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 A167634 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n, having no ascents and no descents of length 1, and having k peaks at odd level. 3
 1, 0, 1, 0, 1, 2, 0, 0, 3, 1, 5, 2, 1, 1, 10, 5, 1, 14, 12, 9, 2, 7, 38, 27, 9, 1, 43, 60, 57, 22, 3, 36, 156, 146, 69, 15, 1, 143, 284, 326, 176, 45, 4, 166, 672, 784, 482, 155, 23, 1, 509, 1320, 1780, 1224, 453, 82, 5, 731, 2981, 4162, 3160, 1354, 313, 33, 1, 1915, 6104 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 COMMENTS Sum of entries in row n is the secondary structure number A004148(n-1) (n >= 2). Row n contains ceiling(n/2) entries (n >= 1). T(n,0) = A167635(n). Sum_{k>=0} k*T(n,k) = A167636(n). LINKS FORMULA G.f.: G=G(t,z) satisfies z(1+z-z^2)G^2-(1+z-z^2)(1+z-tz^2)G + 1+z-tz^2=0. The trivariate g.f. G=G(t,s,z), where t marks odd-level peaks, s marks even-level peaks, and z marks semilength, satisfies aG^2 - bG + c = 0, where a = z(1+z-sz^2), b=(1+z-tz^2)(1+z-sz^2), c=1+z-tz^2. EXAMPLE T(5,1)=3 because we have UUDDUU(UD)DD, UU(UD)DDUUDD, and UUUU(UD)DDDD (the odd-level peaks are shown between parentheses). Triangle starts:    1;    0;    1;    0,  1;    2,  0;    0,  3,  1;    5,  2,  1;    1, 10,  5,  1;   14, 12,  9,  2; MAPLE eq := z*(1+z-z^2)*G^2-(1+z-z^2)*(1+z-t*z^2)*G+1+z-t*z^2 = 0: G := RootOf(eq, G): Gser := simplify(series(G, z = 0, 20)): for n from 0 to 16 do P[n] := sort(expand(coeff(Gser, z, n))) end do: 1; for n to 16 do seq(coeff(P[n], t, j), j = 0 .. ceil((1/2)*n)-1) end do; # yields sequence in triangular form CROSSREFS Cf. A004148, A167635, A167636, A167637. Sequence in context: A188023 A077888 A319668 * A174169 A248174 A125095 Adjacent sequences:  A167631 A167632 A167633 * A167635 A167636 A167637 KEYWORD nonn,tabf AUTHOR Emeric Deutsch, Nov 08 2009 STATUS approved

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Last modified July 8 04:16 EDT 2020. Contains 335504 sequences. (Running on oeis4.)