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a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ... + a(n-1)*a(1) for n >= 4.
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%I #73 Jun 14 2023 17:16:16

%S 1,1,1,3,8,23,68,207,644,2040,6558,21343,70186,232864,778550,2620459,

%T 8872074,30195288,103246502,354508628,1221846856,4225644866,

%U 14659644348,51002664023,177909901566,622093882290,2180123564130,7656055966092

%N a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ... + a(n-1)*a(1) for n >= 4.

%C a(n) is the number of ascent sequences (A022493) of length n-1 such that the nonzero entries are weakly increasing and no two consecutive entries are both 0. For example a(4) = 3 counts 010, 011, 012 and a(5) = 8 counts 0101, 0102, 0110, 0111, 0112, 0120, 0122, 0123. - _David Callan_, Nov 25 2021

%C The o.g.f. y (= x + x^2 + x^3 + ...) of this sequence satisfies y^2 - y = x^3 - x. If y is replaced by -y, then it is the elliptic curve y^2 + y = x^3 - x with LMFDB label 37.a1 (Cremona label 37a1) associated to the Somos-4 sequence via elliptic divisibility sequence A006769. - _Michael Somos_, Apr 18 2023

%H Seiichi Manyama, <a href="/A025262/b025262.txt">Table of n, a(n) for n = 1..1766</a>

%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Barry5/barry223.html">On the Hurwitz Transform of Sequences</a>, Journal of Integer Sequences, Vol. 15 (2012), #12.8.7.

%H Paul Barry, <a href="https://arxiv.org/abs/2306.05025">Integer sequences from elliptic curves</a>, arXiv:2306.05025 [math.NT], 2023.

%H David Callan, <a href="https://arxiv.org/abs/1911.02209">On Ascent, Repetition and Descent Sequences</a>, arXiv:1911.02209 [math.CO], 2019.

%H Michael Somos, <a href="http://grail.eecs.csuohio.edu/~somos/nwic.html">Number Walls in Combinatorics</a>.

%H Fumitaka Yura, <a href="http://arxiv.org/abs/1411.6972">Hankel Determinant Solution for Elliptic Sequence</a>, arXiv:1411.6972 [nlin.SI], 2014; see p. 7.

%F G.f.: (1 - sqrt(1 - 4*x + 4*x^3)) / 2. Satisfies A(x) - A(x)^2 = x - x^3. - _Michael Somos_, Aug 04, 2000

%F Given an integer t >= 1 and initial values u = [a_0, a_1, ..., a_{t-1}], we may define an infinite sequence Phi(u) by setting a_n = a_{n-1} + a_0*a_{n-1} + a_1*a_{n-2} + ... + a_{n-2}*a_1 for n >= t. For example Phi([1]) is the Catalan numbers A000108. The present sequence is Phi([1,1,1]). - _Gary W. Adamson_, Oct 27 2008

%F Row sums of A176703 if offset 0. - _Michael Somos_, Jan 09 2012

%F a(n+2) = A056010(n) if n >= 0.

%F a(n) = Sum_{m=0..floor((n-1)/2)} C(n-2*m-1)*binomial(n-2*m,m)*(-1)^m, where C = A000108 are the Catalan numbers. - _Vladimir Kruchinin_, Jan 26 2013

%F 0 = a(n)*(+16*a(n+1) - 64*a(n+3) + 22*a(n+4)) + a(n+1)*(+32*a(n+2) - 14*a(n+3)) + a(n+2)*(+16*a(n+3) - 10*a(n+4)) + a(n+3)*(+2*a(n+3) + a(n+4)) if n>0. - _Michael Somos_, Jan 18 2015

%F Recurrence: n*a(n) = 2*(2*n-3)*a(n-1) - 2*(2*n-9)*a(n-3). - _Vaclav Kotesovec_, Jan 25 2015

%F a(n) ~ sqrt(3 - 8*r) * (4 - 4*r^2)^n / (4*sqrt(Pi)*n^(3/2)), where r = 2*sin(arccos(-3^(3/2)/8)/3 - Pi/6)/sqrt(3). - _Vaclav Kotesovec_, Jun 05 2022

%e G.f. = x + x^2 + x^3 + 3*x^4 + 8*x^5 + 23*x^6 + 68*x^7 + 207*x^8 + 644*x^9 + ...

%t nmax = 30; aa = ConstantArray[0, nmax]; aa[[1]] = 1; aa[[2]] = 1; aa[[3]] = 1; Do[aa[[n]] = Sum[aa[[k]] * aa[[n - k]], {k, 1, n - 1}], {n, 4, nmax}]; aa (* _Vaclav Kotesovec_, Jan 25 2015 *)

%t Nest[Append[#, #.Reverse[#]] &, {1, 1, 1}, 25] (* _Jan Mangaldan_, Jul 07 2020 *)

%o (PARI) {a(n) = polcoeff( (1 - sqrt(1 - 4*x + 4*x^3 + x * O(x^n))) / 2, n)}; /* _Michael Somos_, Aug 04 2000 */

%Y Cf. A000108, A176703, A056010, A025268, A006769.

%K nonn

%O 1,4

%A _Clark Kimberling_