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.

Showing 1-10 of 11 results. Next

A287041 Column 1 of A286783.

Original entry on oeis.org

5, 77, 1044, 14784, 227877, 3862305, 71983440, 1469813400, 32718512925, 789901955325, 20578796752500, 575836554270600, 17232413940017325, 549370878062313825, 18591830334684129600, 665771181527890746000, 25154357611638416671125, 1000094581801108086418125, 41741166856778766269392500
Offset: 2

Views

Author

Gheorghe Coserea, May 19 2017

Keywords

Links

Crossrefs

Cf. A286783.

Programs

  • 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;
};
Kol(K, N=20) = {
  my(s = A286781_ser(N+K+1, 't+O('t^(K+1))),
     p = (1 + x*s + 2*x^2*s')/(1-x*s)^2);
  vector(N, n, polcoeff(polcoeff(p, K+n), K));
};
Kol(1)

A287042 Column 2 of A286783.

Original entry on oeis.org

7, 234, 5390, 113126, 2371845, 51607716, 1185214452, 28937407212, 752882360571, 20870819679150, 615571317411570, 19277508315195090, 639508594242409065, 22419339547774938120, 828617069130919072200, 32214653364317860157400, 1314580813236368248272975, 56192348229571762396268850
Offset: 3

Views

Author

Gheorghe Coserea, May 19 2017

Keywords

Links

Crossrefs

Cf. A286783.

A287043 Column 3 of A286783.

Original entry on oeis.org

9, 550, 19760, 586425, 16271380, 446964322, 12516198870, 362901875292, 10984979930625, 348559615602900, 11613184968311010, 406385382363237945, 14927815613184463440, 575018321453046128700, 23197450399028711170020, 978734324107274793589440, 43125048817869336467480865
Offset: 4

Views

Author

Gheorghe Coserea, May 19 2017

Keywords

Links

Crossrefs

Cf. A286783.

A287044 Column 4 of A286783.

Original entry on oeis.org

11, 1105, 58275, 2356234, 84487110, 2884205268, 97429005146, 3329662705550, 116630388753489, 4219224762555705, 158324528197417845, 6176733251642200764, 250790095398521046060, 10599852184347703429872, 466235894350970551104804, 21329051731506881800764804
Offset: 5

Views

Author

Gheorghe Coserea, May 19 2017

Keywords

Links

Crossrefs

Cf. A286783.

A286781 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.

Original entry on oeis.org

1, 2, 1, 10, 9, 1, 74, 91, 23, 1, 706, 1063, 416, 46, 1, 8162, 14193, 7344, 1350, 80, 1, 110410, 213953, 134613, 34362, 3550, 127, 1, 1708394, 3602891, 2620379, 842751, 125195, 8085, 189, 1, 29752066, 67168527, 54636792, 20862684, 4009832, 382358, 16576, 268, 1, 576037442, 1375636129, 1223392968, 533394516, 124266346, 15653598, 1023340, 31356, 366, 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 self-energy function in a many-body theory of fermions with two-body interaction (see Molinari link).

Examples

			A(x;t) = 1 + (2 + t)*x + (10 + 9*t + t^2)*x^2 + (74 + 91*t + 23*t^2 + t^3)*x^3 + ...
Triangle starts:
n\k  [0]       [1]       [2]       [3]       [4]      [5]     [6]    [7]  [8]
[0]  1;
[1]  2,        1;
[2]  10,       9,        1;
[3]  74,       91,       23,       1;
[4]  706,      1063,     416,      46,       1;
[5]  8162,     14193,    7344,     1350,     80,      1;
[6]  110410,   213953,   134613,   34362,    3550,    127,    1;
[7]  1708394,  3602891,  2620379,  842751,   125195,  8085,   189,   1;
[8]  29752066, 67168527, 54636792, 20862684, 4009832, 382358, 16576, 268, 1;
[9] ...

Links

Crossrefs

For vertex and polarization functions see A286782 and A286783. For GWA of the self-energy and polarization functions see A286784 and A286785.
Columns k=0-8 give: A000698(k=0), A286786(k=1), A286787(k=2), A286788(k=3), A286789(k=4), A286790(k=5), A286791(k=6), A286792(k=7), A286793(k=8).

Programs

  • Mathematica
    max = 10; 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]];
row[n_] := CoefficientList[Coefficient[y0[x, t], x, n], t];
Table[row[n], {n, 0, max-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;
};
concat(apply(p->Vecrev(p), Vec(A286781_ser(10))))
\\ test: y = A286781_ser(50); y*(1-x*y)^2 == (1 + x*y + 2*x^2*deriv(y,'x)) * (1 - x*y*(1-t))

Formula

y(x;t) = Sum_{n>=0} P_n(t)*x^n satisfies y * (1-x*y)^2 = (1 + x*y + 2*x^2*deriv(y,x)) * (1 - x*y*(1-t)), with y(0;t) = 1, where P_n(t) = Sum_{k=0..n} T(n,k)*t^k, 0<=n, 0<=k<=n.
A000698(n+1)=T(n,0), A101986(n)=T(n,n-1), A000108(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).
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