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 A132277 Triangle read by rows: T(n,k) is number of paths in the first quadrant from (0,0) to (n,0) using steps U=(1,1), D=(1,-1), h=(1,0) and H=(2,0), having exactly k h-steps. 2
 1, 0, 1, 2, 0, 1, 0, 5, 0, 1, 6, 0, 9, 0, 1, 0, 25, 0, 14, 0, 1, 22, 0, 66, 0, 20, 0, 1, 0, 129, 0, 140, 0, 27, 0, 1, 90, 0, 450, 0, 260, 0, 35, 0, 1, 0, 681, 0, 1210, 0, 441, 0, 44, 0, 1, 394, 0, 2955, 0, 2765, 0, 700, 0, 54, 0, 1, 0, 3653, 0, 9625, 0, 5642, 0, 1056, 0, 65, 0, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS T(2n+1,0)=0; T(2n,0)=A006318(n) (the large Schroeder numbers). Row sums yield A128720. Sum(k*T(n,k),k=0..n)=A106053(n+1). LINKS Table of n, a(n) for n=0..77. Paul Barry, On Motzkin-Schröder Paths, Riordan Arrays, and Somos-4 Sequences, J. Int. Seq. (2023) Vol. 26, Art. 23.4.7. FORMULA G.f. G=G(t,z) satisfies G = 1 + tzG + z^2*G + z^2*G^2 (see explicit expression at the Maple program). EXAMPLE T(4,2)=9 because we have hhH, hhUD, hHh, hUDh, Hhh, UDhh, hUhD, UhDh and UhhD. MAPLE G:=((1-t*z-z^2-sqrt((1-2*z-t*z-z^2)*(1+2*z-t*z-z^2)))*1/2)/z^2: Gser:=simplify(series(G, z=0, 15)): for n from 0 to 11 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 11 do seq(coeff(P[n], t, j), j=0..n) end do; # yields sequence in triangular form CROSSREFS Cf. A006318, A128720, A106053. Sequence in context: A276193 A357400 A238618 * A137286 A180048 A128890 Adjacent sequences: A132274 A132275 A132276 * A132278 A132279 A132280 KEYWORD nonn,tabl AUTHOR Emeric Deutsch, Aug 26 2007 STATUS approved

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Last modified February 21 01:50 EST 2024. Contains 370219 sequences. (Running on oeis4.)