A126258 -id:A126258 - cp's OEIS frontend

A034299 Alternating sum transform (PSumSIGN) of A000975.

1, 1, 4, 6, 15, 27, 58, 112, 229, 453, 912, 1818, 3643, 7279, 14566, 29124, 58257, 116505, 233020, 466030, 932071, 1864131, 3728274, 7456536, 14913085, 29826157, 59652328, 119304642, 238609299

Offset: 0

N. J. A. Sloane

			G.f. = 1 + x + 4*x^2 + 6*x^3 + 15*x^4 + 27*x^5 + 58*x^6 + 112*x^7 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Transforms
Index entries for linear recurrences with constant coefficients, signature (1,3,-1,-2).

Cf. A160156.

[(2^(n+5)+(6*n+13)*(-1)^n-9)/36: n in [0..50]]; // G. C. Greubel, Oct 12 2017

CoefficientList[Series[(1/(1-x^2))/(1-x-2x^2),{x,0,40}],x] (* Vincenzo Librandi, Apr 04 2012 *)
Table[(2^(n + 5) + (6 n + 13) (-1)^n - 9)/36, {n, 0, 28}] (* Bruno Berselli, Apr 04 2012 *)
LinearRecurrence[{1,3,-1,-2},{1,1,4,6},30] (* Harvey P. Dale, Jun 11 2019 *)

{a(n) = (32 * 2^n - 9 + (6*n + 13) * (-1)^n) / 36}; /* Michael Somos, Jan 23 2014 */

a(n) = sum{k=0..floor(n/2), A001045(n-2k+1)}. - Paul Barry, Nov 24 2003
G.f.: (1/(1-x^2))/(1-x-2x^2); a(n) = sum{k=0..n+1, A001045(k)*(1-(-1)^floor((n+k)/2))}; - Paul Barry, Apr 16 2005
a(n) = sum_{k, 0<=k<=n} A126258(n,k). - Philippe Deléham, Mar 13 2007
a(n) = 2*a(n-1)+A001057(n+1), with a(0)=1. - Bruno Berselli, Nov 09 2010
a(n) = (2^(n+5)+(6n+13)(-1)^n-9)/36. - Bruno Berselli, Apr 04 2012
a(n) = a(n-1) + 2*a(n-2) + (1 + (-1)^n) / 2. - Michael Somos, Jan 23 2014
A160156(n) = a(2*n). - Michael Somos, Oct 16 2020

A116914 Number of UUDD's, where U=(1,1) and D=(1,-1), in all hill-free Dyck paths of semilength n (a hill in a Dyck path is a peak at level 1).

Original entry on oeis.org

1, 1, 5, 16, 58, 211, 781, 2920, 11006, 41746, 159154, 609324, 2341060, 9021559, 34855741, 134972368, 523689718, 2035462990, 7923732118, 30889008112, 120566373676, 471134916286, 1842964183570, 7216096752496, 28279240308268, 110913181145716, 435333520075796, 1709861650762900

Offset: 2

Emeric Deutsch, May 08 2006

nonn

Catalan transform of A034299. - R. J. Mathar, Jun 29 2009

			a(4)=5 because in the 6 (=A000957(5)) hill-free Dyck paths of semilength 4, namely UU(UUDD)DD, UUUDUDDD, UUD(UUDD)D, UUDUDUDD, U(UUDD)UDD and (UUDD)(UUDD) (U=(1,1), D=(1,-1)) we have altogether 5 UUDD's (shown between parentheses).

G. C. Greubel, Table of n, a(n) for n = 2..1000
David Anderson, E. S. Egge, M. Riehl, L. Ryan, R. Steinke, Y. Vaughan, Pattern Avoiding Linear Extensions of Rectangular Posets, arXiv:1605.06825 [math.CO], 2016.
Colin Defant, Proofs of Conjectures about Pattern-Avoiding Linear Extensions, arXiv:1905.02309 [math.CO], 2019.
E. Deutsch and L. Shapiro, A survey of the Fine numbers, Discrete Math., 241 (2001), 241-265.

Cf. A105640.

R:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( x*(1 + 5*x-(1-x)*Sqrt(1-4*x))/(2*(2+x)^2*Sqrt(1-4*x)) )); // G. C. Greubel, May 08 2019

G:=z*(1+5*z-(1-z)*sqrt(1-4*z))/2/(2+z)^2/sqrt(1-4*z): Gser:=series(G,z=0,31): seq(coeff(Gser,z^n),n=2..28);

Rest[Rest[CoefficientList[Series[x*(1+5*x-(1-x)*Sqrt[1-4*x])/2/(2+x)^2/Sqrt[1-4*x], {x, 0, 40}], x]]] (* Vaclav Kotesovec, Mar 20 2014 *)

my(x='x+O('x^40)); Vec(x*(1+5*x-(1-x)*sqrt(1-4*x))/(2*(2+x)^2 *sqrt(1-4*x))) \\ G. C. Greubel, Feb 08 2017

a=(x*(1+5*x-(1-x)*sqrt(1-4*x))/(2*(2+x)^2*sqrt(1-4*x))).series(x, 40).coefficients(x, sparse=False); a[2:] # G. C. Greubel, May 08 2019

a(n) = Sum_{k=0..floor(n/2)} k*A105640(n,k).
G.f.: x*(1 + 5*x - (1-x)*sqrt(1-4*x))/(2*(2+x)^2*sqrt(1-4*x)).
a(n+2) = A126258(2*n,n). - Philippe Deléham, Mar 13 2007
a(n) ~ 2^(2*n-1)/(9*sqrt(Pi*n)). - Vaclav Kotesovec, Mar 20 2014
D-finite with recurrence +2*(-n+1)*a(n) +3*(-n+6)*a(n-1) +3*(13*n-44)*a(n-2) +10*(2*n-5)*a(n-3)=0. - R. J. Mathar, Jul 26 2022

cp's OEIS Frontend

A034299 Alternating sum transform (PSumSIGN) of A000975.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula