A126120 -id:A126120 - cp's OEIS frontend

A358375 Numbers k such that the k-th standard ordered rooted tree is binary.

1, 4, 18, 25, 137, 262146, 393217, 2097161, 2228225

Offset: 1

Gus Wiseman, Nov 14 2022

We define the n-th standard ordered rooted tree to be obtained by taking the (n-1)-th composition in standard order (graded reverse-lexicographic, A066099) as root and replacing each part with its own standard ordered rooted tree. This ranking is an ordered variation of Matula-Goebel numbers, giving a bijective correspondence between positive integers and unlabeled ordered rooted trees.

			The initial terms and their corresponding trees:
       1: o
       4: (oo)
      18: ((oo)o)
      25: (o(oo))
     137: ((oo)(oo))
  262146: (((oo)o)o)
  393217: (o((oo)o))

Gus Wiseman, Statistics, classes, and transformations of standard compositions

The unordered version is A111299, counted by A001190
These trees are counted by A126120.
A000081 counts unlabeled rooted trees, ranked by A358378.
A358371 and A358372 count leaves and nodes in standard ordered rooted trees.
Cf. A001263, A004249, A005043, A063895, A245824, A284778, A358373, A358374, A358376, A358377.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
srt[n_]:=If[n==1,{},srt/@stc[n-1]];
Select[Range[1000],FreeQ[srt[#],[_]?(Length[#]!=2&)]&]

A166587 A signed variant of the Motzkin numbers.

Original entry on oeis.org

1, 1, -1, 2, -4, 9, -21, 51, -127, 323, -835, 2188, -5798, 15511, -41835, 113634, -310572, 853467, -2356779, 6536382, -18199284, 50852019, -142547559, 400763223, -1129760415, 3192727797, -9043402501, 25669818476, -73007772802

Offset: 0

Paul Barry, Oct 17 2009

Hankel transform is A131713. Binomial transform is A166588.

[a(n+1)] = [1,-1,2,-4,9,...] is the inverse binomial transform of A126120. - Philippe Deléham, Nov 29 2009

			G.f. = 1 + x - x^2 + 2*x^3 - 4*x^4 + 9*x^5 - 21*x^6 + 51*x^7 - 127*x^8 + ...

G. C. Greubel, Table of n, a(n) for n = 0..500

f:= gfun:-rectoproc({3*n*a(n)+(-3-2*n)*a(1+n)+(-3-n)*a(n+2)=0,a(0) = 1, a(1) = 1}, a(n),remember):
map(f, [$0..100]); # Robert Israel, May 17 2016
with(PolynomialTools): CoefficientList(convert(taylor((1 + 3*x - sqrt(1 + 2*x - 3*x^2))/2/x, x = 0, 33), polynom), x); # Taras Goy, Aug 07 2017

CoefficientList[Series[(1 + 3*t - Sqrt[1 + 2*t - 3*t^2])/(2 t), {t, 0, 50}], t] (* G. C. Greubel, May 17 2016 *)

G.f.: (1+3x-sqrt(1+2x-3x^2))/(2x); (1+3x)/(1+2x-x^2/(1+x-x^2/(1+x-x^2/(1+x-x^2/(1+...))))) (continued fraction).
a(n) = 0^n + Sum_{k=0..n} binomial(n-1, k-1)*(-3)^(n-k)*A000108(k).
G.f.: (1+3*x-sqrt(1+2*x-3*x^2))/(2x) = (3-1/G(0))/2 ; G(k) = 1+2*x/(1-x/(1-x/(1+2*x/(1+x/(2+x/G(k+1)))))) ; (continued fraction). - Sergei N. Gladkovskii, Dec 11 2011
Conjecture: n*(n+1)*a(n) + n*(n+1)*a(n-1) - (5*n-3)*(n-2)*a(n-2) + 3*(n-2)*(n-3)*a(n-3) = 0. - R. J. Mathar, Nov 15 2012
G.f. G(x) satisfies (3 x^2 - 2 x^2 - x) G'(x) - (x+1) G(x) + 3 x + 1 = 0, from which follows 3*n*a(n) + (-3-2*n)*a(1+n) + (-3-n)*a(n+2) = 0 as well as Mathar's conjecture. - Robert Israel, May 17 2016
E.g.f.: 1 + Integral (exp(-x) * BesselI(1,2*x) / x) dx. - Ilya Gutkovskiy, Jun 01 2020

A303022 Number of free pure symmetric multifunctions (with empty expressions allowed) with one atom, n positions, and no unitary parts (subexpressions of the form x[y]).

Original entry on oeis.org

1, 1, 1, 2, 5, 12, 27, 63, 152, 376, 939, 2371, 6047, 15577, 40429, 105637, 277625, 733518, 1947126, 5190503, 13888811, 37291968, 100444019, 271316998, 734802247, 1994873116, 5427893149, 14799525982, 40429761365, 110645688034, 303316712450, 832799212777

Offset: 1

Gus Wiseman, Aug 15 2018

nonn

Also the number of orderless Mathematica expressions with one atom, n positions, and no unitary parts.

			The a(6) = 12 Mathematica expressions:
  o[o,o[][]]
  o[o[],o[]]
  o[o,o,o[]]
  o[o,o,o,o]
  o[][o,o[]]
  o[][o,o,o]
  o[][][o,o]
  o[o,o[]][]
  o[o,o,o][]
  o[][o,o][]
  o[o,o][][]
  o[][][][][]

Andrew Howroyd, Table of n, a(n) for n = 1..200

Cf. A000108, A001003, A001006, A007853, A102403, A126120, A318049.

Cf. A303023, A303024, A303025, A303026, A303027.

allOLBF[n_]:=allOLBF[n]=If[n==1,{"o"},Join@@Cases[Table[PR[k,n-k-1],{k,n-1}],PR[h_,g_]:>Join@@Table[Apply@@@Tuples[{allOLBF[h],Select[Union[Sort/@Tuples[allOLBF/@p]],Length[#]!=1&]}],{p,IntegerPartitions[g]}]]];
Table[Length[allOLBF[n]],{n,10}]

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={my(v=[1]); for(n=2, n, my(t=EulerT(v)-v); v=concat(v, v[n-1] + sum(k=1, n-2, v[k]*t[n-k-1]))); v} \\ Andrew Howroyd, Aug 19 2018

Terms a(21) and beyond from Andrew Howroyd, Aug 19 2018

A303027 Number of free pure symmetric multifunctions with one atom, n positions, and no empty or unitary parts (subexpressions of the form x[] or x[y]).

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 3, 5, 7, 15, 28, 47, 90, 175, 319, 607, 1181, 2251, 4325, 8449, 16425, 31992, 62823, 123521, 243047, 480316, 951290, 1886293, 3749341, 7467815, 14893500, 29752398, 59532947, 119274491, 239275400, 480638121, 966571853, 1945901716, 3921699524

Offset: 1

Gus Wiseman, Aug 15 2018

nonn

Also the number of orderless Mathematica expressions with one atom, n positions, and no empty or unitary parts.

			The a(10) = 15 Mathematica expressions:
  o[o,o[o,o[o,o]]]
  o[o,o[o,o][o,o]]
  o[o[o,o],o[o,o]]
  o[o,o][o,o[o,o]]
  o[o,o[o,o]][o,o]
  o[o,o][o,o][o,o]
  o[o,o[o,o,o,o,o]]
  o[o,o,o[o,o,o,o]]
  o[o,o,o,o[o,o,o]]
  o[o,o,o,o,o[o,o]]
  o[o,o][o,o,o,o,o]
  o[o,o,o][o,o,o,o]
  o[o,o,o,o][o,o,o]
  o[o,o,o,o,o][o,o]
  o[o,o,o,o,o,o,o,o]

Andrew Howroyd, Table of n, a(n) for n = 1..200

Cf. A000108, A001003, A001006, A007853, A102403, A126120, A318049.

Cf. A303022, A303023, A303024, A303025, A303026.

allOLZR[n_]:=allOLZR[n]=If[n==1,{"o"},Join@@Cases[Table[PR[k,n-k-1],{k,n-1}],PR[h_,g_]:>Join@@Table[Apply@@@Tuples[{allOLZR[h],Select[Union[Sort/@Tuples[allOLZR/@p]],Length[#]>1&]}],{p,IntegerPartitions[g]}]]];
Table[Length[allOLZR[n]],{n,25}]

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={my(v=[1]); for(n=2, n, my(t=EulerT(v)-v); v=concat(v, sum(k=1, n-2, v[k]*t[n-k-1]))); v} \\ Andrew Howroyd, Aug 19 2018

Terms a(29) and beyond from Andrew Howroyd, Aug 19 2018

A304173 Number of rooted plane trees where every branch that has a predecessor (a branch directly to its left and emanating from the same root) has at least as many leaves as its predecessor.

Original entry on oeis.org

1, 1, 2, 5, 13, 34, 90, 242, 660, 1822, 5085, 14333, 40759, 116817, 337140, 979098, 2859439, 8393113, 24747052, 73262246, 217681621, 648939319, 1940461444, 5818595438, 17492367097, 52712114792, 159193762250, 481754196170, 1460650624068, 4436422703787, 13496947320929

Offset: 1

Gus Wiseman, Aug 16 2018

nonn

			The a(5) = 13 plane trees:
  ((((o)))), (((oo))), (((o)o)), ((o(o))), ((ooo)),
  (((o))o), (o((o))), (o(oo)), ((o)(o)),
  ((o)oo), (o(o)o), (oo(o)),
  (oooo).
Missing from this list is ((oo)o).

Andrew Howroyd, Table of n, a(n) for n = 1..200

Cf. A000081, A000108, A001003, A001006, A003238, A007853, A126120, A291442, A291443, A304175.

pplane[n_]:=If[n==1,{{}},Join@@Table[Select[Tuples[pplane/@c],OrderedQ[Count[#,{},{0,Infinity}]&/@#]&],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[pplane[n]],{n,10}]

seq(n)={my(p=x*y+O(x^2)); for(n=2, n, p=x*(y-1 + 1/prod(k=1, n-1, 1 - y^k*polcoef(p,k,y)))); Vec(subst(p,y,1))} \\ Andrew Howroyd, Jan 22 2021

G.f.: A(x,1) where A(x,y) satisfies A(x,y) = x*(y-1 + 1/(Product_{k>=1} 1 - y^k * [y^k] A(x,y))). - Andrew Howroyd, Jan 22 2021

Terms a(15) and beyond from Andrew Howroyd, Jan 22 2021

A304175 Number of leaf-balanced rooted plane trees with n nodes.

Original entry on oeis.org

1, 1, 2, 5, 12, 27, 59, 128, 277, 597, 1280, 2730, 5794, 12248, 25836, 54508, 115222, 244144, 518104, 1099499, 2330326, 4930089, 10415135, 21992400, 46470911, 98353146, 208580686, 443186181, 942988423, 2007981801, 4276830431, 9109431322, 19404918449, 41357252072, 88236092543

Offset: 1

Gus Wiseman, Aug 16 2018

nonn

A rooted plane tree is leaf-balanced if every branch of the root has the same number of leaves, and every branch of the root is itself leaf-balanced.

			The a(5) = 12 leaf-balanced plane trees:
  ((((o)))), (((oo))), (((o)o)), ((o(o))), ((ooo)),
  (((o))o), (o((o))), ((o)(o)),
  ((o)oo), (o(o)o), (oo(o)),
  (oooo).
Missing from this list are ((oo)o) and (o(oo)).

Andrew Howroyd, Table of n, a(n) for n = 1..500

Cf. A000081, A000108, A001003, A001006, A003238, A007853, A126120, A291442, A291443, A304173.

lbplane[n_]:=If[n==1,{{}},Join@@Table[Select[Tuples[lbplane/@c],SameQ@@(Count[#,{},{0,Infinity}]&/@#)&],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[lbplane[n]],{n,10}]

seq(n)={my(v=vector(n)); v[1]=x/(1-x) + O(x*x^n); for(k=2, n, v[k]=x*sumdiv(k, d, if(dAndrew Howroyd, Dec 13 2020

Terms a(17) and beyond from Andrew Howroyd, Dec 13 2020

A247495 Generalized Motzkin numbers: Square array read by descending antidiagonals, T(n, k) = k![x^k](exp(nx)* BesselI_{1}(2*x)/x), n>=0, k>=0.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 0, 2, 2, 1, 2, 4, 5, 3, 1, 0, 9, 14, 10, 4, 1, 5, 21, 42, 36, 17, 5, 1, 0, 51, 132, 137, 76, 26, 6, 1, 14, 127, 429, 543, 354, 140, 37, 7, 1, 0, 323, 1430, 2219, 1704, 777, 234, 50, 8, 1, 42, 835, 4862, 9285, 8421, 4425, 1514, 364, 65, 9, 1

Offset: 0

Peter Luschny, Dec 11 2014

This two-dimensional array of numbers can be seen as a generalization of the Motzkin numbers A001006 for two reasons: The case n=1 reduces to the Motzkin numbers and the columns are the values of the Motzkin polynomials M_{k}(x) = sum_{j=0..k} A097610(k,j)*x^j evaluated at the nonnegative integers.

			Square array starts:
[n\k][0][1] [2]  [3]   [4]   [5]    [6]     [7]      [8]
[0]   1, 0,  1,   0,    2,    0,     5,      0,      14, ...  A126120
[1]   1, 1,  2,   4,    9,   21,    51,    127,     323, ...  A001006
[2]   1, 2,  5,  14,   42,  132,   429,   1430,    4862, ...  A000108
[3]   1, 3, 10,  36,  137,  543,  2219,   9285,   39587, ...  A002212
[4]   1, 4, 17,  76,  354, 1704,  8421,  42508,  218318, ...  A005572
[5]   1, 5, 26, 140,  777, 4425, 25755, 152675,  919139, ...  A182401
[6]   1, 6, 37, 234, 1514, 9996, 67181, 458562, 3172478, ...  A025230
A000012,A001477,A002522,A079908, ...
.
Triangular array starts:
              1,
             0, 1,
           1, 1, 1,
          0, 2, 2, 1,
        2, 4, 5, 3, 1,
      0, 9, 14, 10, 4, 1,
   5, 21, 42, 36, 17, 5, 1,
0, 51, 132, 137, 76, 26, 6, 1.

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Main diagonal gives A247496.

Cf. A126120, A001006, A000108, A002212, A005572, A182401, A025230, A002522, A079908, A055151, A097610, A247497, A306684.

# RECURRENCE
T := proc(n,k) option remember; if k=0 then 1 elif k=1 then n else
(n*(2*k+1)*T(n,k-1)-(n-2)*(n+2)*(k-1)*T(n,k-2))/(k+2) fi end:
seq(print(seq(T(n,k),k=0..9)),n=0..6);
# OGF (row)
ogf := n -> (1-n*x-sqrt(((n-2)*x-1)*((n+2)*x-1)))/(2*x^2):
seq(print(seq(coeff(series(ogf(n),x,12),x,k),k=0..9)),n=0..6);
# EGF (row)
egf := n -> exp(n*x)*hypergeom([],[2],x^2):
seq(print(seq(k!*coeff(series(egf(n),x,k+2),x,k),k=0..9)),n=0..6);
# MOTZKIN polynomial (column)
A097610 := proc(n,k) if type(n-k,odd) then 0 else n!/(k!*((n-k)/2)!^2* ((n-k)/2+1)) fi end: M := (k,x) -> add(A097610(k,j)*x^j,j=0..k):
seq(print(seq(M(k,n),n=0..9)),k=0..6);
# OGF (column)
col := proc(n, len) local G; G := A247497_row(n); (-1)^(n+1)* add(G[k+1]/(x-1)^(k+1), k=0..n); seq(coeff(series(%, x, len+1),x,j), j=0..len) end: seq(print(col(n,8)), n=0..6); # Peter Luschny, Dec 14 2014

T[0, k_] := If[EvenQ[k], CatalanNumber[k/2], 0];
T[n_, k_] := n^k*Hypergeometric2F1[(1 - k)/2, -k/2, 2, 4/n^2];
Table[T[n - k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Nov 03 2017 *)

def A247495(n,k):
    if n==0: return(k//2+1)*factorial(k)/factorial(k//2+1)^2 if is_even(k) else 0
    return n^k*hypergeometric([(1-k)/2,-k/2],[2],4/n^2).simplify()
for n in (0..7): print([A247495(n,k) for k in range(11)])

T(n,k) = (n*(2*k+1)*T(n,k-1)-(n-2)*(n+2)*(k-1)*T(n,k-2))/(k+2) for k>=2.
T(n,k) = Sum_{j=0..floor(k/2)} n^(k-2*j)*binomial(k,2*j)*binomial(2*j,j)/(j+1).
T(n,k) = n^k*hypergeom([(1-k)/2,-k/2], [2], 4/n^2) for n>0.
T(n,n) = A247496(n).
O.g.f. for row n: (1-n*x-sqrt(((n-2)*x-1)*((n+2)*x-1)))/(2*x^2).
O.g.f. for row n: R(x)/x where R(x) is series reversion of x/(1+n*x+x^2).
E.g.f. for row n: exp(n*x)*hypergeom([],[2],x^2).
O.g.f. for column k: the k-th column consists of the values of the k-th Motzkin polynomial M_{k}(x) evaluated at x = 0,1,2,...; M_{k}(x) = sum_{j=0..k} A097610(k,j)*x^j = sum_{j=0..k} (-1)^j*binomial(k,j)*A001006(j)*(x+1)^(k-j).
O.g.f. for column k: sum_{j=0..k} (-1)^(k+1)*A247497(k,j)/(x-1)^(j+1). - Peter Luschny, Dec 14 2014
O.g.f. for row n: 1/(1 - n*x - x^2/(1 - n*x - x^2/(1 - n*x - x^2/(1 - n*x - x^2/(1 - ...))))), a continued fraction. - Ilya Gutkovskiy, Sep 21 2017
T(n,k) is the coefficient of x^k in the expansion of 1/(k+1) * (1 + n*x + x^2)^(k+1). - Seiichi Manyama, May 07 2019

A318049 Number of first/rest balanced rooted plane trees with n nodes.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 1, 3, 2, 6, 8, 11, 26, 28, 67, 96, 162, 316, 448, 922, 1435, 2572, 4660, 7563, 14397, 23896, 43337, 77097, 133071, 244787, 423093, 767732, 1367412, 2426612, 4408497, 7802348, 14152342, 25365035, 45602031, 82631362, 148246136, 269103870, 485379304

Offset: 1

Gus Wiseman, Aug 13 2018

nonn

A rooted plane tree is first/rest balanced if either (1) it is a single node, or (2a) the number of leaves in the first branch is equal to the number of branches minus one, and (2b) every branch is also first/rest balanced.

Also the number of composable free pure multifunctions (CPMs) with one atom and n positions. A CPM is either (case 1) the leaf symbol "o", or (case 2) an expression of the form h[g_1, ..., g_k] where h and each of the g_i for i = 1, ..., k > 0 are CPMs, and the number of leaves in h is equal to k. The number of positions in a CPM is the number of brackets [...] plus the number of o's.

			The a(12) = 11 first/rest balanced rooted plane trees:
  (o(o(o((oo)oo))))
  (o(o((oo)(oo)o)))
  (o(o((oo)o(oo))))
  (o((oo)(o(oo))o))
  (o((oo)o(o(oo))))
  (o((oo)(oo)(oo)))
  ((oo)(o(o(oo)))o)
  ((oo)o(o(o(oo))))
  ((o(o(oo)))oooo)
  ((oo)(o(oo))(oo))
  ((oo)(oo)(o(oo)))
The a(12) = 11 composable free pure multifunctions:
  o[o[o[o[o][o,o]]]]
  o[o[o[o][o[o],o]]]
  o[o[o[o][o,o[o]]]]
  o[o[o][o[o[o]],o]]
  o[o[o][o,o[o[o]]]]
  o[o[o][o[o],o[o]]]
  o[o][o[o[o[o]]],o]
  o[o][o,o[o[o[o]]]]
  o[o][o[o[o]],o[o]]
  o[o][o[o],o[o[o]]]
  o[o[o[o]]][o,o,o,o]

Andrew Howroyd, Table of n, a(n) for n = 1..100

Cf. A000081, A000108, A001003, A001006, A007853, A126120, A317713, A318046, A318048.

balplane[n_]:=balplane[n]=If[n===1,{{}},Join@@Function[c,Select[Tuples[balplane/@c],Length[Cases[#[[1]],{},{0,Infinity}]]==Length[#]-1&]]/@Join@@Permutations/@IntegerPartitions[n-1]];
Table[Length[balplane[n]],{n,10}]

seq(n)={my(p=x*y+O(x^2)); for(n=1, n\2, p = x*y + x*sum(k=1, n, y^k * polcoef(p,k,y) * (O(x^(2*n-k+1)) + p)^k )); Vec(subst(p + O(x*x^n), y, 1)) } \\ Andrew Howroyd, Jan 22 2021

G.f.: A(x,1) where A(x,y) satisfies A(x,y) = x*(y + Sum_{k>=1} y^k * ([y^k] A(x,y)) * A(x,y)^k). - Andrew Howroyd, Jan 22 2021

Terms a(21) and beyond from Andrew Howroyd, Jan 22 2021

A138349 Moment sequence of tr(A) in USp(4).

Original entry on oeis.org

1, 0, 1, 0, 3, 0, 14, 0, 84, 0, 594, 0, 4719, 0, 40898, 0, 379236, 0, 3711916, 0, 37975756, 0, 403127256, 0, 4415203280, 0, 49671036900, 0, 571947380775, 0, 6721316278650, 0, 80419959684900, 0, 977737404590100, 0, 12058761323277900, 0

Offset: 0

Andrew V. Sutherland, Mar 16 2008

An aerated version of A005700, which is the main entry for this sequence.
If A is a random matrix in the compact group USp(4) (4 X 4 complex matrices which are unitary and symplectic), then a(n)=E[(tr(A))^n] is the n-th moment of the trace of A.
The multiplicity of the trivial representation in the n-th tensor power of the standard representation of USp(4).
Number of returning NESW walks of length n on a 2-d integer lattice remaining in the chamber x>=y>=0, same as A005700(n/2) for n even.
Under a generalized Sato-Tate conjecture, this is the moment sequence of the distribution of scaled Frobenius traces a_p/sqrt(p) (as p varies), for almost all genus 2 curves. - Andrew V. Sutherland, Mar 16 2008

			a(4)=3 because E[(tr(A))^4] = 3 for a random matrix A in USp(4).
a(4)=3 because A126120(4)A126120(8)-A126120(6)^2 = 2*14-5*5 = 3.
a(4)=3 because EEWW, EWEW and ENSW are the returning walks on Z^2 with x>=y>=0.

Bostan, Alin ; Chyzak, Frédéric; van Hoeij, Mark; Kauers, Manuel; Pech, Lucien Hypergeometric expressions for generating functions of walks with small steps in the quarter plane. Eur. J. Comb. 61, 242-275 (2017), Table 3
David J. Grabiner and Peter Magyar, Random walks in Weyl chambers and the decomposition of tensor powers, Journal of Algebraic Combinatorics, vol. 2 (1993), no. 3, pp 239-260.
Kiran S. Kedlaya and Andrew V. Sutherland, Hyperelliptic curves, L-polynomials and random matrices, arXiv:0803.4462 [math.NT], 2008-2010.

Cf. A005700, A126120, A000108.

a(n) = (1/2)Integral_{x=0..Pi,y=0..Pi}(2cos(x)+2cos(y))^n(2cos(x)-2cos(y))^2(2/Pi*sin^2(x))(2/Pi*sin^2(y))dxdy.
a(n) = A126120(n)*A126120(n+4)-A126120(n+2)^2.
a(2n) = A005700(n) = A000108(n)*A000108(n+2)-A000108(n+1)^2, a(2n+1)=0.

A215495 a(4n) = a(4n+2) = a(2n+1) = 2n + 1.

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 3, 7, 5, 9, 5, 11, 7, 13, 7, 15, 9, 17, 9, 19, 11, 21, 11, 23, 13, 25, 13, 27, 15, 29, 15, 31, 17, 33, 17, 35, 19, 37, 19, 39, 21, 41, 21, 43, 23, 45, 23, 47, 25, 49, 25, 51, 27, 53, 27, 55, 29, 57, 29, 59, 31, 61, 31, 63, 33, 65, 33, 67, 35, 69, 35, 71, 37, 73, 37, 75, 39, 77, 39, 79, 41, 81, 41, 83, 43, 85, 43

Offset: 0

Paul Curtz, Aug 13 2012

A214282(n) and -A214283(n) are companions. Separately or together, they have many links with the Catalan's numbers A000108(n). Examples:
A214282(n+1) - 2*A214282(n) = -1, -1,  1,  0, -2, -5,  5,   0, -14, -42,  42,    0, -132, ....
2*A214283(n) - A214283(n+1) =  1,  0, -1, -2,  2,  0, -5, -14,  14,   0, -42, -132, 132, ....
A214282(n) + A214283(n) = 1, 0, -1, 0, 2, 0, -5, 0, 14, 0, -42,... (A126120).
The companion to a(n) is b(n) = -A214283(n)/(1,1,1,1,2,2,5,5,...) = 0, 1, 2, 3, 2, 5, 4, 7, 4, 9, 6, ....
a(n) - b(n) = A056594(n).
Discovered as a(n) = A214282(n+1)/A000108([n/2]). See abs(A129996(n-2)).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,1,0,1,0,-1).

Cf. A130823, A090192.

I:=[1,1,1,3,3,5]; [n le 6 select I[n] else Self(n-2) +Self(n-4) -Self(n-6): n in [1..30]]; // G. C. Greubel, Apr 23 2018

a[n_?EvenQ] := n/2 + Boole[Mod[n, 4] == 0]; a[n_?OddQ] := n; Table[a[n], {n, 0, 86}] (* Jean-François Alcover, Aug 14 2012 *)
LinearRecurrence[{0,1,0,1,0,-1}, {1,1,1,3,3,5}, 50] (* G. C. Greubel, Apr 23 2018 *)

x='x+O('x^30); Vec(( 1+x+2*x^3+x^4+x^5 )/( (x^2+1)*(x-1)^2*(1+x)^2 )) \\ G. C. Greubel, Apr 23 2018

a(n+3) = (A185048(n+3)=2,2,4,2,... ) + 1.
a(n+2) - a(n) = 0, 2, 2, 2. (Period 4).
a(n) = 2*a(n-4) - a(n-8).
a(2*n) = A109613(n).
a(n+1) - a(n) = 2* (-1)^n * A059169(n).
G.f. : ( 1+x+2*x^3+x^4+x^5 ) / ( (x^2+1)*(x-1)^2*(1+x)^2 ). - Jean-François Alcover, Aug 14 2012

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A215495 a(4n) = a(4n+2) = a(2n+1) = 2n + 1.