A128750 -id:A128750 - cp's OEIS frontend

A105864 Expansion of (1/(1-x^2))*c(x/(1-x^2)), where c(x) is the g.f. of A000108.

1, 1, 3, 7, 21, 65, 215, 735, 2585, 9281, 33883, 125383, 469229, 1772801, 6752623, 25902975, 99978865, 388001025, 1513077235, 5926139207, 23301146501, 91942524481, 363957103303, 1444966207967, 5752187960841, 22955311342145

Offset: 0

Paul Barry, Apr 23 2005

Binomial transform is A059279.
Hankel transform is A134751. - Paul Barry, Oct 07 2008
The radius of convergence r of the g.f. A(x) satisfies: r = (1-r^2)/4 = lim_{n->inf} a(n)/a(n+1) = sqrt(5) - 2 = 0.2360679... with A(r) = 1/(2*r) = (sqrt(5) + 2)/2 = 2.1180339... - Paul D. Hanna, Sep 06 2011

Paul Barry, Continued fractions and transformations of integer sequences, JIS 12 (2009) 09.7.6.

Partial sums of A128750.

a[0] = a[1] = 1; a[2] = 3; a[3] = 7; a[n_] := a[n] = (-((n-3)*a[n-4]) - 2*(2*n-3)*a[n-3] + 2*(n-1)*a[n-2] + 2*(2*n-1)*a[n-1])/(n+1); Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Sep 09 2017, using "FindSequenceFunction" *)

{a(n)=polcoeff((1-sqrt(1-4*x/(1-x^2 +O(x^(n+2)))))/(2*x), n)} /* Paul D. Hanna */

G.f.: (1 - sqrt((1 - 4*x - x^2)/(1 - x^2)))/(2*x).
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k) * A000108(n-2*k).
G.f. satisfies: A(x) = 1/(1-x^2) + x*A(x)^2. - Paul D. Hanna, Sep 06 2011
Conjecture: (n+1)*a(n) + 2*(1-2*n)*a(n-1) + 2*(1-n)*a(n-2) + 2*(2*n-3)*a(n-3) + (n-3)*a(n-4) = 0. - R. J. Mathar, Nov 15 2011
G.f.: (1-1/G(0))/(2*x), where G(k) = 1 + 4*x*(4*k+1)/( (1-x^2)*(4*k+2) - x*(1-x^2)*(4*k+2)*(4*k+3)/(x*(4*k+3) + (1-x^2)*(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 26 2013
a(n) ~ 5^(1/4)*(2+sqrt(5))^(n+1)/(4*sqrt(2*Pi)*n^(3/2)). - Vaclav Kotesovec, Sep 16 2013
G.f.: 1/G(x), where G(x) = 1 - x^2 - (x - x^3)/ G(x) (continued fraction). - Nikolaos Pantelidis, Jan 08 2023

A128749 Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n having k ascents of length 1.

Original entry on oeis.org

1, 0, 1, 2, 0, 1, 4, 5, 0, 1, 14, 12, 9, 0, 1, 44, 53, 25, 14, 0, 1, 150, 196, 132, 44, 20, 0, 1, 520, 777, 555, 269, 70, 27, 0, 1, 1850, 3064, 2486, 1260, 485, 104, 35, 0, 1, 6696, 12233, 10902, 6264, 2496, 804, 147, 44, 0, 1, 24602, 49096, 47955, 30108, 13600

Offset: 0

Emeric Deutsch, Mar 31 2007

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 a path is defined to be the number of its steps. An ascent in a path is a maximal sequence of consecutive U steps.

Row sums yield A002212.

			T(3,1)=5 because we have (U)DUUDD, (U)DUUDL, UUDD(U)D, UUD(U)DD and UUD(U)DL (the ascents of length 1 are shown between parentheses).
Triangle starts:
   1;
   0,  1;
   2,  0,  1;
   4,  5,  0,  1;
  14, 12,  9,  0,  1;
  44, 53, 25, 14,  0,  1;

E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203.

Cf. A002212, A085362, A128750.

eq:=z*(1+z-t*z)*G^2-(1-t*z+t*z^2-z^2)*G+1-z=0: G:=RootOf(eq,G): Gser:=simplify(series(G,z=0,15)): for n from 0 to 11 do P[n]:=sort(coeff(Gser,z,n)) od: for n from 0 to 11 do seq(coeff(P[n],t,j),j=0..n) od; # yields sequence in triangular form

T(n,0) = A128750(n).
Sum_{k=0..n} k*T(n,k) = A085362(n-1).
G.f.: G = G(t,z) satisfies z(1 + z - tz)G^2 - (1 - tz + tz^2 - z^2)G + 1 - z = 0.

A179532 a(n) = 2^ceiling(n*(n+1)/3).

Original entry on oeis.org

1, 2, 4, 16, 128, 1024, 16384, 524288, 16777216, 1073741824, 137438953472, 17592186044416, 4503599627370496, 2305843009213693952, 1180591620717411303424, 1208925819614629174706176, 2475880078570760549798248448

Offset: 0

Paul Barry, Jan 08 2011

nonn

Hankel transform of A128750. Satisfies (4,0) Somos-4 recurrence 4*a(n-1)*a(n-3)/a(n-4) = a(n),n>3.

G. C. Greubel, Table of n, a(n) for n = 0..99

[2^Ceiling((n(n+1))/3): n in [0..20]]; // G. C. Greubel, Aug 14 2018

Table[2^Ceiling[(n(n+1))/3],{n,0,20}] (* Harvey P. Dale, Sep 25 2013 *)

vector(20,n,n--; 2^ceil(n*(n+1)/3)) \\ G. C. Greubel, Aug 14 2018

2^A007980.

a(n) = a(-1-n) for all n in Z. - Michael Somos, Aug 14 2018

cp's OEIS Frontend

A105864 Expansion of (1/(1-x^2))*c(x/(1-x^2)), where c(x) is the g.f. of A000108.

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A128749 Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n having k ascents of length 1.

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A179532 a(n) = 2^ceiling(n*(n+1)/3).

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