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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. 9
 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 (list; graph; refs; listen; history; text; internal format)
 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 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 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. Index entries for linear recurrences with constant coefficients, signature (2,1,0,-3,-2,-1). 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 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 (Sage) [( 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 Cf. A000073, A118390. Sequence in context: A116726 A228078 A125180 * A033490 A221950 A116716 Adjacent sequences: A073775 A073776 A073777 * A073779 A073780 A073781 KEYWORD easy,nonn,changed 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

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Last modified September 10 16:42 EDT 2024. Contains 375791 sequences. (Running on oeis4.)