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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. 6

%I

%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.

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

%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 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]).

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

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

%F 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

%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]

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

%e ...

%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. A176703, A056010.

%K nonn

%O 1,4

%A _Clark Kimberling_

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