A007043 -id:A007043 - cp's OEIS frontend

A007060 Number of ways n married couples can sit in a row without any spouses next to each other.

1, 0, 8, 240, 13824, 1263360, 168422400, 30865121280, 7445355724800, 2287168006717440, 871804170613555200, 403779880746418176000, 223346806774106790297600, 145427383048755178635264000, 110105698060190464791596236800, 95914116314126658718742347776000, 95252504853751428295192341381120000

Offset: 0

David Roberts Keeney (David.Roberts.Keeney(AT)directory.Reed.edu)

Limit_{n->oo} a(n)/(2n)! = 1/e.
Also the number of (directed) Hamiltonian paths of the n-cocktail party graph. - Eric W. Weisstein, Dec 16 2013
Also the number of ways to label the cells of a 2 X n grid such that no vertically adjacent cells have adjacent labels. - Sela Fried, May 29 2023

			For n = 2, the a(2) = 8 solutions for the couples {1,2} and {3,4} are {1324, 1423, 2314, 2413, 3142, 3241, 4132, 4231}.

Andrew Woods, Table of n, a(n) for n = 0..100
G. Almkvist, Letter to N. J. A. Sloane, Apr. 1992.
Eric Weisstein's World of Mathematics, Cocktail Party Graph.
Eric Weisstein's World of Mathematics, Hamiltonian Path.

Cf. A000166, A000806, A114938, A177840, A053983, A053984, A193624, A229430, A330266.

seq(add((-1)^i*binomial(n, i)*2^i*(2*n-i)!, i=0..n),n=0..20);

Table[Sum[(-1)^i Binomial[n,i] (2 n - i)! 2^i, {i, 0, n}], {n, 0, 20}]
Table[(2 n)! Hypergeometric1F1[-n, -2 n, -2], {n, 0, 20}]

a(n)=sum(k=0, n, binomial(n, k)*(-1)^(n-k)*(n+k)!*2^(n-k)) \\ Charles R Greathouse IV, May 11 2016

from sympy import binomial, subfactorial
def a(n): return sum([(-1)**(n - k)*binomial(n, k)*subfactorial(2*k) for k in range(n + 1)]) # Indranil Ghosh, Apr 28 2017

a(n) = (Pi*BesselI(n+1/2,1)*(-1)^n+BesselK(n+1/2,1))*exp(-1)*(2/Pi)^(1/2)*2^n*n!. - Mark van Hoeij, Nov 12 2009
a(n) = (-1)^n*2^n*n!*A000806(n), n>0. - Vladeta Jovovic, Nov 19 2009
a(n) = n!*hypergeom([-n, n+1],[],1/2)*(-2)^n. - Mark van Hoeij, Nov 13 2009
a(n) = 2^n * A114938(n). - Toby Gottfried, Nov 22 2010
a(n) = 2*n((2*n-1)*a(n-1) + (2*n-2)*a(n-2)), n > 1. - Aaron Meyerowitz, May 14 2014
From Peter Bala, Mar 06 2015: (Start)
a(n) = Sum_{k = 0..n} (-1)^(n-k)*binomial(n,k)*A000166(2*k).
For n >= 1, Integral_{x = 0..1} (x^2 - 1)^n*exp(x) dx = a(n)*e - A177840(n). Hence lim_{n->oo} A177840(n)/a(n) = e. (End)
a(n) ~ sqrt(Pi) * 2^(2*n+1) * n^(2*n + 1/2) / exp(2*n+1). - Vaclav Kotesovec, Mar 09 2016

More terms from Michel ten Voorde, Apr 11 2001

A264607 Degeneracies of entanglement witness eigenstates for spin 3/2 particles.

Original entry on oeis.org

1, 1, 4, 34, 364, 4269, 52844, 679172, 8976188, 121223668, 1665558544, 23207619274, 327167316436, 4657884819670, 66875794530120, 967202289590280, 14077773784645980, 206058395118133932, 3031188276557963312, 44789055557553810152

Offset: 0

N. J. A. Sloane, Nov 24 2015

nonn

Gheorghe Coserea, Table of n, a(n) for n = 0..200
Hacène Belbachir, Oussama Igueroufa, Combinatorial interpretation of bisnomial coefficients and Generalized Catalan numbers, Proceedings of the 1st International Conference on Algebras, Graphs and Ordered Sets (ALGOS 2020), hal-02918958 [math.cs], 47-54.
Eliahu Cohen, Tobias Hansen, Nissan Itzhaki, From Entanglement Witness to Generalized Catalan Numbers, arXiv:1511.06623 [quant-ph], 2015.
T. L. Curtright, T. S. Van Kortryk, and C. K. Zachos, Spin Multiplicities, hal-01345527, 2016.

For spin S = 1/2, 1, 3/2, 2, 5/2, 3, 7/2, 4, 9/2, 5 we get A000108, A005043, this sequence, A007043, A272391, A264608, A272392, A272393, A272394, A272395.

a[n_]:= 2/Pi*4^(2*n)*Integrate[Sqrt[1-t]*(2*t-1)^(2*n)*Sqrt[t]^(2*n-1),{t,0,1}] (* Thomas Curtright, Jun 22 2016 *)
a[n_]:= c[0, 2 n, 3/2]-c[1, 2 n, 3/2]; c[j_, n_, s_]:= Sum[(-1)^k*Binomial[n, k]*Binomial[j - (2*s + 1)*k + n + n*s - 1, j - (2*s + 1)*k + n*s], {k, 0, Min[n, Floor[(j + n*s)/(2*s + 1)]]}]; Table[a[n], {n, 0, 20}] (* Thomas Curtright, Jul 26 2016 *)
Table[CatalanNumber[3 n]Hypergeometric2F1[-1-3n,-2n,1/2-3n,1/2],{n,0,20}] (* Benedict W. J. Irwin, Sep 27 2016 *)

N = 44; S = 3/2;
M = matrix(N+1, N*numerator(S)+1);
Mget(n, j) = { M[1 + n, 1 + j*denominator(S)] };
Mset(n, j, v) = { M[1 + n, 1 + j*denominator(S)] = v };
Minit() = {
  my(step = 1/denominator(S));
  Mset(0, 0, 1);
  for (n = 1, N, forstep (j = 0, n*S, step,
     my(acc = 0);
     for (k = abs(j-S), min(j+S, (n-1)*S), acc += Mget(n-1, k));
     Mset(n, j, acc)));
};
Minit();
vector(1 + N\denominator(S), n, Mget((n-1)*denominator(S),0)) \\ Gheorghe Coserea, Apr 28 2016

a(n) ~ (2*sqrt(10)/25)*4^(2*n)/(sqrt(Pi)*(2*n)^(3/2)) * (1-21/(40*n)+O(1/n^2)). - Thomas Curtright, Jun 17 2016, updated Jul 16 2016
D-finite with recurrence: 3*n*(3*n - 1)*(3*n + 1)*(5*n - 7)*a(n) = 8*(2*n - 1)*(145*n^3 - 338*n^2 + 238*n - 51)*a(n-1) - 128*(n-1)*(2*n - 3)*(2*n - 1)*(5*n - 2)*a(n-2). - Vaclav Kotesovec, Jun 24 2016
a(n) = (1/Pi)*int((sin(4x)/sin(x))^(2n)*(sin(x))^2,x,0,2 Pi). - Thomas Curtright, Jun 24 2016
a(n) = Catalan(3*n)*2F1(-1-3*n,-2*n;1/2-3*n;1/2). - Benedict W. J. Irwin, Sep 27 2016

More terms from Gheorghe Coserea, Apr 28 2016

A264608 Degeneracies of entanglement witness eigenstates for spin 3 particles.

Original entry on oeis.org

1, 0, 1, 1, 7, 31, 175, 981, 5719, 33922, 204687, 1251460, 7737807, 48297536, 303922983, 1926038492, 12281450455, 78741558512, 507301771543, 3282586312161, 21323849229781, 139012437340660, 909161626641121, 5963576112550771, 39223341189188339, 258619428254117476, 1709124801693650075

Offset: 0

N. J. A. Sloane, Nov 24 2015

nonn

			A(x) = 1 + x^2 + x^3 + 7*x^4 + 31*x^5 + 175*x^6 + 981*x^7 + ...

Gheorghe Coserea, Table of n, a(n) for n = 0..401
Eliahu Cohen, Tobias Hansen, Nissan Itzhaki, From Entanglement Witness to Generalized Catalan Numbers, arXiv:1511.06623 [quant-ph], 2015.
T. L. Curtright, T. S. Van Kortryk, and C. K. Zachos, Spin Multiplicities, hal-01345527, 2016.
T. L. Curtright, T. S. Van Kortryk, and C. K. Zachos, Spin Multiplicities, arXiv:1607.05849 [hep-th], 2016.

For spin S = 1/2, 1, 3/2, 2, 5/2, 3, 7/2, 4, 9/2, 5 we get A000108, A005043, A264607, A007043, A272391, this sequence, A272392, A272393, A272394, A272395.

Cf. A348210 (column k=3).

a[n_]:= 2/Pi*Integrate[Sqrt[(1-t)/t]*(64t^3-80t^2+24t-1)^n, {t, 0, 1}] (* Thomas Curtright, Jun 23 2016 *)
a[n_]:= c[0, n, 3]-c[1, n, 3]; c[j_, n_, s_]:= Sum[(-1)^k*Binomial[n, k]*Binomial[j - (2*s + 1)*k + n + n*s - 1, j - (2*s + 1)*k + n*s], {k, 0, Floor[(j + n*s)/(2*s + 1)]}]; Table[a[n], {n, 0, 20}] (* Thomas Curtright, Jul 26 2016 *)
a[n_]:= mult[0, n, 3]; mult[j_,n_,s_]:=Sum[(-1)^(k+1)*Binomial[n,k]*Binomial[n*s+j-(2*s+1)*k+n- 1,n*s+j-(2*s+1)*k+1],{k,0,Floor[(n*s+j+1)/(2*s+1)]}] (* Thomas Curtright, Jun 14 2017 *)

N = 26; S = 3;
M = matrix(N+1, N*numerator(S)+1);
Mget(n, j) =  { M[1 + n, 1 + j*denominator(S)] };
Mset(n, j, v) = { M[1 + n, 1 + j*denominator(S)] = v };
Minit() = {
  my(step = 1/denominator(S));
  Mset(0, 0, 1);
  for (n = 1, N, forstep (j = 0, n*S, step,
     my(acc = 0);
     for (k = abs(j-S), min(j+S, (n-1)*S), acc += Mget(n-1, k));
     Mset(n, j, acc)));
};
Minit();
vector(1 + N\denominator(S), n, Mget((n-1)*denominator(S),0)) \\ Gheorghe Coserea, Apr 28 2016

seq(N) = {
  my(a = vector(N), s); a[2]=1; a[3]=1; a[4]=7; a[5]=31;
  for (n=6, N, s = ((n-1)*(2*n - 5)*(2*n - 1)*(3*n - 4)*(4*n - 3)*(37*n - 38)*a[n-1] + 7*(n-1)*(2*n - 3)*(3*n - 1)*(92*n^3 - 404*n^2 + 509*n - 150)*a[n-2] - 49*(n-2)*(n-1)*(2*n - 5)*(2*n - 1)*(3*n - 4)*(4*n - 3)*a[n-3] - 343*(n-3)*(n-2)*(n-1)*(2*n - 3)*(2*n - 1)*(3*n - 1)*a[n-4]);
    a[n] = s/(3*n*(2*n - 5)*(2*n - 3)*(3*n - 4)*(3*n - 1)*(3*n + 1)));
  concat(1,a);
};
seq(26) \\ Gheorghe Coserea, Aug 07 2018

a(n) ~ (1/8^(3/2))*7^n/(sqrt(Pi)*n^(3/2)) * (1-27/(32*n)+O(1/n^2)). - Thomas Curtright, Jun 17 2016, updated Jul 26 2016
D-finite with recurrence 3*n*(2*n - 5)*(2*n - 3)*(3*n - 4)*(3*n - 1)*(3*n + 1)*a(n) = (n-1)*(2*n - 5)*(2*n - 1)*(3*n - 4)*(4*n - 3)*(37*n - 38)*a(n-1) + 7*(n-1)*(2*n - 3)*(3*n - 1)*(92*n^3 - 404*n^2 + 509*n - 150)*a(n-2) - 49*(n-2)*(n-1)*(2*n - 5)*(2*n - 1)*(3*n - 4)*(4*n - 3)*a(n-3) - 343*(n-3)*(n-2)*(n-1)*(2*n - 3)*(2*n - 1)*(3*n - 1)*a(n-4). - Vaclav Kotesovec, Jun 24 2016
a(n) = (1/Pi)*int((sin(7x)/sin(x))^n*(sin(x))^2,x,0,2Pi). - Thomas Curtright, Jun 24 2016
From Gheorghe Coserea, Aug 07 2018: (Start)
G.f. y=A(x) satisfies:
0 = x^3*(x + 1)^4*(49*x^2 - 14*x - 27)^2*y^8 + 2*x^3*(x + 1)^3*(35*x + 23)*(49*x^2 - 14*x - 27)*y^6 + x^2*(x + 1)^2*(1421*x^3 + 1652*x^2 + 393*x - 54)*y^4 + x*(x + 1)*(147*x^3 + 175*x^2 + 51*x - 1)*y^2 + x*(2*x + 1)^2.
0 = x^2*(x + 1)*(7*x - 1)*(7*x + 1)*(49*x^2 - 70*x + 5)*(49*x^2 - 14*x - 27)*y''' + x*(1058841*x^7 - 1092455*x^6 - 1212505*x^5 + 627347*x^4 + 222999*x^3 - 6657*x^2 - 5015*x + 405)*y'' + 2*(1058841*x^7 - 1428595*x^6 - 725102*x^5 + 224322*x^4 + 24157*x^3 + 6909*x^2 - 720*x + 60)*y' + 14*x*(50421*x^5 - 84035*x^4 - 19894*x^3 - 2058*x^2 + 665*x - 75)*y.
(End)

More terms from Gheorghe Coserea, Apr 28 2016

A272393 Degeneracies of entanglement witness eigenstates for n spin 4 irreducible representations.

Original entry on oeis.org

1, 0, 1, 1, 9, 51, 369, 2661, 19929, 151936, 1178289, 9259812, 73593729, 590475744, 4776464121, 38912018796, 318971849625, 2629040965776, 21774894337449, 181136924953317, 1512731101731499, 12678230972826340, 106600213003114719

Offset: 0

Gheorghe Coserea, Apr 28 2016

nonn

Gheorghe Coserea, Table of n, a(n) for n = 0..401
Eliahu Cohen, Tobias Hansen, and Nissan Itzhaki, From Entanglement Witness to Generalized Catalan Numbers, arXiv:1511.06623 [quant-ph], 2015.
T. L. Curtright, T. S. Van Kortryk, and C. K. Zachos, Spin Multiplicities, hal-01345527, 2016.

For spin S = 1/2, 1, 3/2, 2, 5/2, 3, 7/2, 4, 9/2, 5 we get A000108, A005043, A264607, A007043, A272391, A264608, A272392, this sequence, A272394, A272395.

Cf. A348210 (column k=4).

a[n_]:= 2/Pi*Integrate[Sqrt[(1-t)/t]*((4t-1)(4t(4t-3)^2-1))^n, {t, 0, 1}] (* Thomas Curtright, Jun 24 2016 *)
a[n_]:= c[0, n, 4]-c[1, n, 4]; c[j_, n_, s_]:= Sum[(-1)^k*Binomial[n, k]*Binomial[j - (2*s + 1)*k + n + n*s - 1, j - (2*s + 1)*k + n*s], {k, 0, Floor[(j + n*s)/(2*s + 1)]}]; Table[a[n], {n, 0, 20}] (* Thomas Curtright, Jul 26 2016 *)
a[n_]:= mult[0, n, 4]
mult[j_,n_,s_]:=Sum[(-1)^(k+1)*Binomial[n,k]*Binomial[n*s+j-(2*s+1)*k+n- 1,n*s+j-(2*s+1)*k+1],{k,0,Floor[(n*s+j+1)/(2*s+1)]}] (* Thomas Curtright, Jun 14 2017 *)

N = 26; S = 4;
M = matrix(N+1, N*numerator(S)+1);
Mget(n, j) = { M[1 + n, 1 + j*denominator(S)] };
Mset(n, j, v) = { M[1 + n, 1 + j*denominator(S)] = v };
Minit() = {
  my(step = 1/denominator(S));
  Mset(0, 0, 1);
  for (n = 1, N, forstep (j = 0, n*S, step,
     my(acc = 0);
     for (k = abs(j-S), min(j+S, (n-1)*S), acc += Mget(n-1, k));
     Mset(n, j, acc)));
};
Minit();
vector(1 + N\denominator(S), n, Mget((n-1)*denominator(S),0))

a(n) ~ (3/40)^(3/2)*9^n/(sqrt(Pi)*n^(3/2)) * (1-129/(160*n)+O(1/n^2)). - Thomas Curtright, Jun 17 2016, updated Jul 26 2016
D-finite with recurrence 8*n*(2*n - 5)*(2*n - 3)*(2*n - 1)*(4*n - 1)*(4*n + 1)*(5*n - 17)*(5*n - 12)*(5*n - 8)*(5*n - 7)*a(n) = (n-1)*(2*n - 5)*(2*n - 3)*(5*n - 17)*(5*n - 12)*(5*n - 3)*(5*n - 2)*(2101*n^3 - 6506*n^2 + 6608*n - 2200)*a(n-1) + 9*(n-1)*(2*n - 5)*(2*n - 1)*(4*n - 1)*(5*n - 17)*(5*n - 8)*(5*n - 7)*(385*n^3 - 1659*n^2 + 2008*n - 492)*a(n-2) - 81*(n-2)*(n-1)*(2*n - 3)*(5*n - 12)*(5*n - 3)*(5*n - 2)*(1020*n^4 - 8088*n^3 + 21761*n^2 - 22557*n + 7264)*a(n-3) - 729*(n-3)*(n-2)*(n-1)*(2*n - 5)*(2*n - 1)*(4*n - 1)*(5*n - 17)*(5*n - 8)*(5*n - 7)*(5*n - 2)*a(n-4) + 6561*(n-4)*(n-3)*(n-2)*(n-1)*(2*n - 3)*(2*n - 1)*(5*n - 12)*(5*n - 7)*(5*n - 3)*(5*n - 2)*a(n-5). - Vaclav Kotesovec, Jun 24 2016
a(n) = (1/Pi)*int((sin(9x)/sin(x))^n*(sin(x))^2,x,0,2Pi). - Thomas Curtright, Jun 24 2016

A272391 Degeneracies of entanglement witness eigenstates for 2n spin 5/2 irreducible representations.

Original entry on oeis.org

1, 1, 6, 111, 2666, 70146, 1949156, 56267133, 1670963202, 50720602314, 1566629938776, 49080774275121, 1555873464248076, 49814409137161480, 1608523756282054800, 52323002586904505427, 1712956041168844662002

Offset: 0

Gheorghe Coserea, Apr 28 2016

nonn

Gheorghe Coserea, Table of n, a(n) for n = 0..200
Hacène Belbachir, Oussama Igueroufa, Combinatorial interpretation of bisnomial coefficients and Generalized Catalan numbers, Proceedings of the 1st International Conference on Algebras, Graphs and Ordered Sets (ALGOS 2020), hal-02918958 [math.cs], 47-54.
Eliahu Cohen, Tobias Hansen, Nissan Itzhaki, From Entanglement Witness to Generalized Catalan Numbers, arXiv:1511.06623 [quant-ph], 2015.
Thomas Curtright, Thomas Van Kortryk, and Cosmas Zachos, Spin Multiplicities, hal-01345527, 2016.

For spin S = 1/2, 1, 3/2, 2, 5/2, 3, 7/2, 4, 9/2, 5 we get A000108, A005043, A264607, A007043, this sequence, A264608, A272392, A272393, A272394, A272395.

a[n_] := 2/Pi * 2^(2 * n) * Integrate[Sqrt[1 - t] * ((4 * t - 1)(4 * t - 3))^(2 * n) * Sqrt[t]^(2 * n - 1), {t, 0, 1}] (* Thomas Curtright, Jun 23 2016 *)
a[n_] := c[0, 2 n, 5/2] - c[1, 2 n, 5/2]; c[j_, n_, s_]:= Sum[(-1)^k * Binomial[n, k] * Binomial[j - (2 * s + 1) * k + n + n * s - 1, j - (2 * s + 1) * k + n * s], {k, 0, Min[n, Floor[(j + n * s)/(2 * s + 1)]]}]; Table[a[n], {n, 0, 20}] (* Thomas Curtright, Jul 26 2016 *)

N = 34; S = 5/2;
M = matrix(N+1, N*numerator(S)+1);
Mget(n, j) =  { M[1 + n, 1 + j*denominator(S)] };
Mset(n, j, v) = { M[1 + n, 1 + j*denominator(S)] = v };
Minit() = {
  my(step = 1/denominator(S));
  Mset(0, 0, 1);
  for (n = 1, N, forstep (j = 0, n*S, step,
     my(acc = 0);
     for (k = abs(j-S), min(j+S, (n-1)*S), acc += Mget(n-1, k));
     Mset(n, j, acc)));
};
Minit();
vector(1 + N\denominator(S), n, Mget((n-1)*denominator(S),0))

a(n) ~ (6*sqrt(210)/1225)*6^(2*n)/(sqrt(Pi)*(2*n)^(3/2)) * (1-123/(280n)+O(1/n^2)). - Thomas Curtright, Jun 17 2016, updated Jul 26 2016
Recurrence: 5*n*(5*n - 8)*(5*n - 3)*(5*n - 2)*(5*n - 1)*(5*n + 1)*(7*n - 16)*(7*n - 10)*(7*n - 9)*a(n) = 6*(2*n - 1)*(5*n - 8)*(7*n - 16)*(499359*n^6 - 2314137*n^5 + 4264709*n^4 - 3984323*n^3 + 1983172*n^2 - 496780*n + 48720)*a(n-1) - 864*(n-1)*(2*n - 3)*(2*n - 1)*(7*n - 2)*(25480*n^5 - 160398*n^4 + 375142*n^3 - 401079*n^2 + 192819*n - 33500)*a(n-2) + 31104*(n-2)*(n-1)*(2*n - 5)*(2*n - 3)*(2*n - 1)*(5*n - 3)*(7*n - 9)*(7*n - 3)*(7*n - 2)*a(n-3). - Vaclav Kotesovec, Jun 24 2016
a(n) = (1/Pi)*int((sin(6x)/sin(x))^(2n)*(sin(x))^2,x,0,2 Pi). - Thomas Curtright, Jun 24 2016

A272392 Degeneracies of entanglement witness eigenstates for 2n spin 7/2 irreducible representations.

Original entry on oeis.org

1, 1, 8, 260, 11096, 518498, 25593128, 1312660700, 69270071480, 3736677346685, 205125498479384, 11421904528488264, 643564228586076344, 36624864117451994600, 2102142593641513473240, 121548403269918189484872, 7073453049221266117909752, 413976401197504361048673896

Offset: 0

Gheorghe Coserea, Apr 28 2016

nonn

Gheorghe Coserea, Table of n, a(n) for n = 0..200
Eliahu Cohen, Tobias Hansen, Nissan Itzhaki, From Entanglement Witness to Generalized Catalan Numbers, arXiv:1511.06623 [quant-ph], 2015.
T. L. Curtright, T. S. Van Kortryk, and C. K. Zachos, Spin Multiplicities, hal-01345527, 2016.

For spin S = 1/2, 1, 3/2, 2, 5/2, 3, 7/2, 4, 9/2, 5 we get A000108, A005043, A264607, A007043, A272391, A264608, this sequence, A272393, A272394, A272395.

a[n_]:= c[0, 2*n, 7/2]-c[1, 2*n, 7/2]; c[j_, n_, s_]:= Sum[(-1)^k*Binomial[n, k]*Binomial[j - (2*s + 1)*k + n + n*s - 1, j - (2*s + 1)*k + n*s], {k, 0, Min[n, Floor[(j + n*s)/(2*s + 1)]]}]; Table[a[n], {n, 0, 20}] (* Thomas Curtright, Jul 26 2016 *)

N = 44; S = 7/2;
M = matrix(N+1, N*numerator(S)+1);
Mget(n, j) =  { M[1 + n, 1 + j*denominator(S)] };
Mset(n, j, v) = { M[1 + n, 1 + j*denominator(S)] = v };
Minit() = {
  my(step = 1/denominator(S));
  Mset(0, 0, 1);
  for (n = 1, N, forstep (j = 0, n*S, step,
     my(acc = 0);
     for (k = abs(j-S), min(j+S, (n-1)*S), acc += Mget(n-1, k));
     Mset(n, j, acc)));
};
Minit();
vector(1 + N\denominator(S), n, Mget((n-1)*denominator(S),0)) \\ Gheorghe Coserea, Apr 28 2016

a(n) ~ (2*sqrt(42)/441)*8^(2*n)/(sqrt(Pi)*(2*n)^(3/2)) * (1-23/(56*n)+O(1/n^2)). - Thomas Curtright and Cosmas Zachos, Jun 17 2016, updated Jul 26 2016
Recurrence: 7*n*(3*n - 7)*(3*n - 4)*(7*n - 19)*(7*n - 12)*(7*n - 11)*(7*n - 5)*(7*n - 4)*(7*n - 3)*(7*n - 2)*(7*n - 1)*(7*n + 1)*(9*n - 29)*(9*n - 20)*(9*n - 13)*(9*n - 11)*a(n) = 32*(2*n - 1)*(3*n - 7)*(7*n - 19)*(7*n - 12)*(7*n - 11)*(9*n - 29)*(9*n - 20)*(218437803*n^9 - 1510747767*n^8 + 4498401903*n^7 - 7551222032*n^6 + 7855986297*n^5 - 5239178603*n^4 + 2233354977*n^3 - 584916638*n^2 + 85090380*n - 5216400)*a(n-1) - 6144*(n-1)*(2*n - 3)*(2*n - 1)*(7*n - 19)*(9*n - 29)*(9*n - 2)*(460622295*n^10 - 5800755303*n^9 + 31804940376*n^8 - 99676215732*n^7 + 197077947989*n^6 - 255958437117*n^5 + 220361564054*n^4 - 123775781978*n^3 + 43301190686*n^2 - 8505466270*n + 711711000)*a(n-2) + 262144*(n-2)*(n-1)*(2*n - 5)*(2*n - 3)*(2*n - 1)*(3*n - 1)*(7*n - 5)*(9*n - 11)*(9*n - 2)*(2988657*n^7 - 36693972*n^6 + 183168228*n^5 - 477680566*n^4 + 695101884*n^3 - 556549424*n^2 + 223584828*n - 34734735)*a(n-3) - 16777216*(n-3)*(n-2)*(n-1)*(2*n - 7)*(2*n - 5)*(2*n - 3)*(2*n - 1)*(3*n - 4)*(3*n - 1)*(7*n - 12)*(7*n - 5)*(7*n - 4)*(9*n - 20)*(9*n - 11)*(9*n - 4)*(9*n - 2)*a(n-4). - Vaclav Kotesovec, Jun 24 2016
a(n) = (1/Pi)*int((sin(8x)/sin(x))^(2n)*(sin(x))^2,x,0,2 Pi). - Thomas Curtright, Jun 24 2016

A272394 Degeneracies of entanglement witness eigenstates for 2n spin 9/2 irreducible representations.

Original entry on oeis.org

1, 1, 10, 505, 33670, 2457190, 189442252, 15177798415, 1251216059950, 105443928375598, 9043123211156440, 786701771691580227, 69253844083218535300, 6157639918607211895000, 552193624489443516667344, 49885368826043082235592687

Offset: 0

Gheorghe Coserea, Apr 28 2016

nonn

Gheorghe Coserea, Table of n, a(n) for n = 0..200
Eliahu Cohen, Tobias Hansen, Nissan Itzhaki, From Entanglement Witness to Generalized Catalan Numbers, arXiv:1511.06623 [quant-ph], 2015.
T. L. Curtright, T. S. Van Kortryk, and C. K. Zachos, Spin Multiplicities, hal-01345527, 2016.
Vaclav Kotesovec, Recurrence (of order 5)

For spin S = 1/2, 1, 3/2, 2, 5/2, 3, 7/2, 4, 9/2, 5 we get A000108, A005043, A264607, A007043, A272391, A264608, A272392, A272393, this sequence, A272395.

a[n_]:= 2/Pi*Integrate[Sqrt[(1-t)/t]*(4t)^n*((16t^2-20t+5)((4t-1)^2-4t))^(2n), {t, 0, 1}] (* Thomas Curtright, Jun 24 2016 *)
a[n_]:= c[0, 2 n, 9/2]-c[1, 2 n, 9/2]
c[j_, n_, s_]:= Sum[(-1)^k*Binomial[n, k]*Binomial[j - (2*s + 1)*k + n + n*s - 1, j - (2*s + 1)*k + n*s],{k, 0, Min[n, Floor[(j + n*s)/(2*s + 1)]]}] (* Thomas Curtright, Jul 26 2016 *)

N = 33; S = 9/2;
M = matrix(N+1, N*numerator(S)+1);
Mget(n, j) =  { M[1 + n, 1 + j*denominator(S)] };
Mset(n, j, v) = { M[1 + n, 1 + j*denominator(S)] = v };
Minit() = {
  my(step = 1/denominator(S));
  Mset(0, 0, 1);
  for (n = 1, N, forstep (j = 0, n*S, step,
     my(acc = 0);
     for (k = abs(j-S), min(j+S, (n-1)*S), acc += Mget(n-1, k));
     Mset(n, j, acc)));
};
Minit();
vector(1 + N\denominator(S), n, Mget((n-1)*denominator(S),0))

a(n) = (1/Pi)*int((sin(10x)/sin(x))^(2n)*(sin(x))^2,x,0,2 Pi). - Thomas Curtright, Jun 24 2016

a(n) ~ (2*sqrt(66)/1089)*10^(2n)/(sqrt(Pi)*(2n)^(3/2))(1-35/(88n) + O(1/n^2)). - Thomas Curtright, Jul 26 2016

A272395 Degeneracies of entanglement witness eigenstates for n spin 5 irreducible representations.

Original entry on oeis.org

1, 0, 1, 1, 11, 76, 671, 5916, 54131, 504316, 4779291, 45898975, 445798221, 4371237794, 43213522209, 430241859971, 4310236148075, 43417944574136, 439495074016427, 4468208369691396, 45605656313488271, 467140985042718910

Offset: 0

Gheorghe Coserea, Apr 28 2016

nonn

The Mathematica formula for a(n) as the difference of two generalized binomial coefficients is adapted from the Appendix of the Mendonça link. - Thomas Curtright, Jul 27 2016

Gheorghe Coserea, Table of n, a(n) for n = 0..401
Eliahu Cohen, Tobias Hansen, Nissan Itzhaki, From Entanglement Witness to Generalized Catalan Numbers, arXiv:1511.06623 [quant-ph], 2015.
T. L. Curtright, T. S. Van Kortryk, and C. K. Zachos, Spin Multiplicities, hal-01345527, 2016.
Vaclav Kotesovec, Recurrence (of order 6)
J. R. G. Mendonça, Exact eigenspectrum of the symmetric simple exclusion process on the complete, complete bipartite and related graphs, Journal of Physics A: Mathematical and Theoretical 46:29 (2013) 295001. arXiv:1207.4106 [cond-mat.stat-mech], 2012-2013.

For spin S = 1/2, 1, 3/2, 2, 5/2, 3, 7/2, 4, 9/2, 5 we get A000108, A005043, A264607, A007043, A272391, A264608, A272392, A272393, A272394, this sequence.

a[n_]:= 2/Pi*Integrate[Sqrt[(1-t)/t]*(1024t^5-2304t^4+1792t^3-560t^2 +60t-1)^n, {t, 0, 1}] (* Thomas Curtright, Jun 24 2016 *)
a[n_]:= c[0, n, 5]-c[1, n, 5]
c[j_, n_, s_]:= Sum[(-1)^k*Binomial[n, k]*Binomial[j - (2*s + 1)*k + n + n*s - 1, j - (2*s + 1)*k + n*s],{k, 0, Floor[(j + n*s)/(2*s + 1)]}] (* Thomas Curtright, Jul 26 2016 *)
a[n_]:= mult[0, n, 5]
mult[j_,n_,s_]:=Sum[(-1)^(k+1)*Binomial[n,k]*Binomial[n*s+j-(2*s+1)*k+n- 1,n*s+j-(2*s+1)*k+1],{k,0,Floor[(n*s+j+1)/(2*s+1)]}] (* Thomas Curtright, Jun 14 2017 *)

N = 22; S = 5;
M = matrix(N+1, N*numerator(S)+1);
Mget(n, j) =  { M[1 + n, 1 + j*denominator(S)] };
Mset(n, j, v) = { M[1 + n, 1 + j*denominator(S)] = v };
Minit() = {
  my(step = 1/denominator(S));
  Mset(0, 0, 1);
  for (n = 1, N, forstep (j = 0, n*S, step,
     my(acc = 0);
     for (k = abs(j-S), min(j+S, (n-1)*S), acc += Mget(n-1, k));
     Mset(n, j, acc)));
};
Minit();
vector(1 + N\denominator(S), n, Mget((n-1)*denominator(S),0))

c(j, n) = sum(k=0, min((j + 5*n)\11, n), (-1)^k*binomial(n, k)*binomial(j - 11*k + n + 5*n - 1, j - 11*k + n*5))
a(n)=c(0, n)-c(1, n) \\ Charles R Greathouse IV, Jul 28 2016; adapted from Curtright's Mathematica code

a(n)=(1/Pi)*int((sin(11x)/sin(x))^n*(sin(x))^2,x,0,2Pi). - Thomas Curtright, Jun 24 2016

a(n) ~ (1/20)^(3/2)*11^n/(sqrt(Pi)*n^(3/2))(1-63/(80n)+O(1/n^2)). - Thomas Curtright, Jul 26 2016

A348210 Varma's Kosta numbers of semi-standard tableaux: array A(n>=2, k>=0) read by rising antidiagonals.

Original entry on oeis.org

0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 6, 5, 1, 0, 1, 15, 16, 7, 1, 0, 1, 36, 65, 31, 9, 1, 0, 1, 91, 260, 175, 51, 11, 1, 0, 1, 232, 1085, 981, 369, 76, 13, 1, 0, 1, 603, 4600, 5719, 2661, 671, 106, 15, 1, 0, 1, 1585, 19845, 33922, 19929, 5916, 1105, 141, 17, 1, 0, 1, 4213, 86725, 204687, 151936, 54131, 11516, 1695, 181, 19, 1, 0

Offset: 2

R. J. Mathar, Oct 07 2021

(More characteristic NAME desired.)

Each row is a polynomial in k, which implies that the inverse binomial transformation of each row is a finite sequence and that the row can be represented by a rational generating function (A348211).

			The array starts in row n=2 with columns k>=0 as:
  0   0    0    0     0     0      0      0 ...
  1   1    1    1     1     1      1      1 ...
  1   3    5    7     9    11     13     15 ...
  1   6   16   31    51    76    106    141 ...
  1  15   65  175   369   671   1105   1695 ...
  1  36  260  981  2661  5916  11516  20385 ...
  1  91 1085 5719 19929 54131 124501 254255 ...
Antidiagonal rows begin as:
  0;
  1,   0;
  1,   1,    0;
  1,   3,    1,    0;
  1,   6,    5,    1,    0;
  1,  15,   16,    7,    1,    0;
  1,  36,   65,   31,    9,    1,   0;
  1,  91,  260,  175,   51,   11,   1,   0;
  1, 232, 1085,  981,  369,   76,  13,   1,  0;
  1, 603, 4600, 5719, 2661,  671, 106,  15,  1,  0;

G. C. Greubel, Antidiagonals n = 2..52, flattened
D.-N. Verma, Towards Classifying Finite Point-Set Configurations, 1997, Unpublished. [Scanned copy of annotated version of preprint given to me by the author in 1997. - _N. J. A. Sloane_, Oct 03 2021]

Cf. A005043 (column k=1), A007043 (k=2), A264608 (k=3), A272393 (k=4), A005408 (row n=4), A005891 (n=5), A005917 (n=6), A348211 (condensed g.f.)

A:= func< n,k | (&+[(-1)^(j+1)*Binomial(n,j)*Binomial((n-2*j)*k+n-j-2,n-3)/2 : j in [0..Floor((n-1)/2)]]) >;
A348210:= func< n,k | A(n-k,k) >;
[A348210(n,k): k in [0..n-2], n in [2..13]]; // G. C. Greubel, Feb 28 2024

A348210 := proc(n,k)
    local a,j ;
    a := 0 ;
    for j from 0 to floor((n-1)/2) do
            a := a+ (-1)^j *binomial(n,j) *binomial( (n-2*j)*k+n-j-2,n-3) ;
    end do:
    -a/2 ;
end proc:
seq( seq( A348210(d-k,k),k=0..d-2),d=2..12) ;

A[n_, k_] := (-1/2)*Sum[(-1)^j*Binomial[n, j]*Binomial[(n - 2*j)*k + n - j - 2, n - 3], {j, 0, Floor[(n - 1)/2]}];
Table[A[n - k, k], {n, 2, 13}, {k, 0, n - 2}] // Flatten (* Jean-François Alcover, Mar 06 2023 *)

def A(n,k): return sum( (-1)^(j+1)*binomial(n,j)*binomial((n-2*j)*k+n-j-2,n-3) for j in range(1+(n-1)//2) )/2
def A348210(n,k): return A(n-k, k)
flatten([[A348210(n,k) for k in range(n-1)] for n in range(2,13)]) # G. C. Greubel, Feb 28 2024

A(n,k) = (-1/2)*Sum_{j=0..floor((n-1)/2)} (-1)^j *binomial(n,j) *binomial((n-2*j)*k+n-j-2,n-3).
A(7,k) = 1 + 7*k*(k+1)*(11*k^2+11*k+8)/12.
A(8,k) = (2*k+1)*(4*k^2+6*k+3)*(4*k^2+2*k+1)/3.
A(9,k) = 1 + k*(k+1)*(289*k^4+578*k^3+581*k^2+292*k+108)/16.

A335323 First lower diagonal of Parker's triangle A047812.

Original entry on oeis.org

0, 1, 3, 7, 11, 18, 26, 38, 52, 73, 97, 131, 172, 227, 293, 381, 486, 623, 788, 998, 1251, 1571, 1954, 2432, 3006, 3714, 4561, 5600, 6838, 8345, 10139, 12306, 14879, 17973, 21633, 26011, 31181, 37334, 44579, 53170, 63257, 75171, 89130, 105554, 124750, 147269

Offset: 1

Petros Hadjicostas, May 31 2020

nonn

Apparently, this sequence was originally intended to be A7043 (now A007043), but for some reason it was crossed out on p. 4 of the annotated copy of Guy's 1992 preprint.

a(n) is the number of partitions of (n-2)*(n+1) into at most n parts each no bigger than n. Thus, a(n) is the coefficient of q^((n-2)*(n+1)) in the q-binomial coefficient [2*n, n].

			a(1) = 0 because it does not make sense to talk about the partitions of (1-2)*(1+1) = -2.
a(2) = 1 because we have only the empty partition for (2-2)*(2+1) = 0.
a(3) = 3 because we have the following partitions of (3-2)*(3+1) = 4 into no more than 3 parts each no bigger than 3: 1+3 = 1+1+2 = 2+2.
a(4) = 7 because we have the following partitions of (4-2)*(4+1) = 10 into no more than 4 parts each no bigger than 4: 2+4+4 = 3+3+4 = 1+1+4+4 = 1+2+3+4 = 1+3+3+3 = 2+2+2+4 = 2+2+3+3.
The PARI function partitions((n-2)*(n+1), n, n) can generate these partitions.

Alois P. Heinz, Table of n, a(n) for n = 1..500
R. K. Guy, Letter to N. J. A. Sloane, Aug. 1992.
R. K. Guy, Parker's permutation problem involves the Catalan numbers, preprint, 1992. (Annotated scanned copy)
R. K. Guy, Parker's permutation problem involves the Catalan numbers, Amer. Math. Monthly 100 (1993), 287-289.

Cf. A007042, A007043, A007044, A007045, A047812, A051643.

b:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(n<0
      or t*i b((n-2)*(n+1), n$2):
seq(a(n), n=1..50);  # Alois P. Heinz, May 31 2020

b[n_, i_, t_] := b[n, i, t] = If[n == 0, 1, If[n < 0 || t i < n, 0, b[n, i - 1, t] + b[n - i, Min[i, n - i], t - 1]]];
a[n_] := b[(n-2)(n+1), n, n];
Array[a, 50] (* Jean-François Alcover, Nov 27 2020, after Alois P. Heinz *)

T(n, k) = polcoeff(prod(j=0, n-1, (1-q^(2*n-j))/(1-q^(j+1)) ), k*(n+1) );
for(n=1, 43, print1(T(n, n-2), ", "))

cp's OEIS Frontend

A007060 Number of ways n married couples can sit in a row without any spouses next to each other.

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A264607 Degeneracies of entanglement witness eigenstates for spin 3/2 particles.

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A264608 Degeneracies of entanglement witness eigenstates for spin 3 particles.

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A272393 Degeneracies of entanglement witness eigenstates for n spin 4 irreducible representations.

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A272391 Degeneracies of entanglement witness eigenstates for 2n spin 5/2 irreducible representations.

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A272392 Degeneracies of entanglement witness eigenstates for 2n spin 7/2 irreducible representations.

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A272394 Degeneracies of entanglement witness eigenstates for 2n spin 9/2 irreducible representations.

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