cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 11-17 of 17 results.

A286783 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.

Original entry on oeis.org

1, 3, 15, 5, 105, 77, 7, 945, 1044, 234, 9, 10395, 14784, 5390, 550, 11, 135135, 227877, 113126, 19760, 1105, 13, 2027025, 3862305, 2371845, 586425, 58275, 1995, 15, 34459425, 71983440, 51607716, 16271380, 2356234, 147560, 3332, 17, 654729075, 1469813400, 1185214452, 446964322, 84487110, 7888876, 333564, 5244, 19, 13749310575, 32718512925, 28937407212, 12516198870, 2884205268, 358182846, 23006928, 690480, 7875, 21
Offset: 0

Views

Author

Gheorghe Coserea, May 14 2017

Keywords

Comments

Row n>0 contains n terms.
T(n,k) is the number of Feynman's diagrams with k fermionic loops in the order n of the perturbative expansion in dimension zero for the polarization function in a many-body theory of fermions with two-body interaction (see Molinari link).

Examples

			A(x;t) = 1 + 3*x + (15 + 5*t)*x^2 + (105 + 77*t + 7*t^2)*x^3 + ...
Triangle starts:
n\k  [0]       [1]       [2]       [3]       [4]      [5]     [6]   [7]
[0]  1;
[1]  3;
[2]  15,       5;
[3]  105,      77,       7;
[4]  945,      1044,     234,      9;
[5]  10395,    14784,    5390,     550,      11;
[6]  135135,   227877,   113126,   19760,    1105,    13;
[7]  2027025,  3862305,  2371845,  586425,   58275,   1995,   15;
[8]  34459425, 71983440, 51607716, 16271380, 2356234, 147560, 3332, 17;
[9]  ...

Links

Programs

  • Mathematica
    max = 11; y0[x_, t_] = 1; y1[x_, t_] = 0; For[n = 1, n <= max, n++, y1[x_, t_] = (1 + x*y0[x, t] + 2*x^2*D[y0[x, t], x])*(1 - x*y0[x, t]*(1 - t))/(1 - x*y0[x, t])^2 + O[x]^n // Normal; y0[x_, t_] = y1[x, t] // Simplify];
s = y0[x, t];
se = (1 + x*s + 2*x^2*D[s, x])/(1 - x*s)^2  + O[x]^max // Normal;
row[n_] := row[n] = CoefficientList[Coefficient[se, x, n], t];
T[0, 0] = 1; T[n_, k_] := row[n][[k + 1]];
Table[T[n, k], {n, 0, max-1}, {k, 0, If[n == 0, 0, n-1]}] // Flatten (* Jean-François Alcover, May 19 2017, adapted from PARI *)
  • PARI
    A286781_ser(N,t='t) = {
  my(x='x+O('x^N), y0=1+O('x^N), y1=0, n=1);
  while(n++,
    y1 = (1 + x*y0 + 2*x^2*y0')*(1 - x*y0*(1-t))/(1-x*y0)^2;
    if (y1 == y0, break()); y0 = y1;);
  y0;
};
A286783_ser(N,t='t) = {
  my(s=A286781_ser(N,t)); (1 + x*s + 2*x^2*deriv(s,'x))/(1-x*s)^2;
};
concat(apply(p->Vecrev(p), Vec(A286783_ser(10))))

Formula

A(x;t) = Sum_{n>=0} P_n(t)*x^n = (1 + x*s + 2*x^2*deriv(s,x))/(1-x*s)^2, where s(x;t) = A286781(x;t) and P_n(t) = Sum_{k=0..n-1} T(n,k)*t^k for n>0.
A001147(n+1)=T(n,0), A001700(n)=P_n(-1), A286794(n)=P_n(1).

A286784 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.

Original entry on oeis.org

1, 1, 1, 2, 4, 1, 5, 15, 9, 1, 14, 56, 56, 16, 1, 42, 210, 300, 150, 25, 1, 132, 792, 1485, 1100, 330, 36, 1, 429, 3003, 7007, 7007, 3185, 637, 49, 1, 1430, 11440, 32032, 40768, 25480, 7840, 1120, 64, 1, 4862, 43758, 143208, 222768, 179928, 77112, 17136, 1836, 81, 1, 16796, 167960, 629850, 1162800, 1162800, 651168, 203490, 34200, 2850, 100, 1
Offset: 0

Views

Author

Gheorghe Coserea, May 14 2017

Keywords

Comments

T(n,k) is the number of Feynman's diagrams with k fermionic loops in the order n of the perturbative expansion in dimension zero for the GW approximation of the self-energy function in a many-body theory of fermions with two-body interaction (see Molinari link).

Examples

			A(x;t) = 1 + (1 + t)*x + (2 + 4*t + t^2)*x^2 + (5 + 15*t + 9*t^2 + t^3)*x^3 + ...
Triangle starts:
n\k  [0]   [1]    [2]     [3]     [4]     [5]    [6]    [7]   [8] [9]
[0]  1;
[1]  1,    1;
[2]  2,    4,     1;
[3]  5,    15,    9,      1;
[4]  14,   56,    56,     16,     1;
[5]  42,   210,   300,    150,    25,     1;
[6]  132,  792,   1485,   1100,   330,    36,    1;
[7]  429,  3003,  7007,   7007,   3185,   637,   49,    1;
[8]  1430, 11440, 32032,  40768,  25480,  7840,  1120,  64,   1;
[9]  4862, 43758, 143208, 222768, 179928, 77112, 17136, 1836, 81, 1;
[10] ...

Links

Crossrefs

Cf. A286781, A286782, A286783.

Programs

  • Magma
    /* As triangle */ [[(Binomial(2*n, n+m)*Binomial(n+1, m))/(n+1): m in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Sep 23 2018
  • Mathematica
    Flatten@Table[Binomial[2 n, n + m] Binomial[n + 1, m] / (n + 1), {n, 0, 10}, {m, 0, n}] (* Vincenzo Librandi, Sep 23 2018 *)
  • Maxima
    T(n,m):=(binomial(2*n,n+m)*binomial(n+1,m))/(n+1); /* Vladimir Kruchinin, Sep 23 2018 */
  • PARI
    A286784_ser(N,t='t) = my(x='x+O('x^N)); serreverse(Ser(x*(1-x)^2/(1+(t-1)*x)))/x;
concat(apply(p->Vecrev(p), Vec(A286784_ser(12))))
\\ test: y=A286784_ser(50); y*(1-x*y)^2 == 1 + ('t-1)*x*y

Formula

y(x;t) = Sum_{n>=0} P_n(t)*x^n satisfies y*(1-x*y)^2 = 1 + (t-1)*x*y, where P_n(t) = Sum_{k=0..n} T(n,k)*t^k.
A000108(n) = T(n,0), A001791(n) = T(n,1), A002055(n+3) = T(n,2), A000290(n) = T(n,n-1), A006013(n) = P_n(1), A003169(n+1) = P_n(2).
T(n,m) = C(2*n,n+m)*C(n+1,m)/(n+1). - Vladimir Kruchinin, Sep 23 2018

A286785 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.

Original entry on oeis.org

1, 2, 5, 2, 14, 14, 2, 42, 72, 27, 2, 132, 330, 220, 44, 2, 429, 1430, 1430, 520, 65, 2, 1430, 6006, 8190, 4550, 1050, 90, 2, 4862, 24752, 43316, 33320, 11900, 1904, 119, 2, 16796, 100776, 217056, 217056, 108528, 27132, 3192, 152, 2, 58786, 406980, 1046520, 1302336, 854658, 301644, 55860, 5040, 189, 2, 208012, 1634380, 4903140, 7354710, 6056820, 2826516, 743820, 106260, 7590, 230, 2
Offset: 0

Views

Author

Gheorghe Coserea, May 15 2017

Keywords

Comments

Row n>0 contains n terms.
T(n,k) is the number of Feynman's diagrams with k fermionic loops in the order n of the perturbative expansion in dimension zero for the GW approximation of the polarization function in a many-body theory of fermions with two-body interaction (see Molinari link).

Examples

			A(x;t) = 1 + 2*x + (5 + 2*t)*x^2 + (14 + 14*t + 2*t^2)*x^3 + ...
Triangle starts:
   n\k |     0       1       2       3       4      5     6    7  8
  -----+-----------------------------------------------------------
   0   |     1;
   1   |     2;
   2   |     5,      2;
   3   |    14,     14,      2;
   4   |    42,     72,     27,      2;
   5   |   132,    330,    220,     44,      2;
   6   |   429,   1430,   1430,    520,     65,     2;
   7   |  1430,   6006,   8190,   4550,   1050,    90,    2;
   8   |  4862,  24752,  43316,  33320,  11900,  1904,  119,   2;
   9   | 16796, 100776, 217056, 217056, 108528, 27132, 3192, 152, 2;

Links

Crossrefs

Cf. A000108, A286781, A286782, A286783.

Programs

  • Maxima
    T(n,k):=(binomial(n-1,k)*binomial(2*(n+1),n-k))/(n+1); /* Vladimir Kruchinin, Jan 14 2022 */
  • PARI
    A286784_ser(N,t='t) = my(x='x+O('x^N)); serreverse(Ser(x*(1-x)^2/(1+(t-1)*x)))/x;
A286785_ser(N,t='t) = 1/(1-x*A286784_ser(N,t))^2;
concat(apply(p->Vecrev(p), Vec(A286785_ser(12))))

Formula

y(x;t) = Sum_{n>=0} P_n(t)*x^n = 1/(1-x*s)^2, where s(x;t) = A286784(x;t) and P_n(t) = Sum_{k=0..n-1} T(n,k)*t^k for n>0.
A000108(n+1) = T(n,0), A002058(n+3) = T(n,1), A014106(n-1) = T(n,n-2), A006013(n) = P_n(1), A211789(n+1) = P_n(2).
T(n,k) = C(n-1,k)*C(2*n+2,n-k)/(n+1). - Vladimir Kruchinin, Jan 14 2022

A286795 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.

Original entry on oeis.org

1, 1, 4, 3, 27, 31, 5, 248, 357, 117, 7, 2830, 4742, 2218, 314, 9, 38232, 71698, 42046, 9258, 690, 11, 593859, 1216251, 837639, 243987, 30057, 1329, 13, 10401712, 22877725, 17798029, 6314177, 1071809, 81963, 2331, 15, 202601898, 472751962, 404979234, 166620434, 35456432, 3857904, 196532, 3812, 17, 4342263000, 10651493718, 9869474106, 4561150162, 1149976242, 160594860, 11946360, 426852, 5904, 19
Offset: 0

Views

Author

Gheorghe Coserea, May 21 2017

Keywords

Comments

Row n>0 contains n terms.
"The series expansion of the solution counts skeleton vertex diagrams with dressed propagators and bare interactions." (see G^2v-skeleton expansion in Molinari link)

Examples

			A(x;t) = 1 + x + (4 + 3*t)*x^2 + (27 + 31*t + 5*t^2)*x^3 + ...
Triangle starts:
n\k  [0]       [1]       [2]       [3]      [4]      [5]    [6]   [7]
[0]  1;
[1]  1;
[2]  4,        3;
[3]  27,       31,       5;
[4]  248,      357,      117,      7;
[5]  2830,     4742,     2218,     314,     9;
[6]  38232,    71698,    42046,    9258,    690,     11;
[7]  593859,   1216251,  837639,   243987,  30057,   1329,  13;
[8]  10401712, 22877725, 17798029, 6314177, 1071809, 81963, 2331, 15;
[9] ...

Links

Crossrefs

Cf. A286781, A286782, A286783, A286784, A286785.

Programs

  • Mathematica
    max = 11; y0[x_, t_] = 1; y1[x_, t_] = 0; For[n = 1, n <= max, n++, y1[x_, t_] = ((1 + x*(1 + 2 t + x t^2) y0[x, t]^2 + t (1 - t)*x^2*y0[x, t]^3 + 2 x^2 y0[x, t] D[y0[x, t], x]))/(1 + 2 x*t) + O[x]^n // Normal; y0[x_, t_] = y1[x, t]];
row[n_] := CoefficientList[Coefficient[y0[x, t], x, n], t];
Table[row[n], {n, 0, max - 1}] // Flatten (* Jean-François Alcover, May 23 2017, adapted from PARI *)
  • PARI
    A286795_ser(N, t='t) = {
  my(x='x+O('x^N), y0=1, y1=0, n=1);
  while(n++,
    y1 = (1 + x*(1 + 2*t + x*t^2)*y0^2 + t*(1-t)*x^2*y0^3 + 2*x^2*y0*y0');
    y1 = y1 / (1+2*x*t); if (y1 == y0, break()); y0 = y1;); y0;
};
concat(apply(p->Vecrev(p), Vec(A286795_ser(11))))
\\ test: y=A286795_ser(50); 0 == 1 - (1 + 2*x*t)*y + x*(1 + 2*t + x*t^2)*y^2 + t*(1-t)*x^2*y^3 + 2*x^2*y*y'

Formula

y(x;t) = Sum_{n>=0} P_n(t)*x^n satisfies 0 = 1 - (1 + 2*x*t)*y + x*(1 + 2*t + x*t^2)*y^2 + t*(1-t)*x^2*y^3 + 2*x^2*y*deriv(y,x), with y(0;t)=1, where P_n(t) = Sum_{k=0..n-1} T(n,k)*t^k for n>0.
A000699(n+1) = T(n,0), 1 = P_n(-1), A049464(n+1) = P_n(1).

A286798 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section. .

Original entry on oeis.org

1, 1, 4, 2, 27, 22, 248, 264, 30, 2830, 3610, 830, 8, 38232, 55768, 18746, 1078, 593859, 961740, 414720, 46986, 576, 10401712, 18326976, 9457788, 1593664, 62682, 112, 202601898, 382706674, 226526362, 49941310, 3569882, 45296, 4342263000, 8697475368, 5740088706, 1540965514, 160998750, 4909674, 16896, 101551822350, 213865372020, 154271354280, 48205014786, 6580808784, 337737294, 4200032, 2560
Offset: 0

Views

Author

Gheorghe Coserea, May 21 2017

Keywords

Comments

Row n>0 contains floor(2*(n+1)/3) terms.

Examples

			A(x;t) = 1 + x + (4 + 2*t)*x^2 + (27 + 22*t)*x^3 + (248 + 264*t + 30*t^2)*x^4 +
Triangle starts:
n\k  [0]        [1]        [2]        [3]       [4]      [5]
[0]  1;
[1]  1;
[2]  4,         2;
[3]  27,        22;
[4]  248,       264,       30;
[5]  2830,      3610,      830,       8;
[6]  38232,     55768,     18746,     1078;
[7]  593859,    961740,    414720,    46986,    576;
[8]  10401712,  18326976,  9457788,   1593664,  62682,   112;
[9]  202601898, 382706674, 226526362, 49941310, 3569882, 45296;
[10] ...

Links

Crossrefs

Cf. A286781, A286782, A286783, A286784, A286785.

Programs

  • Mathematica
    max = 12; y0[x_, t_] = 1; y1[x_, t_] = 0; For[n = 1, n <= max, n++, y1[x_, t_] = 1 + x y0[x, t]^2 + 3 t x^3 y0[x, t]^2 D[y0[x, t], x] + x^2 (2 y0[x, t] D[y0[x, t], x] + t (2 y0[x, t]^3 - D[y0[x, t], x] + y0[x, t] D[y0[x, t], x])) + O[x]^n // Normal // Simplify; y0[x_, t_] = y1[x, t]];
P[n_, t_] := Coefficient[y0[x, t] , x, n];
row[n_] := CoefficientList[P[n, t], t];
Table[row[n], {n, 0, max}] // Flatten (* Jean-François Alcover, May 24 2017, adapted from PARI *)
  • PARI
    A286795_ser(N, t='t) = {
  my(x='x+O('x^N), y0=1, y1=0, n=1);
  while(n++,
    y1 = (1 + x*(1 + 2*t + x*t^2)*y0^2 + t*(1-t)*x^2*y0^3 + 2*x^2*y0*y0');
    y1 = y1 / (1+2*x*t); if (y1 == y0, break()); y0 = y1;); y0;
};
A286798_ser(N,t='t) = {
  my(v = A286795_ser(N,t)); subst(v, 'x, serreverse(x/(1-x*t*v)));
};
concat(apply(p->Vecrev(p), Vec(A286798_ser(12))))
\\ test: y=A286798_ser(50); x^2*y' == (1 - y + x*y^2 + 2*x^2*t*y^3)/(t - (2+t)*y - 3*x*t*y^2)

Formula

y(x;t) = Sum_{n>=0} P_n(t)*x^n satisfies x^2*deriv(y,x) = (1 - y + x*y^2 + 2*x^2*t*y^3)/(t - (2+t)*y - 3*x*t*y^2), with y(0;t) = 1, where P_n(t) = Sum_{k=0..floor((2*n-1)/3)} T(n,k)*t^k for n>0.
A000699(n+1)=T(n,0), A000108(n)=P_n(-1), A286799(n)=P_n(1).

A286800 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.

Original entry on oeis.org

1, 1, 2, 7, 6, 63, 74, 10, 729, 974, 254, 8, 10113, 15084, 5376, 406, 161935, 264724, 117424, 14954, 320, 2923135, 5163276, 2697804, 481222, 23670, 112, 58547761, 110483028, 65662932, 14892090, 1186362, 21936, 1286468225, 2570021310, 1695874928, 461501018, 51034896, 1866986, 11264, 30747331223, 64547199082, 46461697760, 14603254902, 2055851560, 116329886, 1905888, 2560
Offset: 1

Views

Author

Gheorghe Coserea, May 22 2017

Keywords

Comments

Row n>0 contains floor(2*(n+1)/3) terms.

Examples

			A(x;t) = x + (1 + 2*t)*x^2 + (7 + 6*t)*x^3 + (63 + 74*t + 10*t^2)*x^4 + ...
Triangle starts:
n\k  [0]       [1]        [2]       [3]       [4]      [5]
[1]  1;
[2]  1,        2;
[3]  7,        6;
[4]  63,       74,        10;
[5]  729,      974,       254,      8;
[6]  10113,    15084,     5376,     406;
[7]  161935,   264724,    117424,   14954,    320;
[8]  2923135,  5163276,   2697804,  481222,   23670,   112;
[9]  58547761, 110483028, 65662932, 14892090, 1186362, 21936;
[10] ...

Links

Crossrefs

Cf. A286781, A286782, A286783, A286784, A286785.

Programs

  • Mathematica
    max = 12; y0[0, ] = y1[0, ] = 0; y0[x_, t_] = x; y1[x_, t_] = 0; For[n = 1, n <= max, n++, y1[x_, t_] = Normal[(1/(-1 + y0[x, t]))*x*(-1 - y0[x, t]^2 - 2*y0[x, t]*(-1 + D[y0[x, t], x]) + t*x*(-1 + y0[x, t])*(2*(-1 + y0[x, t])^2 + (x*(-1 + y0[x, t]) + y0[x, t])*D[y0[x, t], x])) + O[x]^n]; y0[x_, t_] = y1[x, t]];
row[n_] := CoefficientList[SeriesCoefficient[y0[x, t], {x, 0, n}], t];
Flatten[Table[row[n], {n, 0, max-1}]] (* Jean-François Alcover, May 24 2017, adapted from PARI *)
  • PARI
    A286795_ser(N, t='t) = {
  my(x='x+O('x^N), y0=1, y1=0, n=1);
  while(n++,
    y1 = (1 + x*(1 + 2*t + x*t^2)*y0^2 + t*(1-t)*x^2*y0^3 + 2*x^2*y0*y0');
    y1 = y1 / (1+2*x*t); if (y1 == y0, break()); y0 = y1;); y0;
};
A286798_ser(N,t='t) = {
  my(v = A286795_ser(N,t)); subst(v, 'x, serreverse(x/(1-x*t*v)));
};
A286800_ser(N, t='t) = {
  my(v = A286798_ser(N,t)); 1-1/subst(v, 'x, serreverse(x*v^2));
};
concat(apply(p->Vecrev(p), Vec(A286800_ser(12))))
\\ test: y=A286800_ser(50); x*y' == (1-y) * (2*t*x^2*(1-y)^2 + x*(1-y) - y) / (t*x^2*(1-y)^2 - t*x*y*(1-y) - 2*y)

Formula

y(x;t) = Sum_{n>0} P_n(t)*x^n satisfies x*deriv(y,x) = (1-y) * (2*t*x^2*(1-y)^2 + x*(1-y) - y) / (t*x^2*(1-y)^2 - t*x*y*(1-y) - 2*y), with y(0;t) = 0, where P_n(t) = Sum_{k=0..floor((2*n-1)/3)} T(n,k)*t^k for n>0.
A049464(n) = T(n,0), P_n(-1) = (-1)^(n-1), A287029(n) = P_n(1).

A286796 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.

Original entry on oeis.org

1, 1, 1, 4, 5, 1, 27, 40, 14, 1, 248, 419, 200, 30, 1, 2830, 5308, 3124, 700, 55, 1, 38232, 78070, 53620, 15652, 1960, 91, 1, 593859, 1301088, 1007292, 356048, 60550, 4704, 140, 1, 10401712, 24177939, 20604768, 8430844, 1787280, 194854, 10080, 204, 1, 202601898, 495263284, 456715752, 209878440, 52619854, 7322172, 545908, 19800, 285, 1, 4342263000, 11085720018, 10921213644, 5516785032, 1579263840, 264576774, 25677652, 1372228, 36300, 385, 1
Offset: 0

Views

Author

Gheorghe Coserea, May 21 2017

Keywords

Examples

			A(x;t) = 1 + (1 + t)*x + (4 + 5*t + t^2)*x^2 + (27 + 40*t + 14*t^2 + t^3)*x^3 + ...
Triangle starts:
n\k  [0]       [1]       [2]       [3]      [4]      [5]     [6]    [7]  [8]
[0]  1;
[1]  1;        1;
[2]  4,        5,        1;
[3]  27,       40,       14,       1;
[4]  248,      419,      200,      30,      1;
[5]  2830,     5308,     3124,     700,     55,      1;
[6]  38232,    78070,    53620,    15652,   1960,    91,     1;
[7]  593859,   1301088,  1007292,  356048,  60550,   4704,   140,   1;
[8]  10401712, 24177939, 20604768, 8430844, 1787280, 194854, 10080, 204, 1;
[9]  ...

Links

Crossrefs

Cf. A286781, A286782, A286783, A286784, A286785.

Programs

  • Mathematica
    max = 11; y0[x_, t_] = 1; y1[x_, t_] = 0; For[n = 1, n <= max, n++, y1[x_, t_] = Normal[(1 + x*(1 + 2*t + x*t^2)*y0[x, t]^2 + t*(1 - t)*x^2*y0[x, t]^3 + 2*x^2*y0[x, t]*D[y0[x, t], x])/(1 + 2*x*t) + O[x]^n]; y0[x_, t_] = y1[x, t]];
row[n_] := CoefficientList[SeriesCoefficient[y0[x, t]/(1 - x*t*y0[x, t]), {x, 0, n}], t];
Flatten[Table[row[n], {n, 0, max-1}]] (* Jean-François Alcover, May 23 2017, adapted from PARI *)
  • PARI
    A286795_ser(N, t='t) = {
  my(x='x+O('x^N), y0=1, y1=0, n=1);
  while(n++,
    y1 = (1 + x*(1 + 2*t + x*t^2)*y0^2 + t*(1-t)*x^2*y0^3 + 2*x^2*y0*y0');
    y1 = y1 / (1+2*x*t); if (y1 == y0, break()); y0 = y1;); y0;
};
A286796_ser(N,t='t) = my(v=A286795_ser(N,t)); v/(1-x*t*v);
concat(apply(p->Vecrev(p), Vec(A286796_ser(11))))

Formula

A(x;t) = Sum_{n>=0} P_n(t)*x^n = v/(1-x*t*v), where v(x;t) = A286795(x;t) and P_n(t) = Sum_{k=0..n} T(n,k)*t^k.
A000699(n+1)=T(n,0), A000330(n)=T(n,n-1), A286797(n)=P_n(1) and P_n(-1)=0 for n>0.
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