A232500 -id:A232500 - cp's OEIS frontend

A274708 A statistic on orbital systems over n sectors: the number of orbitals with k peaks.

1, 1, 2, 4, 2, 4, 2, 12, 15, 3, 10, 8, 2, 38, 68, 30, 4, 26, 30, 12, 2, 121, 272, 183, 49, 5, 70, 104, 60, 16, 2, 384, 1026, 912, 372, 72, 6, 192, 350, 260, 100, 20, 2, 1214, 3727, 4095, 2220, 650, 99, 7, 534, 1152, 1050, 520, 150, 24, 2, 3822, 13200, 17178, 11600, 4510, 1032, 130, 8

Offset: 0

Peter Luschny, Jul 10 2016

The definition of an orbital system is given in A232500 (see also the illustration there). The number of orbitals over n sectors is counted by the swinging factorial A056040.
An orbital w has a 'peak' at i+1 when signum(w[i]) < signum(w[i+1]) and signum(w[i+1]) > signum(w[i+2]).
A097692 is a subtriangle.

			Triangle read by rows, n>=0. The length of row n is floor((n+1)/2) for n>=1.
[ n] [k=0,1,2,...]               [row sum]
[ 0] [  1]                           1
[ 1] [  1]                           1
[ 2] [  2]                           2
[ 3] [  4,    2]                     6
[ 4] [  4,    2]                     6
[ 5] [ 12,   15,   3]               30
[ 6] [ 10,    8,   2]               20
[ 7] [ 38,   68,  30,   4]         140
[ 8] [ 26,   30,  12,   2]          70
[ 9] [121,  272, 183,  49,  5]     630
[10] [ 70,  104,  60,  16,  2]     252
[11] [384, 1026, 912, 372, 72, 6] 2772
[12] [192,  350, 260, 100, 20, 2]  924
T(6, 2) = 2 because the two orbitals [-1, 1, -1, 1, -1, 1] and [1, -1, 1, -1, 1, -1] have 2 peaks.

Peter Luschny, Orbitals

Cf. A025565 (even col. 0), A056040 (row sum), A097692, A232500.

Other orbital statistics: A241477 (first zero crossing), A274706 (absolute integral), A274709 (max. height), A274710 (number of turns), A274878 (span), A274879 (returns), A274880 (restarts), A274881 (ascent).

# uses[unit_orbitals from A274709]
# Brute force counting
def orbital_peaks(n):
    if n == 0: return [1]
    S = [0]*((n+1)//2)
    for u in unit_orbitals(n):
        L = [1 if sgn(u[i]) < sgn(u[i+1]) and sgn(u[i+1]) > sgn(u[i+2]) else 0 for i in (0..n-3)]
        S[sum(L)] += 1
    return S
for n in (0..12): print(orbital_peaks(n))

A274888 Triangle read by rows: the q-analog of the swinging factorial which is defined as q-multinomial([floor(n/2), n mod 2, floor(n/2)]).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 2, 4, 5, 6, 5, 4, 2, 1, 1, 1, 2, 3, 3, 3, 3, 2, 1, 1, 1, 2, 4, 7, 10, 13, 16, 17, 17, 16, 13, 10, 7, 4, 2, 1, 1, 1, 2, 3, 5, 5, 7, 7, 8, 7, 7, 5, 5, 3, 2, 1, 1, 1, 2, 4, 7, 12, 17, 24, 31, 39, 45, 51, 54, 56, 54, 51, 45, 39, 31, 24, 17, 12, 7, 4, 2, 1

Offset: 0

Peter Luschny, Jul 19 2016

The q-swing_factorial(n) is a univariate polynomial over the integers with degree floor((n+1)/2)^2 + ((n+1) mod 2) and at least floor(n/2) irreducible factors.
Evaluated at q=1 q-swing_factorial(n) gives the swinging factorial A056040(n).
Combinatorial interpretation: The definition of an orbital system is given in A232500 and in the link 'Orbitals'. The number of orbitals over n sectors is counted by the swinging factorial.
The major index of an orbital is the sum of the positions of steps which are immediately followed by a step with strictly smaller value. This statistic is an extension of the major index statistic given in A063746 which reappears in the even numbered rows here. This reflects the fact that the swinging factorial can be seen as an extension of the central binomial. As in the case of the central binomial also in the case of the swinging factorial the major index coincides with its q-analog.

			The polynomials start:
[0] 1
[1] 1
[2] q + 1
[3] (q + 1) * (q^2 + q + 1)
[4] (q^2 + 1) * (q^2 + q + 1)
[5] (q^2 + 1) * (q^2 + q + 1) * (q^4 + q^3 + q^2 + q + 1)
[6] (q + 1) * (q^2 - q + 1) * (q^2 + 1) * (q^4 + q^3 + q^2 + q + 1)
The coefficients of the polynomials start:
[n] [k=0,1,2,...] [row sum]
[0] [1] [1]
[1] [1] [1]
[2] [1, 1] [2]
[3] [1, 2, 2, 1] [6]
[4] [1, 1, 2, 1, 1] [6]
[5] [1, 2, 4, 5, 6, 5, 4, 2, 1] [30]
[6] [1, 1, 2, 3, 3, 3, 3, 2, 1, 1] [20]
[7] [1, 2, 4, 7, 10, 13, 16, 17, 17, 16, 13, 10, 7, 4, 2, 1] [140]
[8] [1, 1, 2, 3, 5, 5, 7, 7, 8, 7, 7, 5, 5, 3, 2, 1, 1] [70]
T(5, 4) = 6 because the 2 orbitals [-1,-1,1,1,0] and [-1,0,1,1,-1] have at position 4 and the 4 orbitals [0,-1,1,-1,1], [1,-1,0,-1,1], [1,-1,1,-1,0] and [1,0,1,-1,-1] at positions 1 and 3 a down step.

Peter Luschny, Orbitals

Cf. A056040 (row sums), A274887 (q-factorial), A063746 (q-central_binomial).

Other orbital statistics: A241477 (first zero crossing), A274706 (absolute integral), A274708 (peaks), A274709 (max. height), A274710 (number of turns), A274878 (span), A274879 (returns), A274880 (restarts), A274881 (ascent).

QSwingFactorial_coeffs := proc(n) local P,a,b;
a := mul((p^(n-i)-1)/(p^(i+1)-1),i=0..iquo(n,2)-1);
b := ((p^(iquo(n,2)+1)-1)/(p-1))^((1-(-1)^n)/2);
P := simplify(a*b); seq(coeff(P,p,j),j=0..degree(P)) end:
for n from 0 to 9 do print(QSwingFactorial_coeffs(n)) od;
# Alternatively (recursive):
with(QDifferenceEquations):
QSwingRec := proc(n,q) local r; if n = 0 then return 1 fi:
if irem(n,2) = 0 then r := (1+q^(n/2))/QBrackets(n/2,q)
else r := QBrackets(n,q) fi; r*QSwingRec(n-1,q) end:
Trow := proc(n) expand(QSimplify(QSwingRec(n,q)));
seq(coeff(%,q,j),j=0..degree(%)) end: seq(Trow(n),n=0..10);

p[n_] := QFactorial[n, q] / QFactorial[Quotient[n, 2], q]^2
Table[CoefficientList[p[n] // FunctionExpand, q], {n,0,9}] // Flatten

from sage.combinat.q_analogues import q_factorial
def q_swing_factorial(n, q=None):
    return q_factorial(n)//q_factorial(n//2)^2
for n in (0..8): print(q_swing_factorial(n).list())

# uses[unit_orbitals from A274709]
# Brute force counting
def orbital_major_index(n):
    S = [0]*(((n+1)//2)^2 + ((n+1) % 2))
    for u in unit_orbitals(n):
        L = [i+1 if u[i+1] < u[i] else 0 for i in (0..n-2)]
        # i+1 because u is 0-based whereas convention assumes 1-base.
        S[sum(L)] += 1
    return S
for n in (0..9): print(orbital_major_index(n))

q_swing_factorial(n) = q_factorial(n)/q_factorial(floor(n/2))^2.
q_swing_factorial(n) = q_binomial(n-eta(n),floor((n-eta(n))/2))*q_int(n)^eta(n) with eta(n) = (1-(-1)^n)/2.
Recurrence: q_swing_factorial(0,q) = 1 and for n>0 q_swing_factorial(n,q) = r*q_swing_factorial(n-1,q) with r = (1+q^(n/2))/[n/2;q] if n is even else r = [n;q]. Here [a;q] are the q_brackets.
The generating polynomial for row n is P_n(p) = ((p^(floor(n/2)+1)-1)/(p-1))^((1-(-1)^n)/2)*Product_{i=0..floor(n/2)-1}((p^(n-i)-1)/(p^(i+1)-1)).

A241543 a(n) = A241477(n, n).

Original entry on oeis.org

1, 1, 2, 2, 2, 6, 4, 20, 10, 70, 28, 252, 84, 924, 264, 3432, 858, 12870, 2860, 48620, 9724, 184756, 33592, 705432, 117572, 2704156, 416024, 10400600, 1485800, 40116600, 5348880, 155117520, 19389690, 601080390, 70715340, 2333606220, 259289580, 9075135300

Offset: 0

Peter Luschny, Apr 25 2014

nonn

See A241477 and A232500 for the combinatorial definitions.

Cf. A057977, A241477, A232500.

A241543 := proc(n)
    if n < 2 then 1
  else 2*iquo(n,2)*(n-2)!/iquo(n,2)!^2
    fi end:
seq(A241543(n), n=0..37);

a(n) = 2*floor(n/2)*(n-2)!/floor(n/2)!^2 for n>=2.

a(n+2) = 2*A057977(n) for n>=0. - Peter Luschny, Jul 17 2016

A241810 Number of balanced orbitals over n sectors.

Original entry on oeis.org

1, 1, 0, 0, 2, 6, 0, 6, 8, 36, 0, 88, 58, 376, 0, 1096, 526, 4476, 0, 14200, 5448, 57284, 0, 190206, 61108, 764812, 0, 2615268, 723354, 10499504, 0, 36677626, 8908546, 147110276, 0, 522288944, 113093022

Offset: 0

Peter Luschny, Apr 29 2014

For the combinatorial definitions see A232500. An orbital is balanced if its integral is 0. The integral of an orbital w over n sectors is Sum_{k=1..n} Sum_{i=1..k} w(i) where w(i) are the jumps of the orbital represented by -1, 0, 1.

Cf. A232500, A242087.

np[z_]:=Module[{i,j},For[i=Length[z],i>1&&z[[i-1]]>=z[[i]],i--];For[j=Length[z],z[[j]]<=z[[i-1]],j--];Join[Take[z,i-2],{z[[j]]},Reverse[Drop[ReplacePart[z,z[[i-1]],j],i-1]]]];o=Table[1,{16}];
n=0;f=0;Print[1];Print[1];While[n<16,n++;f=1-f;If[OddQ[f*n],Print[0],p=Join[-Take[o,n],{f},Take[o,n-f]];c=0;Do[If[Accumulate[Accumulate[p]][[-1]]==0,c++];p=np[p],{(2*n+1-f)!/(2*n!^2)}];Print[2*c]];n=n-f]
(* Hans Havermann, May 10 2014 *)

def A241810(n):
    if n == 0: return 1
    A = 0
    T = [0] if is_odd(n) else []
    for i in (1..n//2):
        T.append(-1); T.append(1)
    for p in Permutations(T):
        P = 0; S = 0
        for k in (0..n-1):
            P += p[k]; S += P
        if S == 0: A += 1
    return A
[A241810(n) for n in (0..32)]

a(2*n) = A204459(2, n).
a(2*n+1) = A242087(n).
a(4*n) = A063074(n) = A029895(2*n) = A067059(2*n, 2*n).
a(4*n+2) = 0 for all n (proved by H. Havermann).

More terms from Hans Havermann, May 10 2014

a(35), a(36) from Hans Havermann, May 23 2014

A242087 Number of balanced orbitals over an odd number of sectors.

Original entry on oeis.org

1, 0, 6, 6, 36, 88, 376, 1096, 4476, 14200, 57284, 190206, 764812, 2615268, 10499504, 36677626, 147110276, 522288944

Offset: 0

Peter Luschny, May 04 2014

See A241810 and A232500 for the combinatorial definitions.

np[z_]:=Module[{i,j},For[i=Length[z],i>1&&z[[i-1]]>=z[[i]],i--]; For[j=Length[z],z[[j]]<=z[[i-1]],j--]; Join[Take[z,i-2],{z[[j]]}, Reverse[Drop[ReplacePart[z,z[[i-1]],j],i-1]]]]; o=Table[1,{16}];
Print[1]; Do[p=Join[-Take[o,n],{0},Take[o,n]]; c=0; Do[If[Accumulate[Accumulate[p]][[-1]]==0,c++]; p=np[p],{(2*n+1)!/(2*n!^2)}]; Print[2*c],{n,16}]
(* Hans Havermann, May 10 2014 *)

def A242087(n):
    if n == 0: return 1
    A = 0; T = [0]
    for i in (1..n):
        T.append(-1); T.append(1)
    for p in Permutations(T):
        P = 0; S = 0
        for k in (0..2*n):
            P += p[k]; S += P
        if S == 0: A += 1
    return A
[A242087(n) for n in (0..10)]

a(n) = A241810(2*n+1).

More terms from Hans Havermann, May 10 2014

a(17) from Hans Havermann, May 23 2014

A276666 a(n) = (n-1)*Catalan(n).

Original entry on oeis.org

-1, 0, 2, 10, 42, 168, 660, 2574, 10010, 38896, 151164, 587860, 2288132, 8914800, 34767720, 135727830, 530365050, 2074316640, 8119857900, 31810737420, 124718287980, 489325340400, 1921133836440, 7547311500300, 29667795388452, 116686713634848, 459183826803800

Offset: 0

Peter Luschny, Sep 12 2016

sign

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

A024483 is a variant of this sequence.

Cf. A000108, A000984, A001622, A051631, A056040, A057977, A232500.

Concatenation([-1], List([1..30], n-> 2*Binomial(2*n-1, n+1))); # G. C. Greubel, Aug 29 2019

[(n-1)*Catalan(n): n in [0..30]]; // Vincenzo Librandi, Sep 13 2016

f := (1-3*x)/(x*sqrt(1-4*x))-1/x:
series(f,x,29): seq(coeff(%,x,n), n=0..26);
A276666 := n -> (n^2-1)*(2*n)!/(n+1)!^2:
seq(A276666(n), n=0..26);

Table[(n - 1) CatalanNumber[n], {n, 0, 30}] (* Vincenzo Librandi, Sep 13 2016 *)

a(n) = if(n==0,-1, 2*binomial(2*n-1, n+1)); \\ G. C. Greubel, Aug 29 2019

A276666 = lambda n: (n - 1) * catalan_number(n)
[A276666(n) for n in range(27)]

a(n) = [x^n] (1-3*x)/(x*sqrt(1-4*x))-1/x.
a(n) = 4^n*(n-1)*hypergeom([3/2, -n], [2], 1).
a(n) = 4^n*(n-1)*JacobiP(n,1,-1/2-n,-1)/(n+1).
a(n) = (2*n)! [x^(2^n)]( BesselI(2,2*x) - (1+1/x)*BesselI(1,2*x) ).
a(n) = binomial(2*n,n) - 2*Catalan(n). (See Geoffrey Critzer's formula in A024483).
a(n) = A056040(2*n) - 2*A057977(2*n).
a(n) = A056040(2*n)*(1-2/(n+1)) = (n^2-1)*(2*n)!/(n+1)!^2.
a(n) = A232500(2*n).
a(n) = a(n-1)*2*(n-1)*(2*n-1)/((n-2)*(n+1)) for n > 2. - Chai Wah Wu, Sep 12 2016
a(n) = A024483(n+1) for n>0. - R. J. Mathar, Sep 13 2016
a(n) = A000984(n+1)-3*A000984(n). - Ezhilarasu Velayutham, Aug 27 2019
From Amiram Eldar, Mar 22 2022: (Start)
Sum_{n>=2} 1/a(n) = 5/6 - Pi/(9*sqrt(3)).
Sum_{n>=2} (-1)^n/a(n) = 26*sqrt(5)*log(phi)/25 - 7/10, where phi is the golden ratio (A001622). (End)

A275325 Triangle read by rows: number of orbitals over n sectors which have a Catalan decomposition into k parts.

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 6, 0, 4, 2, 0, 20, 10, 0, 10, 8, 2, 0, 70, 56, 14, 0, 28, 28, 12, 2, 0, 252, 252, 108, 18, 0, 84, 96, 54, 16, 2, 0, 924, 1056, 594, 176, 22, 0, 264, 330, 220, 88, 20, 2, 0, 3432, 4290, 2860, 1144, 260, 26, 0, 858, 1144, 858, 416, 130, 24, 2

Offset: 0

Peter Luschny, Aug 15 2016

The definition of an orbital system is given in A232500.
The Catalan decomposition of an orbital w is a list of orbitals which are alternately entirely above or below the main circle ('above' and 'below' in the weak sense) such that their concatenation equals w. If a zero is on the border of two orbitals then it is allocated to the first one. By convention T(0,0) = 1.
The number of orbitals over n sectors is counted by the swinging factorial A056040.

			Table starts:
[ n] [k=0,1,2,...] [row sum]
[ 0] [1] 1
[ 1] [0, 1] 1
[ 2] [0, 2] 2
[ 3] [0, 6] 6
[ 4] [0, 4, 2] 6
[ 5] [0, 20, 10] 30
[ 6] [0, 10, 8, 2] 20
[ 7] [0, 70, 56, 14] 140
[ 8] [0, 28, 28, 12, 2] 70
[ 9] [0, 252, 252, 108, 18] 630
[10] [0, 84, 96, 54, 16, 2] 252
[11] [0, 924, 1056, 594, 176,  22] 2772
[12] [0, 264, 330, 220, 88, 20, 2] 924
For example T(2*n, n) = 2 counts the Catalan decompositions
[[-1, 1], [1, -1], [-1, 1], ..., [(-1)^n, (-1)^(n+1)]] and
[[1, -1], [-1, 1], [1, -1], ..., [(-1)^(n+1), (-1)^n]].

Peter Luschny, Orbitals

Cf. A056040 (row sum), A241543, A232500, A266722, A275326.

Other orbital statistics: A241477 (first zero crossing), A274706 (absolute integral), A274709 (max. height), A274710 (number of turns), A274878 (span), A274879 (returns), A274880 (restarts), A274881 (ascent).

# uses[unit_orbitals from A274709]
# Brute force counting
from itertools import accumulate
def catalan_factors(P):
    def bisect(orb):
        i = 1
        A = list(accumulate(orb))
        if orb[1] > 0 if orb[0] == 0 else orb[0] > 0:
            while i < len(A) and A[i] >= 0: i += 1
        else:
            while i < len(A) and A[i] <= 0: i += 1
        return i
    R = []
    while P:
        i = bisect(P)
        R.append(P[:i])
        P = P[i:]
    return R
def orbital_factors(n):
    if n == 0: return [1]
    if n == 1: return [0, 1]
    S = [0]*(n//2 + 1)
    for o in unit_orbitals(n):
        S[len(catalan_factors(o))] += 1
    return S
for n in (0..9): print(orbital_factors(n))

T(n,1) = 2*floor((n+2)/2)*n!/floor((n+2)/2)!^2 = A241543(n+2) for n>=2.
For odd n>1 T(n,1) = Sum_{k>=0} T(n+1,k).
A056040(n) - T(n,1) = A232500(n) for n>=2.
Main diagonal: T(n, floor(n/2)) = A266722(n) for n>1.
A275326(n,k) = ceiling(T(n,k)/2).

A275333 Triangle read by rows, the break statistic on orbital systems over n sectors.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 1, 1, 2, 1, 1, 3, 3, 6, 6, 6, 3, 3, 1, 1, 2, 3, 3, 3, 3, 2, 1, 1, 4, 4, 8, 12, 16, 16, 20, 16, 16, 12, 8, 4, 4, 1, 1, 2, 3, 5, 5, 7, 7, 8, 7, 7, 5, 5, 3, 2, 1, 1, 5, 5, 10, 15, 25, 30, 40, 45, 55, 55, 60, 55, 55, 45, 40, 30, 25, 15, 10, 5, 5

Offset: 0

Peter Luschny, Jul 23 2016

The definition of an orbital system is given in A232500. The number of orbitals over n sectors is counted by the swinging factorial A056040.

The break index of an orbital is the sum of the positions of the up steps that are immediately followed by a step which is not an up step. This statistic is an extension of the major index statistic given in A063746 which appears as the even numbered rows here. This reflects the fact that the swinging factorial can be seen as an extension of the central binomial. The break index is different from the major index of the swinging factorial (which is in A274888).

			The length of row n is floor(n^2/4 + 1). Triangle starts:
[n] [k=0,1,2,...] [row sum]
[0] [1] 1
[1] [1] 1
[2] [1, 1] 2
[3] [2, 2, 2] 6
[4] [1, 1, 2, 1, 1] 6
[5] [3, 3, 6, 6, 6, 3, 3] 30
[6] [1, 1, 2, 3, 3, 3, 3, 2, 1, 1] 20
[7] [4, 4, 8, 12, 16, 16, 20, 16, 16, 12, 8, 4, 4] 140
[8] [1, 1, 2, 3, 5, 5, 7, 7, 8, 7, 7, 5, 5, 3, 2, 1, 1] 70
[9] [5, 5, 10, 15, 25, 30, 40, 45, 55, 55, 60, 55, 55, 45, 40, 30,25,15,10,5,5] 630
T(5, 5) = 3 because the three orbitals [1, -1, -1, 1, 0], [1, -1, 0, 1, -1] and [1, 0, -1, 1, -1] have at position 1 and position 4 an up-step which is immediately followed by a step which is not an up-step.

Peter Luschny, Orbitals

Cf. A056040 (row sum), A063746 (sub triangle), A274888 (q-swinging factorial).

Other orbital statistics: A241477 (first zero crossing), A274706 (absolute integral), A274708 (peaks), A274709 (max. height), A274710 (number of turns), A274878 (span), A274879 (returns), A274880 (restarts), A274881 (ascent).

# uses[unit_orbitals from A274709]
# Brute force counting
def orbital_break_index(n):
    S = [0]*(n^2//4 + 1)
    for u in unit_orbitals(n):
        L = [i+1 if u[i] == 1 and u[i+1] != 1 else 0 for i in (0..n-2)]
        #    i+1 because u is 0-based
        S[sum(L)] += 1
    return S
for n in (0..9): print(orbital_break_index(n))

cp's OEIS Frontend

A274708 A statistic on orbital systems over n sectors: the number of orbitals with k peaks.

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A274888 Triangle read by rows: the q-analog of the swinging factorial which is defined as q-multinomial([floor(n/2), n mod 2, floor(n/2)]).

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A241543 a(n) = A241477(n, n).

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A241810 Number of balanced orbitals over n sectors.

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A242087 Number of balanced orbitals over an odd number of sectors.

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A276666 a(n) = (n-1)*Catalan(n).

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A275325 Triangle read by rows: number of orbitals over n sectors which have a Catalan decomposition into k parts.

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