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A073778 Sum_{k=0..n} T(k)*T(n-k), convolution of A000073 with itself. 6
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

0,4

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

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

D. Birmajer, J. Gil and M. Weiner, Linear recurrence sequences and their convolutions via Bell polynomials, arXiv:1405.7727 [math.CO], 2014  and J. Int. Seq. 18 (2015) # 15.1.2.

Dairyko, Michael; Tyner, Samantha; Pudwell, Lara; Wynn, Casey. 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, Helmut Prodinger, A Note on the Conjecture of Ramirez and Sirvent, J. Int. Seq. 17 (2014) # 14.5.8

J. L. Ramirez and V. 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^2/(1 - x - x^2 - x^3)^2.

a(n) = Sum_{k=0..n-2} (k+1)*Sum_{j=0..k} binomial(j,n-3*k+2*j-2)*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^2/(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^2/(1-x-x^2-x^3)^2, {x, 0, 40}], x]

PROG

(Sage) taylor( x/(1-x-x^2-x^3)^2 , x, 0, 32) # Zerinvary Lajos, Jun 02 2009

(Maxima)

a(n):=sum((k+1)*sum(binomial(j, n-3*k+2*j-2)*binomial(k, j), j, 0, k) , k, 0, n-2); /* Vladimir Kruchinin, Dec 14 2011 */

CROSSREFS

Cf. A000073, A118390.

Sequence in context: A116726 A228078 A125180 * A033490 A221950 A116716

Adjacent sequences:  A073775 A073776 A073777 * A073779 A073780 A073781

KEYWORD

easy,nonn

AUTHOR

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

STATUS

approved

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Last modified January 23 03:02 EST 2021. Contains 340384 sequences. (Running on oeis4.)