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A025262 a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ...+ a(n-1)*a(1) for n >= 4. 8
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

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.

M. Somos, Number Walls in Combinatorics.

Fumitaka Yura, Hankel Determinant Solution for Elliptic Sequence, arXiv:1411.6972 [nlin.SI], (25-November-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

Comment from Gary W. Adamson, Oct 27 2008: 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]).

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

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

Conjecture: n*a(n) +(n+1)*a(n-1) +10*(-2*n+5)*a(n-2) +2*(2*n-9)*a(n-3) +10*(2*n-11)*a(n-4)=0. - R. J. Mathar, Nov 26 2012

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

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 *)

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.

Sequence in context: A199103 A057198 * A056010 A002712 A192679 A193418

Adjacent sequences:  A025259 A025260 A025261 * A025263 A025264 A025265

KEYWORD

nonn

AUTHOR

Clark Kimberling

STATUS

approved

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Last modified December 4 23:10 EST 2016. Contains 278755 sequences.