it), Aug 10 2002
EXTENSIONS
Two initial zeros inserted by Hans J. H. Tuenter, Oct 20 2021
STATUS
approved