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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

%I #91 Dec 15 2021 01:33:14

%S 0,0,0,0,1,2,5,12,26,56,118,244,499,1010,2027,4040,8004,15776,30956,

%T 60504,117845,228818,443057,855732,1649022,3171128,6086626,11662252,

%U 22309543,42614178,81286743,154856528,294660040,560052736,1063367384,2017030256

%N a(n) = Sum_{k=0..n} T(k)*T(n-k), where T is A000073; convolution of A000073 with itself.

%C 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

%C 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

%H Michael De Vlieger, <a href="/A073778/b073778.txt">Table of n, a(n) for n = 0..3770</a>, terms n = 2..1002 from Vincenzo Librandi.

%H D. Birmajer, J. Gil and M. Weiner, <a href="http://arxiv.org/abs/1405.7727">Linear recurrence sequences and their convolutions via Bell polynomials</a>, arXiv:1405.7727 [math.CO], 2014; see <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Gil/gil3.html">also</a>, J. Int. Seq. 18 (2015) # 15.1.2.

%H Michael Dairyko, Lara Pudwell, Samantha Tyner and Casey Wynn. <a href="https://doi.org/10.37236/2099">Non-contiguous pattern avoidance in binary trees</a>. Electron. J. Combin. 19 (2012), no. 3, Paper 22, 21 pp. MR2967227. - From _N. J. A. Sloane_, Feb 01 2013

%H Emrah Kilic and Helmut Prodinger, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Kilic/kilic12.html">A Note on the Conjecture of Ramirez and Sirvent</a>, J. Int. Seq. 17 (2014) # 14.5.8.

%H J. L. Ramirez and V. F. Sirvent, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Ramirez/ramirez4.html">Incomplete Tribonacci Numbers and Polynomials</a>, Journal of Integer Sequences, Vol. 17, 2014, #14.4.2.

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,0,-3,-2,-1).

%F G.f.: x^4/(1 - x - x^2 - x^3)^2.

%F 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

%F (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

%p 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

%t CoefficientList[Series[x^4/(1-x-x^2-x^3)^2, {x, 0, 40}], x]

%o (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

%o (Maxima)

%o a(n):= sum((k+1)*sum(binomial(j,n-3*k+2*j-4)*binomial(k,j), j,0,k), k,0,n-4);

%o makelist(a(n), n, 0, 30); /* _Vladimir Kruchinin_, Dec 14 2011 */

%o (PARI) T(n) = ([0, 1, 0; 0, 0, 1; 1, 1, 1]^n)[1, 3]; \\ A000073

%o a(n) = sum(k=0, n, T(k)*T(n-k)); \\ _Michel Marcus_, Oct 20 2021

%Y Cf. A000073, A118390.

%K easy,nonn

%O 0,6

%A Mario Catalani (mario.catalani(AT)unito.it), Aug 10 2002

%E Two initial zeros inserted by _Hans J. H. Tuenter_, Oct 20 2021

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Last modified June 7 05:23 EDT 2024. Contains 373144 sequences. (Running on oeis4.)