A025262 a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ... + a(n-1)*a(1) for n >= 4.

1, 1, 1, 3, 8, 23, 68, 207, 644, 2040, 6558, 21343, 70186, 232864, 778550, 2620459, 8872074, 30195288, 103246502, 354508628, 1221846856, 4225644866, 14659644348, 51002664023, 177909901566, 622093882290, 2180123564130, 7656055966092

Offset

1,4

Comments

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

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

Links

Seiichi Manyama, Table of n, a(n) for n = 1..1766

Paul Barry, On the Hurwitz Transform of Sequences, Journal of Integer Sequences, Vol. 15 (2012), #12.8.7.

Paul Barry, Integer sequences from elliptic curves, arXiv:2306.05025 [math.NT], 2023.

David Callan, On Ascent, Repetition and Descent Sequences, arXiv:1911.02209 [math.CO], 2019.

Michael Somos, Number Walls in Combinatorics.

Fumitaka Yura, Hankel Determinant Solution for Elliptic Sequence, arXiv:1411.6972 [nlin.SI], 2014; see p. 7.

Formula

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

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

Row sums of A176703 if offset 0. - Michael Somos, Jan 09 2012

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

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

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

Recurrence: n*a(n) = 2*(2*n-3)*a(n-1) - 2*(2*n-9)*a(n-3). - Vaclav Kotesovec, Jan 25 2015

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

Example

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 + ...

Mathematica

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 *)
Nest[Append[#, #.Reverse[#]] &, {1, 1, 1}, 25] (* Jan Mangaldan, Jul 07 2020 *)

Prog

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

Crossrefs

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

Sequence in context: A230122 A199103 A057198 * A056010 A002712 A192679

Adjacent sequences: A025259 A025260 A025261 * A025263 A025264 A025265

Keyword

nonn

Author

Clark Kimberling

Status

approved