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