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A247296 Number of uhd and uHd in all weighted lattice paths B(n).

%I A247296 #21 Sep 08 2022 08:46:09
%S A247296 0,0,0,0,1,3,7,18,45,112,281,706,1778,4490,11363,28814,73199,186257,
%T A247296 474635,1211122,3094171,7913765,20261142,51921920,133171656,341836748,
%U A247296 878104607,2257208148,5805964495,14942942127,38480449261,99145105834,255573465001,659114680270
%N A247296 Number of uhd and uHd in all weighted lattice paths B(n).
%C A247296 B(n) is the set of lattice paths of weight n that start in (0,0), end on the horizontal axis and never go below this axis, whose steps are of the following four kinds: h = (1,0) of weight 1, H = (1,0) of weight 2, u = (1,1) of weight 2, and d = (1,-1) of weight 1. The weight of a path is the sum of the weights of its steps.
%C A247296 a(n) = A110320(n-3) + A110320(n-4) (n>=5).
%H A247296 G. C. Greubel, <a href="/A247296/b247296.txt">Table of n, a(n) for n = 0..1000</a>
%H A247296 M. Bona and A. Knopfmacher, <a href="http://dx.doi.org/10.1007/s00026-010-0060-7">On the probability that certain compositions have the same number of parts</a>, Ann. Comb., 14 (2010), 291-306.
%F A247296 G.f.: G = z^4*(1 + z)*g/(1 - z - z^2 - 2*z^3*g), where g = 1 + z*g + z^2*g + z^3*g^2.
%F A247296 D-finite with recurrence +(n-1)*(202*n-903)*a(n) +(-250*n^2+1095*n-691)*a(n-1) +(-510*n^2+4095*n-8039)*a(n-2) +(-558*n^2+4287*n-7831)*a(n-3) +(-106*n^2+1587*n-4575)*a(n-4) +(154*n-547)*(n-7)*a(n-5)=0. - _R. J. Mathar_, Jul 24 2022
%e A247296 a(6)=7 because among the 37 (=A004148(7)) members of B(6) only (uhd)hh, h(uhd)h, hh(uhd), H(uhd), (uhd)H, (uHd)h, and h(uHd) contain uhd or uHd (shown between parentheses).
%e A247296 G.f. = x^4 + 3*x^5 + 7*x^6 + 18*x^7 + 45*x^8 + 112*x^9 + 281*x^10 + ...
%p A247296 eqg := g = 1+z*g+z^2*g+z^3*g^2: g := RootOf(eqg, g): H := z^4*(1+z)*g/(1-z-z^2-2*z^3*g): Hser := series(H, z = 0, 40): seq(coeff(Hser, z, n), n = 0 .. 35);
%t A247296 a[ n_] := With[{t = (1 - 3 x + x^2) (1 + x + x^2)}, SeriesCoefficient[ x (x + 1) (-1 + (1 - x - x^2) / Sqrt[t]) / 2, {x, 0, n}]]; (* _Michael Somos_, Sep 16 2014 *)
%o A247296 (PARI) x='x+O('x^30); concat(vector(4), Vec(x*(x+1)*(-1 +(1-x-x^2 )/sqrt((1-3*x+x^2)*(1+x+x^2)))/2)) \\ _G. C. Greubel_, Aug 05 2018
%o A247296 (Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); [0,0,0,0] cat Coefficients(R!(x*(x+1)*(-1 +(1-x-x^2 )/sqrt((1-3*x+x^2)*(1+x+x^2)) )/2)); // _G. C. Greubel_, Aug 05 2018
%Y A247296 Cf. A004148, A110320, A247294.
%K A247296 nonn
%O A247296 0,6
%A A247296 _Emeric Deutsch_, Sep 16 2014