A264607 -id:A264607 - cp's OEIS frontend

A264608 Degeneracies of entanglement witness eigenstates for spin 3 particles.

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

A349934 Array read by ascending antidiagonals: A(n, s) is the n-th s-Catalan number.

Original entry on oeis.org

1, 2, 1, 5, 3, 1, 14, 15, 4, 1, 42, 91, 34, 5, 1, 132, 603, 364, 65, 6, 1, 429, 4213, 4269, 1085, 111, 7, 1, 1430, 30537, 52844, 19845, 2666, 175, 8, 1, 4862, 227475, 679172, 383251, 70146, 5719, 260, 9, 1, 16796, 1730787, 8976188, 7687615, 1949156, 204687, 11096, 369, 10, 1

Offset: 1

Stefano Spezia, Dec 06 2021

			The array begins:
n\s |  1    2     3      4      5
----+----------------------------
  1 |  1    1     1      1      1 ...
  2 |  2    3     4      5      6 ...
  3 |  5   15    34     65    111 ...
  4 | 14   91   364   1085   2666 ...
  5 | 42  603  4269  19845  70146 ...
  ...

William Linz, s-Catalan numbers and Littlewood-Richardson polynomials, arXiv:2110.12095 [math.CO], 2021. See p. 2.

Cf. A000012 (n=1), A220892 (n=4).
Cf. A000108 (s=1), A099251 (s=2), A264607 (s=3).
Cf. A000027, A007318, A008287, A027907, A035343, A063260.
Cf. A349933.

T[n_,k_,s_]:=If[k==0,1,Coefficient[(Sum[x^i,{i,0,s}])^n,x^k]]; A[n_,s_]:=T[2n,s n,s]-T[2n,s n+1,s]; Flatten[Table[A[n-s+1,s],{n,10},{s,n}]]

T(n, k, s) = polcoef((sum(i=0, s, x^i))^n, k);
A(n, s) = T(2*n, s*n, s) - T(2*n, s*n+1, s); \\ Michel Marcus, Dec 10 2021

A(n, s) = T(2*n, s*n, s) - T(2*n, s*n+1, s), where T(n, k, s) is the s-binomial coefficient defined as the coefficient of x^k in (Sum_{i=0..s} x^i)^n.
A(2, n) = A000027(n+1).
A(3, n) = A006003(n+1).

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