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A211278 a(n) is the number of n-lettered words in the alphabet {1, 2, 3} with as many occurrences of the substring (consecutive subword) [1, 1] as of [2, 3].

%I A211278 #28 Apr 10 2025 23:22:25
%S A211278 1,3,7,16,38,95,248,668,1838,5131,14470,41112,117475,337203,971515,
%T A211278 2807744,8136090,23630215,68768210,200481036,585381973,1711647959,
%U A211278 5011157073,14687848012,43095321203,126565380735,372030471493,1094437253428,3221999290418,9492019319771,27981390048004
%N A211278 a(n) is the number of n-lettered words in the alphabet {1, 2, 3} with as many occurrences of the substring (consecutive subword) [1, 1] as of [2, 3].
%C A211278 Define a triangle by T(n, n) = n + 1 and T(n, 0) = 1 for n >= 0, and T(r, c) = T(r, c-1) + T(r-1, c-1) + T(r-2, c-1). The sum of the terms in row n is a(n). - _J. M. Bergot_, Mar 01 2013
%H A211278 Shalosh B. Ekhad and Doron Zeilberger, <a href="http://arxiv.org/abs/1112.6207">Automatic Solution of Richard Stanley's Amer. Math. Monthly Problem #11610 and ANY Problem of That Type</a>, arXiv preprint arXiv:1112.6207 [math.CO], 2011. See subpages for rigorous derivations of g.f., recurrence, asymptotics for this sequence.
%F A211278 Conjecture: n*a(n) - 4*n*a(n-1) +2*(n+3)*a(n-2) + 4*(n-3)*a(n-3) + 3*(-n+2)*a(n-4) = 0. - _R. J. Mathar_, Jun 09 2013
%F A211278 From _Mélika Tebni_, Mar 20 2025: (Start)
%F A211278 G.f.: (1 + sqrt((1 + x) / (1 - 3*x))) / (1 - x)^2 / 2.
%F A211278 a(n) = Sum_{k=0..n} binomial(n+1, k+1)*abs(Sum_{j=0..floor(k/2)} (-1)^j*binomial(k, j)).
%F A211278 E.g.f.: (g(x) + Integral_{x=-oo..oo} g(x) dx)*exp(x) where g(x) = Integral_{x=-oo..oo} BesselI(0, 2*x) dx + (1 + BesselI(0, 2*x)) / 2. (End)
%p A211278 gf := (1 + sqrt((1 + x) / (1 - 3*x))) / (1 - x)^2 / 2:
%p A211278 A211278 := n -> coeff(series(gf, x, n+1), x, n):
%p A211278 seq(A211278(n), n = 0 .. 30); # _Mélika Tebni_, Mar 20 2025
%o A211278 (Python)
%o A211278 from math import comb as C
%o A211278 def a(n):
%o A211278     return sum(C(n+1, k+1)*abs(sum((-1)**j*C(k, j) for j in range(k//2+1))) for k in range(n+1))
%o A211278 print([a(n) for n in range(31)]) # _Mélika Tebni_, Mar 20 2025
%K A211278 nonn
%O A211278 0,2
%A A211278 _N. J. A. Sloane_, Apr 07 2012