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 A191785 Triangle read by rows: T(n,k) is the number of length n left factors of Dyck paths having k triple-rises (triple-rise = three consecutive (1,1)-steps). 2
 1, 1, 2, 2, 1, 4, 1, 1, 5, 3, 1, 1, 9, 6, 3, 1, 1, 12, 11, 7, 3, 1, 1, 22, 21, 14, 8, 3, 1, 1, 30, 38, 28, 16, 9, 3, 1, 1, 55, 71, 58, 35, 18, 10, 3, 1, 1, 77, 123, 110, 75, 41, 20, 11, 3, 1, 1, 141, 230, 220, 154, 93, 47, 22, 12, 3, 1, 1, 201, 389, 410, 314, 196, 111, 53, 24, 13, 3, 1, 1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Row n>=2 contains n-1 entries. Sum of entries in row n is binomial(n,floor(n/2))=A001405(n). T(n,0)=A191786(n). Sum(k*T(n,k),k>=0)=A191787(n). LINKS Table of n, a(n) for n=0..79. FORMULA G.f.: G=G(t,z) is given by G = C + z*C*(t*(G - 1 - z - z^2*G) + 1 + z + z^2*G), where C=C(t,z) is the solution of the equation z^2*(t+z^2-t*z^2)*C^2 - (1 - z^2 + t*z^2)*C + 1 = 0. EXAMPLE T(6,2)=3 because we have UD(U[UU)U], (U[UU)U]DD, and (U[UU)U]DU, where U=(1,1) and D=(1,-1) (the triple-rises are shown between parentheses). Triangle starts: 1; 1; 2; 2,1; 4,1,1; 5,3,1,1; 9,6,3,1,1; MAPLE eqC := z^2*(t+z^2-t*z^2)*C^2-(1-z^2+t*z^2)*C+1 = 0: C := RootOf(eqC, C): G := C*(1+z+z^2-t*z-t*z^2)/(1-t*z*C-z^3*C+t*z^3*C): Gser := simplify(series(G, z = 0, 15)): for n from 0 to 13 do P[n] := sort(coeff(Gser, z, n)) end do: 1; 1; for n from 2 to 13 do seq(coeff(P[n], t, k), k = 0 .. n-2) end do; # yields sequence in triangular form CROSSREFS Cf. A001405, A191786, A191787 Sequence in context: A071430 A092514 A106641 * A320500 A140957 A197250 Adjacent sequences: A191782 A191783 A191784 * A191786 A191787 A191788 KEYWORD nonn,tabf AUTHOR Emeric Deutsch, Jun 18 2011 STATUS approved

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Last modified May 27 02:17 EDT 2024. Contains 372847 sequences. (Running on oeis4.)