A106191 -id:A106191 - cp's OEIS frontend

A137717 Hankel transform of A106191.

1, -4, 4, 8, -32, 32, 64, -256, 256, 512, -2048, 2048, 4096, -16384, 16384, 32768, -131072, 131072, 262144, -1048576, 1048576, 2097152, -8388608, 8388608, 16777216, -67108864, 67108864, 134217728, -536870912, 536870912

Offset: 0

Paul Barry, Feb 08 2008

sign

Hankel transform of A132310. [From Paul Barry, Apr 26 2009]

Index entries for linear recurrences with constant coefficients, signature (-2,-4).

Apart from signs, essentially the same as A096252.

LinearRecurrence[{-2,-4},{1,-4},30] (* Harvey P. Dale, Oct 05 2017 *)

G.f.: (1-2x)/(1+2x+4x^2).
a(n)=Product{k=0..n, (3*cos(2*pi*(k-1)/3)/2-5/4-2*0^k)^(n-k)};
a(n) = 2^n*A061347(n+2) = -2a(n-1)-4a(n-2). - R. J. Mathar, Feb 21 2008

A127301 Matula-Goebel signatures for plane general trees encoded by A014486.

Original entry on oeis.org

1, 2, 4, 3, 8, 6, 6, 7, 5, 16, 12, 12, 14, 10, 12, 9, 14, 19, 13, 10, 13, 17, 11, 32, 24, 24, 28, 20, 24, 18, 28, 38, 26, 20, 26, 34, 22, 24, 18, 18, 21, 15, 28, 21, 38, 53, 37, 26, 37, 43, 29, 20, 15, 26, 37, 23, 34, 43, 67, 41, 22, 29, 41, 59, 31, 64, 48, 48, 56, 40, 48, 36

Offset: 0

Antti Karttunen, Jan 16 2007

nonn

This sequence maps A000108(n) oriented (plane) rooted general trees encoded in range [A014137(n-1)..A014138(n)] of A014486 to A000081(n+1) distinct non-oriented rooted general trees, encoded by their Matula-Goebel numbers. The latter encoding is explained in A061773.
A005517 and A005518 give the minimum and maximum value occurring in each such range.
Primes occur at positions given by A057548 (not in order, and with duplicates), and similarly, semiprimes, A001358, occur at positions given by A057518, and in general, A001222(a(n)) = A057515(n).
If the signature-permutation of a Catalan automorphism SP satisfies the condition A127301(SP(n)) = A127301(n) for all n, then it preserves the non-oriented form of a general tree, which implies also that it is Łukasiewicz-word permuting, satisfying A129593(SP(n)) = A129593(n) for all n >= 0. Examples of such automorphisms include A072796, A057508, A057509/A057510, A057511/A057512, A057164, A127285/A127286 and A127287/A127288.
A206487(n) tells how many times n occurs in this sequence. - Antti Karttunen, Jan 03 2013

			A000081(n+1) distinct values occur each range [A014137(n-1)..A014138(n-1)]. As an example, A014486(5) = 44 (= 101100 in binary = A063171(5)), encodes the following plane tree:
.....o
.....|
.o...o
..\./.
...*..
Matula-Goebel encoding for this tree gives a code number A000040(1) * A000040(A000040(1)) = 2*3 = 6, thus a(5)=6.
Likewise, A014486(6) = 50 (= 110010 in binary = A063171(6)) encodes the plane tree:
.o
.|
.o...o
..\./.
...*..
Matula-Goebel encoding for this tree gives a code number A000040(A000040(1)) * A000040(1) = 3*2 = 6, thus a(6) is also 6, which shows these two trees are identical if one ignores their orientation.

Antti Karttunen, Table of n, a(n) for n = 0..6917
OEIS Wiki, Łukasiewicz words
Index entries for sequences related to Łukasiewicz
Index entries for sequences related to Matula-Goebel numbers

a(A014138(n)) = A007097(n+1), a(A014137(n)) = A000079(n+1) for all n.
a(|A106191(n)|) = A033844(n-1) for all n >= 1.
Cf. A001222, A005517, A005518, A057515, A057518, A057548, A127302, A129593, A153826, A209638, A243491, A243492, A243494, A243496.
For standard instead of binary encoding we have A358506.
A000108 counts ordered rooted trees, unordered A000081.
A014486 lists binary encodings of ordered rooted trees.
Cf. A001263, A061775, A196050, A206487, A358505.

mgnum[t_]:=If[t=={},1,Times@@Prime/@mgnum/@t];
binbalQ[n_]:=n==0||With[{dig=IntegerDigits[n,2]},And@@Table[If[k==Length[dig],SameQ,LessEqual][Count[Take[dig,k],0],Count[Take[dig,k],1]],{k,Length[dig]}]];
bint[n_]:=If[n==0,{},ToExpression[StringReplace[StringReplace[ToString[IntegerDigits[n,2]/.{1->"{",0->"}"}],","->""],"} {"->"},{"]]];
Table[mgnum[bint[n]],{n,Select[Range[0,1000],binbalQ]}] (* Gus Wiseman, Nov 22 2022 *)

(define (A127301 n) (*A127301 (A014486->parenthesization (A014486 n)))) ;; A014486->parenthesization given in A014486.
(define (*A127301 s) (if (null? s) 1 (fold-left (lambda (m t) (* m (A000040 (*A127301 t)))) 1 s)))

A001222(a(n)) = A057515(n) for all n.

A122237 a(n) = A057548(A082358(n)).

Original entry on oeis.org

1, 3, 8, 7, 22, 21, 18, 17, 20, 64, 63, 59, 58, 62, 50, 49, 46, 45, 48, 61, 57, 55, 54, 196, 195, 190, 189, 194, 176, 175, 171, 170, 174, 193, 188, 185, 184, 148, 147, 143, 142, 146, 134, 133, 130, 129, 132, 145, 141, 139, 138, 192, 187, 173, 169, 183, 181, 167

Offset: 0

Antti Karttunen, Sep 14 2006

nonn

Iterates: A106191, A122241, A122244. Cf. also A122227, A080067.

A137719 Sequence based on the pattern [3n, 3n, 3n, 3n+2, 3n+1, 3n+2].

Original entry on oeis.org

0, 2, 1, 2, 3, 3, 3, 5, 4, 5, 6, 6, 6, 8, 7, 8, 9, 9, 9, 11, 10, 11, 12, 12, 12, 14, 13, 14, 15, 15, 15, 17, 16, 17, 18, 18, 18, 20, 19, 20, 21, 21, 21, 23, 22, 23, 24, 24, 24, 26, 25, 26, 27, 27, 27, 29, 28, 29, 30, 30, 30, 32, 31, 32, 33, 33, 33, 35, 34

Offset: 0

Paul Barry, Feb 08 2008

Powers of 2 in a scaled version of the Hankel transform of A106191.

M. F. Hasler, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,1,1,0,-1).

Cf. A106191, A137718.

[&+[(2*n-i) mod 3: i in [0..Floor(n/2)]]: n in [0..80]]; // Wesley Ivan Hurt, Mar 21 2016

A137719:=n->add(2*n-i mod 3, i=0..floor(n/2)): seq(A137719(n), n=0..100); # Wesley Ivan Hurt, Mar 21 2016

Table[Sum[Mod[2 n - i, 3], {i, 0, Floor[n/2]}], {n, 0, 80}] (* Wesley Ivan Hurt, Mar 21 2016 *)

apply( A137719(n)={(n=divrem(n-1,6))[1]*3+min(n[2]+2*!n[2],3)}, [0..30]) \\ M. F. Hasler, Oct 27 2019

a(n) = log(abs(A137718(n)))/log(2).
From R. J. Mathar, Feb 10 2008: (Start)
O.g.f.: 1/(2*(x-1)^2) + (x-1)/(3*(x^2+x+1)) - 1/(4*(x+1)) - 1/(12*(x-1)).
a(n) = 3 + a(n-6). (End)
From Colin Barker, Jun 27 2013: (Start)
a(n) = a(n-2) + a(n-3) - a(n-5).
G.f.: x*(x+2) / ((x-1)^2*(x+1)*(x^2+x+1)). (End)
a(n) = Sum_{i=0..floor(n/2)} (2n-i mod 3). - Wesley Ivan Hurt, Mar 22 2016
a(n) = A004526(n+1) + A079978(n). - R. J. Mathar, Oct 27 2019

A371965 a(n) is the sum of all peaks in the set of Catalan words of length n.

Original entry on oeis.org

0, 0, 0, 1, 6, 27, 111, 441, 1728, 6733, 26181, 101763, 395693, 1539759, 5997159, 23381019, 91244934, 356427459, 1393585779, 5453514729, 21358883439, 83718027429, 328380697629, 1288947615849, 5062603365999, 19896501060225, 78239857877649, 307831771279549, 1211767933187601

Offset: 0

Stefano Spezia, Apr 14 2024

nonn

			a(3) = 1 because there is 1 Catalan word of length 3 with one peak: 010.
a(4) = 6 because there are 6 Catalan words of length 4 with one peak: 0010, 0100, 0101, 0110, 0120, and 0121 (see Figure 10 at p. 19 in Baril et al.).

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Jean-Luc Baril, Pamela E. Harris, Kimberly J. Harry, Matt McClinton, and José L. Ramírez, Enumerating runs, valleys, and peaks in Catalan words, arXiv:2404.05672 [math.CO], 2024. See Corollary 4.7, p. 19.

Cf. A371963, A371964.

Cf. A002054, A275289.

a:= proc(n) option remember; `if`(n<3, 0,
      a(n-1)+binomial(2*n-3, n-3))
    end:
seq(a(n), n=0..28);  # Alois P. Heinz, Apr 15 2024
# Second Maple program:
A371965 := series((exp(2*x)*BesselI(0,2*x)-1)/2-exp(x)*(int(BesselI(0,2*x)*exp(x), x)), x = 0, 29):
seq(n!*coeff(A371965, x, n), n = 0 .. 28); # Mélika Tebni, Jun 15 2024

CoefficientList[Series[(1-3x-(1-x)Sqrt[1-4x])/(2(1-x) Sqrt[1-4x]),{x,0,28}],x]

from math import comb
def A371965(n): return sum(comb((n-i<<1)-3,n-i-3) for i in range(n-2)) # Chai Wah Wu, Apr 15 2024

G.f.: (1 - 3*x - (1 - x)*sqrt(1 - 4*x))/(2*(1 - x)*sqrt(1 - 4*x)).
a(n) = Sum_{i=1..n-1} binomial(2*(n-i)-1,n-i-2).
a(n) ~ 2^(2*n)/(6*sqrt(Pi*n)).
a(n)/A371963(n) ~ 1.
a(n) - a(n-1) = A002054(n-2).
From Mélika Tebni, Jun 15 2024: (Start)
E.g.f.: (exp(2*x)*BesselI(0,2*x)-1)/2 - exp(x)*Integral_{x=-oo..oo} BesselI(0,2*x)*exp(x) dx.
a(n) = binomial(2*n,n)*(1/2 + hypergeom([1,n+1/2],[n+1],4)) + i/sqrt(3) - 0^n/2.
a(n) = (3*A106191(n) + A006134(n) + 4*0^n) / 8.
a(n) = A281593(n) - (A000984(n) + 0^n) / 2. (End)
Binomial transform of A275289. - Alois P. Heinz, Jun 20 2025

A080675 a(n) = (5*4^n - 8)/6.

Original entry on oeis.org

2, 12, 52, 212, 852, 3412, 13652, 54612, 218452, 873812, 3495252, 13981012, 55924052, 223696212, 894784852, 3579139412, 14316557652, 57266230612, 229064922452, 916259689812, 3665038759252, 14660155037012, 58640620148052, 234562480592212, 938249922368852

Offset: 1

N. J. A. Sloane, Mar 02 2003

These numbers have a simple binary pattern: 10,1100,110100,11010100,1101010100, ... i.e., the n-th term has a binary expansion 1(10){n-1}0, that is, there are n-1 10's between the most significant 1 and the least significant 0.

Vincenzo Librandi, Table of n, a(n) for n = 1..170
Andrei Asinowski, Cyril Banderier, Benjamin Hackl, On extremal cases of pop-stack sorting, Permutation Patterns (Zürich, Switzerland, 2019).
Index entries for linear recurrences with constant coefficients, signature (5,-4).

a(n) = A072197(n-1) - 1 = A014486(|A106191(n)|). a(n) = A079946(A020988(n-2)) for n>=2. Cf. also A122229.

[(5*4^n-8)/6: n in [1..40] ]; // Vincenzo Librandi, Apr 28 2011

(5*4^Range[30]-8)/6 (* or *) LinearRecurrence[{5,-4},{2,12},30] (* Harvey P. Dale, Oct 16 2012 *)

a(n)=(5*4^n-8)/6 \\ Charles R Greathouse IV, Oct 07 2015

a(1)=2, a(2)=12, a(n)=5*a(n-1)-4*a(n-2). - Harvey P. Dale, Oct 16 2012

Further comments added by Antti Karttunen, Sep 14 2006

A122228 Iterates of A122227, starting from 0.

Original entry on oeis.org

0, 1, 3, 8, 20, 55, 160, 493, 1579, 5212, 17595, 60462, 210749, 743284, 2647461, 9509504

Offset: 0

Antti Karttunen, Sep 14 2006

nonn

Cf. A122229, A122230, A106191.

(define (A122228 n) (if (zero? n) 0 (A122227 (A122228 (- n 1)))))

A106190 Triangle read by rows: T(n,k) = binomial(2(n-k),n-k)/(1-2(n-k)).

Original entry on oeis.org

1, -2, 1, -2, -2, 1, -4, -2, -2, 1, -10, -4, -2, -2, 1, -28, -10, -4, -2, -2, 1, -84, -28, -10, -4, -2, -2, 1, -264, -84, -28, -10, -4, -2, -2, 1, -858, -264, -84, -28, -10, -4, -2, -2, 1, -2860, -858, -264, -84, -28, -10, -4, -2, -2, 1, -9724, -2860, -858, -264, -84, -28, -10, -4, -2, -2, 1, -33592, -9724, -2860, -858

Offset: 0

Paul Barry, Apr 24 2005

Sequence array for expansion of sqrt(1-4x).
Row sums are A106191. Diagonal sums are A106192. Sequence array for A002420. Inverse of number triangle A106187.
Riordan array (sqrt(1-4x),x).

			Triangle begins
1;
-2,1;
-2,-2,1;
-4,-2,-2,1;
-10,-4,-2,-2,1;
-28,-10,-4,-2,-2,1;

T[n_, k_] := Binomial[2(n - k), n - k]/(1 - 2(n - k)); Flatten[ Table[ T[n, k], {n, 0, 10}, {k, 0, n}]] (* Robert G. Wilson v, Apr 25 2005 *)

More terms from Robert G. Wilson v, Apr 25 2005

A137720 Expansion of sqrt(1-4x)/(1-3x).

Original entry on oeis.org

1, 1, 1, -1, -13, -67, -285, -1119, -4215, -15505, -56239, -202309, -724499, -2589521, -9254363, -33111969, -118725597, -426892131, -1539965973, -5575175319, -20260052337, -73908397851, -270657727593, -994938310059

Offset: 0

Paul Barry, Feb 08 2008

Hankel transform is A120617. In general, sqrt(1-4*x)/(1-k*x) has Hankel transform with g.f. of (1-2*x)/(1+2*(k+2)*x+4*x^2).

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A126966, A106191, A157674.

CoefficientList[Series[Sqrt[1-4*x]/(1-3*x), {x, 0, 20}], x] (* Vaclav Kotesovec, Jul 31 2014 *)
FullSimplify[Table[I*3^(-1/2+n) + 2^(1+2*n)*Gamma[1/2+n] * Hypergeometric2F1Regularized[1, 1/2+n, 2+n, 4/3]/(3*Sqrt[Pi]), {n, 0, 20}]] (* Vaclav Kotesovec, Jul 31 2014 *)

x='x+O('x^50); Vec(sqrt(1-4*x)/(1-3*x)) \\ G. C. Greubel, Mar 21 2017

a(n) = Sum_{k=0..n} 3^k*C(2*n-2*k,n-k)/(1-(2*n-2*k)).
D-finite with recurrence: n*a(n) + (6-7*n)*a(n-1) + 6*(2*n-3)*a(n-2) = 0. - R. J. Mathar, Nov 16 2011
a(n) ~ -2^(2*n+1) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Jul 31 2014
a(n) = (-1)^n * A157674(2*n+1). - Vaclav Kotesovec, Jul 31 2014

A137718 A scaled Hankel transform.

Original entry on oeis.org

1, -4, 2, 4, -8, 8, 8, -32, 16, 32, -64, 64, 64, -256, 128, 256, -512, 512, 512, -2048, 1024, 2048, -4096, 4096, 4096, -16384, 8192, 16384, -32768, 32768, 32768, -131072, 65536, 131072, -262144, 262144, 262144

Offset: 0

Paul Barry, Feb 08 2008

A137717=2^floor(n/2)*a(n) is the Hankel transform of A106191.

Cf. A137719.

a(n)=A137717(n)/2^floor(n/2).

Empirical g.f.: -(4*x^3-4*x^2+4*x-1) / (4*x^4+2*x^2+1). - Colin Barker, Jun 27 2013

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A137717 Hankel transform of A106191.

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A127301 Matula-Goebel signatures for plane general trees encoded by A014486.

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A122237 a(n) = A057548(A082358(n)).

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A137719 Sequence based on the pattern [3n, 3n, 3n, 3n+2, 3n+1, 3n+2].

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A371965 a(n) is the sum of all peaks in the set of Catalan words of length n.

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A080675 a(n) = (5*4^n - 8)/6.

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A122228 Iterates of A122227, starting from 0.

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A106190 Triangle read by rows: T(n,k) = binomial(2(n-k),n-k)/(1-2(n-k)).

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A137720 Expansion of sqrt(1-4x)/(1-3x).