A129170 -id:A129170 - cp's OEIS frontend

A129171 Sum of the heights of the peaks in all skew Dyck paths of semilength n.

0, 1, 6, 32, 165, 840, 4251, 21443, 107946, 542680, 2725635, 13679997, 68623176, 344090307, 1724754180, 8642952000, 43300971885, 216895107480, 1086253033035, 5439405705125, 27234492215400, 136345625309965, 682531666024170

Offset: 0

Emeric Deutsch, Apr 07 2007

nonn

A skew Dyck path is a path in the first quadrant which begins at the origin, ends on the x-axis, consists of steps U=(1,1) (up), D=(1,-1) (down) and L=(-1,-1) (left) so that up and left steps do not overlap. The length of the path is defined to be the number of its steps.

			a(2)=6 because in the 3 skew Dyck paths of semilength 2, namely UDUD, UUDD and UUDL, the heights of the peaks are 1,1,2 and 2.

G. C. Greubel and Vincenzo Librandi, Table of n, a(n) for n = 0..1000(terms 1..300 from Vincenzo Librandi)
E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203.

Cf. A129170.

G:=z*(3-3*z-sqrt(1-6*z+5*z^2))/(1-6*z+5*z^2)/2: Gser:=series(G,z=0,30): seq(coeff(Gser,z,n),n=0..27);

CoefficientList[Series[x*(3 - 3*x - Sqrt[1 - 6*x + 5*x^2])/(1 - 6*x + 5*x^2)/2, {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 20 2012 *)

z='z+O('z^25); concat([0], Vec(z*(3-3*z-sqrt(1-6*z+5*z^2))/(1-6*z+5*z^2)/2)) \\ G. C. Greubel, Feb 10 2017

a(n) = Sum_{k=0,..,n} k*A129170(n,k).
G.f.: z*(3-3*z-sqrt(1-6*z+5*z^2))/(1-6*z+5*z^2)/2. - corrected by Vaclav Kotesovec, Oct 20 2012
Recurrence: (n-1)*a(n) = (11*n-19)*a(n-1) - 5*(7*n-17)*a(n-2) + 25*(n-3)*a(n-3) . - Vaclav Kotesovec, Oct 20 2012
a(n) ~ 3*5^(n-1)/2*(1-sqrt(5)/(6*sqrt(Pi*n))) . - Vaclav Kotesovec, Oct 20 2012

A128888 Table with g.f. [1-xn-sqrt(x^2n^2-2nx+1+4x^2-4x)]/(2*x).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 3, 8, 10, 0, 1, 4, 15, 36, 36, 0, 1, 5, 24, 84, 176, 137, 0, 1, 6, 35, 160, 510, 912, 543, 0, 1, 7, 48, 270, 1152, 3279, 4928, 2219, 0, 1, 8, 63, 420, 2240, 8768, 21975, 27472, 9285, 0, 1, 9, 80, 616, 3936, 19605, 69504, 151905, 156864

Offset: 0

R. J. Mathar, Apr 19 2007

Column m=2 is essentially the same as A005563 or A067998 or A106230. Row n=1 is essentially the same as A025238 and A002212. The table is read along diagonals and provides the Taylor coefficient of x^m in column m. It also is the slice t=1 through the trivariate g.f. defined in A129170, which provides an implicit proof that all values are nonnegative.

			Table with rows n>=0 and columns m>=0 starts
1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
1, 1, 3, 10, 36, 137, 543, 2219, 9285, 39587, 171369, ...
1, 2, 8, 36, 176, 912, 4928, 27472, 156864, 912832, 5394176, ...
1, 3, 15, 84, 510, 3279, 21975, 151905, 1075425, 7758777, 56839965, ...
1, 4, 24, 160, 1152, 8768, 69504, 568064, 4753920, 40537088, 350963712, ...
1, 5, 35, 270, 2240, 19605, 178535, 1675495, 16095765, 157527055, 1565170985, ...
1, 6, 48, 420, 3936, 38832, 398208, 4205904, 45459840, 500488512, 5593373184, ...
1, 7, 63, 616, 6426, 70427, 801423, 9387917, 112501809, 1372985957, 17007257421,...

Cf. A005563, A002212, A129170.

H := proc(n,x) (-x*n+1-(x^2*n^2-2*n*x+1+4*x^2-4*x)^(1/2))/(2*x) ; end: T := proc(n,m) coeftayl( H(n,x),x=0,m) ; end: for diag from 0 to 20 do for m from 0 to diag do n := diag-m ; printf("%d, ",T(n,m)) ; od ; od;

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A129171 Sum of the heights of the peaks in all skew Dyck paths of semilength n.

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A128888 Table with g.f. [1-xn-sqrt(x^2n^2-2nx+1+4x^2-4x)]/(2*x).

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cp's OEIS Frontend

A129171 Sum of the heights of the peaks in all skew Dyck paths of semilength n.

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Author

Keywords

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Examples

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Programs

Maple

Mathematica

PARI

Formula

A128888 Table with g.f. [1-x*n-sqrt(x^2*n^2-2*n*x+1+4*x^2-4*x)]/(2*x).

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Maple

A128888 Table with g.f. [1-xn-sqrt(x^2n^2-2nx+1+4x^2-4x)]/(2*x).