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 A191788 Triangle read by rows: T(n,k) is the number of length n left factors of Dyck paths having k base pyramids. A base pyramid is a factor of the form U^j D^j (j>0), starting at the horizontal axis.Here U=(1,1) and D=(1,-1). 2
 1, 1, 1, 1, 2, 1, 3, 2, 1, 6, 3, 1, 11, 5, 3, 1, 21, 9, 4, 1, 40, 17, 8, 4, 1, 76, 31, 13, 5, 1, 146, 62, 26, 12, 5, 1, 279, 113, 45, 18, 6, 1, 539, 228, 94, 39, 17, 6, 1, 1036, 419, 165, 64, 24, 7, 1, 2011, 845, 348, 140, 57, 23, 7, 1, 3883, 1568, 618, 237, 89, 31, 8, 1, 7566, 3160, 1298, 521, 205, 81, 30, 8, 1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS Row n contains 1+floor(n/2) entries. Sum of entries in row n is binomial(n,floor(n/2))=A001405(n). T(n,0) = A191789(n). Sum(k*T(n,k), k>=0) = A191790(n). LINKS Table of n, a(n) for n=0..80. FORMULA G.f.: G(t,z)=(1-z^2)/((1-z*c)*(1-z^2*c+z^4*c-t*z^2)), where c=(1-sqrt(1-4*z^2))/(2*z^2). EXAMPLE T(6,2)=3 because we have (UD)(UD)UU, (UD)(UUDD), and (UUDD)(UD) (the base pyramids are shown between parentheses). Triangle starts: 1; 1; 1,1; 2,1; 3,2,1; 6,3,1; 11,5,3,1; 21,9,4,1; MAPLE c := ((1-sqrt(1-4*z^2))*1/2)/z^2: G := (1-z^2)/((1-z*c)*(1-z^2*c+z^4*c-t*z^2)): Gser := simplify(series(G, z = 0, 20)): for n from 0 to 16 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 16 do seq(coeff(P[n], t, k), k = 0 .. floor((1/2)*n)) end do; # yields sequence in triangular form CROSSREFS Cf. A001405, A191789, A191790 Sequence in context: A305299 A308701 A191528 * A070979 A363272 A054098 Adjacent sequences: A191785 A191786 A191787 * A191789 A191790 A191791 KEYWORD nonn,tabf AUTHOR Emeric Deutsch, Jun 18 2011 STATUS approved

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Last modified May 21 17:21 EDT 2024. Contains 372738 sequences. (Running on oeis4.)