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A247296 Number of uhd and uHd in all weighted lattice paths B(n). 2
0, 0, 0, 0, 1, 3, 7, 18, 45, 112, 281, 706, 1778, 4490, 11363, 28814, 73199, 186257, 474635, 1211122, 3094171, 7913765, 20261142, 51921920, 133171656, 341836748, 878104607, 2257208148, 5805964495, 14942942127, 38480449261, 99145105834, 255573465001, 659114680270 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,6
COMMENTS
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.
a(n) = A110320(n-3) + A110320(n-4) (n>=5).
LINKS
M. Bona and A. Knopfmacher, On the probability that certain compositions have the same number of parts, Ann. Comb., 14 (2010), 291-306.
FORMULA
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.
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
EXAMPLE
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).
G.f. = x^4 + 3*x^5 + 7*x^6 + 18*x^7 + 45*x^8 + 112*x^9 + 281*x^10 + ...
MAPLE
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);
MATHEMATICA
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 *)
PROG
(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
(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
CROSSREFS
Sequence in context: A291734 A291229 A036883 * A191652 A191826 A318899
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Sep 16 2014
STATUS
approved

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Last modified March 3 00:46 EST 2024. Contains 370499 sequences. (Running on oeis4.)