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A167637 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 even level. 3
1, 0, 0, 1, 1, 0, 0, 1, 1, 2, 2, 0, 1, 3, 3, 1, 5, 8, 4, 0, 5, 13, 12, 6, 1, 15, 32, 27, 8, 0, 21, 59, 61, 33, 10, 1, 51, 134, 147, 76, 15, 0, 85, 267, 327, 208, 75, 15, 1, 188, 584, 771, 528, 186, 26, 0, 344, 1209, 1734, 1329, 585, 150, 21, 1, 730, 2608, 4008, 3344, 1595, 408 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,10

COMMENTS

Sum of entries in row n is the secondary structure number A004148(n-1) (n>=2).

Row n contains 1 + floor(n/2) entries.

T(n,0) = A167638(n).

Sum_{k>=0} k*T(n,k) = A167639(n).

LINKS

Table of n, a(n) for n=0..69.

FORMULA

G.f.: G=G(t,z) satisfies z(1+z-tz^2)G^2-(1+z-z^2)(1+z-tz^2)G + 1+z-z^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(6,2)=3 because we have U(UD)DUUU(UD)DDD, UUU(UD)DDDU(UD)D, and UUU(UD)DU(UD)DDD (the even-level peaks are shown between parentheses).

Triangle starts:

  1;

  0;

  0,  1;

  1,  0;

  0,  1,  1;

  2,  2,  0;

  1,  3,  3,  1;

  5,  8,  4,  0;

  5, 13, 12,  6,  1;

MAPLE

eq := z*(1+z-t*z^2)*G^2-(1+z-z^2)*(1+z-t*z^2)*G+1+z-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: for n from 0 to 16 do seq(coeff(P[n], t, j), j = 0 .. floor((1/2)*n)) end do; # yields sequence in triangular form

CROSSREFS

Cf. A004148, A167634, A167638, A167639.

Sequence in context: A276554 A297323 A257654 * A109754 A220074 A059259

Adjacent sequences:  A167634 A167635 A167636 * A167638 A167639 A167640

KEYWORD

nonn,tabf

AUTHOR

Emeric Deutsch, Nov 08 2009

STATUS

approved

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Last modified February 17 14:12 EST 2018. Contains 299296 sequences. (Running on oeis4.)