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A047665 Expansion of (1/sqrt(1-6*x+x^2)-1/(1-x))/2. 7
0, 1, 6, 31, 160, 841, 4494, 24319, 132864, 731281, 4048726, 22523359, 125797984, 704966809, 3961924126, 22321190911, 126027618304, 712917362209, 4039658528934, 22924714957471, 130271906898720, 741188107113961, 4221707080583086, 24070622500965631, 137369104574280960, 784622537295845041 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Previous name was: Main diagonal of square array defined in A047662.

a(n) = total number of weak plateaus in all Schroeder n-paths. A weak plateau is a subpath of the form UFF..FD where there are 0 or more Fs. For example, a(2)=6 counts the following weak plateaus (in parentheses) in the 6 Schroeder 2-paths: (UFD), U(UD)D, FF, (UD)F, F(UD), (UD)(UD). - David Callan, Aug 16 2006

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Y. Ding, R. R. X. Du, Counting Humps in Motzkin Paths, arXiv:1109.2661 (2011) Eq. (4.2).

D. E. Knuth and N. J. A. Sloane, Correspondence, December 1999

FORMULA

2*a(n)+1 = A001850(n).

a(n)-a(n-1) = A002002(n).

a(n) = sum(k=0..n, sum(l=0..n, T(k, l) ) ), array T as in A008288.

a(n) = Sum_{j=1..n} C(2*j-1, j-1)*C(n+j, 2*j). - Stefan Hollos (stefan(AT)exstrom.com), Jul 21 2004

Recurrence: n*(2*n-3)*a(n) = (2*n-1)*(7*n-10)*a(n-1) - (2*n-3)*(7*n-4)*a(n-2) + (n-2)*(2*n-1)*a(n-3). - Vaclav Kotesovec, Oct 08 2012

a(n) ~ sqrt(8+6*sqrt(2))*(3+2*sqrt(2))^n/(8*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 08 2012

a(n) = (hyper2F1(-n,n+1,1,-1)-1)/2 = (hyper2F1(-n, -n, 1, 2)-1)/2. - Peter Luschny, May 19 2015

MAPLE

seq(add(multinomial(n+k, n-k, k, k)/2, k=1..n), n=1..22); # Zerinvary Lajos, Oct 18 2006

a:=n->add(add(binomial(n, j)*binomial(n, k)*binomial(k, j), j=0..n), k=1..n): seq(a(n)/2, n=1..22); # Zerinvary Lajos, Jun 02 2007

MATHEMATICA

Table[SeriesCoefficient[(1/Sqrt[1-6*x+x^2]-1/(1-x))/2, {x, 0, n}], {n, 1, 20}] (* Vaclav Kotesovec, Oct 08 2012 *)

PROG

(PARI) x='x+O('x^66); Vec((1/sqrt(1-6*x+x^2)-1/(1-x))/2) \\ Joerg Arndt, May 04 2013

(Sage)

a = lambda n: (hypergeometric([-n, n+1], [1], -1)-1)/2

[simplify(a(n)) for n in (1..25)] # Peter Luschny, May 19 2015

CROSSREFS

Cf. A001850, A002002  (Schroeder paths interpretation).

Cf. A008288 (Delannoy numbers triangle).

Sequence in context: A038223 A022034 A277669 * A003128 A058146 A015449

Adjacent sequences:  A047662 A047663 A047664 * A047666 A047667 A047668

KEYWORD

nonn

AUTHOR

Don Knuth, N. J. A. Sloane

EXTENSIONS

Prepended 0, set offset to 0 and new name using a comment of Emeric Deutsch from Dec 25 2003 by Peter Luschny, May 20 2015

STATUS

approved

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Last modified February 18 23:26 EST 2018. Contains 299330 sequences. (Running on oeis4.)