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%I #62 Nov 15 2021 12:03:16
%S 0,1,6,31,160,841,4494,24319,132864,731281,4048726,22523359,125797984,
%T 704966809,3961924126,22321190911,126027618304,712917362209,
%U 4039658528934,22924714957471,130271906898720,741188107113961,4221707080583086,24070622500965631,137369104574280960,784622537295845041
%N Expansion of (1/sqrt(1-6*x+x^2)-1/(1-x))/2.
%C Previous name was: Main diagonal of square array defined in A047662.
%C a(n) is the 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
%H Vincenzo Librandi, <a href="/A047665/b047665.txt">Table of n, a(n) for n = 0..200</a> (corrected by Sean A. Irvine, Jan 18 2019)
%H Y. Ding and R. R. X. Du, <a href="http://arxiv.org/abs/1109.2661">Counting Humps in Motzkin Paths</a>, arXiv:1109.2661 [math.CO], 2011, Eq. (4.2).
%H D. E. Knuth and N. J. A. Sloane, <a href="/A047662/a047662.pdf">Correspondence, December 1999</a>
%H Matthew Roughan, <a href="https://arxiv.org/abs/1810.10373">Surreal Birthdays and Their Arithmetic</a>, arXiv:1810.10373 [math.HO], 2018.
%F 2*a(n)+1 = A001850(n).
%F a(n)-a(n-1) = A002002(n).
%F a(n) = Sum_{k=0..n} Sum_{j=0..n} A008288(k, j).
%F 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
%F D-finite with 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
%F a(n) ~ sqrt(8+6*sqrt(2))*(3+2*sqrt(2))^n/(8*sqrt(Pi*n)). - _Vaclav Kotesovec_, Oct 08 2012
%F a(n) = (hyper2F1(-n,n+1,1,-1)-1)/2 = (hyper2F1(-n, -n, 1, 2)-1)/2. - _Peter Luschny_, May 19 2015
%F a(n) = Sum_{k=1..n} binomial(n,k)^2 * 2^(k-1). - _Ilya Gutkovskiy_, Nov 15 2021
%p seq(add(multinomial(n+k,n-k,k,k)/2,k=1..n),n=1..22); # _Zerinvary Lajos_, Oct 18 2006
%p 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
%t Table[SeriesCoefficient[(1/Sqrt[1-6*x+x^2]-1/(1-x))/2,{x,0,n}],{n,1,20}] (* _Vaclav Kotesovec_, Oct 08 2012 *)
%o (PARI) x='x+O('x^66); Vec((1/sqrt(1-6*x+x^2)-1/(1-x))/2) \\ _Joerg Arndt_, May 04 2013
%o (Sage)
%o a = lambda n: (hypergeometric([-n, n+1], [1], -1)-1)/2
%o [simplify(a(n)) for n in (1..25)] # _Peter Luschny_, May 19 2015
%Y Cf. A001850, A002002 (Schroeder paths interpretation).
%Y Cf. A008288 (Delannoy numbers triangle).
%K nonn
%O 0,3
%A _Don Knuth_, _N. J. A. Sloane_
%E 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
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