A049464 -id:A049464 - cp's OEIS frontend

A005416 Vertex diagrams of order 2n.

1, 1, 6, 50, 518, 6354, 89782, 1435330, 25625910, 505785122, 10944711398, 257834384850, 6572585595622, 180334118225650, 5300553714899094, 166206234856979810, 5538980473666776854, 195527829569946627138, 7288988096561232432070

Offset: 0

N. J. A. Sloane

			G.f. = 1 + x + 6*x^2 + 50*x^3 + 518*x^4 + 6354*x^5 + 89782*x^6 + 1435330*x^7 + ...

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..100
D. J. Broadhurst, Four-loop Dyson-Schwinger-Johnson anatomy, arXiv:hep-ph/9909336, 1999.
P. Cvitanovic, Asymptotic estimates and gauge invariance, Nuclear Phys. B 127 (1977), 176-188.
R. J. Martin and M. J. Kearney, An exactly solvable self-convolutive recurrence, arXiv:1103.4936 [math.CO], 2011.
R. J. Martin and M. J. Kearney, An exactly solvable self-convolutive recurrence, Aequat. Math., 80 (2010), 291-318. see p. 292.

Cf. A000698, A001147, A049464.

m = 19; s[x_] = Sum[(2*n)!/(2^n*n!)*x^n, {n, 0, m}]; gf[x_] = (s[x] - 1)/(s[x]^2*x); Most[CoefficientList[Series[gf[x], {x, 0, m}], x]] (* Jean-François Alcover, Aug 31 2011, after g.f. *)

{a(n) = my(A); if( n<0, 0, A = sum( k=0, n+1, (2*k)! / k! /2^k * x^k, x^2 * O(x^n)); polcoeff( (A - 1) / (x * A^2), n))}; /* Michael Somos, Oct 11 2006 */

{a(n) = my(A); if( n<1, n==0, A = vector(n); A[1] = 1; for( k=2, n, A[k] = (2 * k - 3) * A[k-1] + sum( j=1, k-1, A[j] * A[k-j])); (2*n - 1) * A[n])}; /* Michael Somos, Jul 24 2011 */

Let s_n = (2*n)!/(2^n*n!) (A001147), S(x) = Sum_{n >= 0} s_n*x^n; sequence has g.f. A(x) satisfying 1 - 1/S(x) = x*A(x)*S(x).

a(n) = (2*n - 1) * A000698(n). [Martin and Kearney]

A286797 Row sums of A286796.

Original entry on oeis.org

1, 2, 10, 82, 898, 12018, 187626, 3323682, 65607682, 1424967394, 33736908874, 864372576626, 23825543471234, 703074672632018, 22118247888976170, 739081808704195650, 26146116129400483842, 976382058777174451650, 38386296866727499728522, 1584986693941237056394386

Offset: 0

Gheorghe Coserea, May 21 2017

nonn

Gheorghe Coserea, Table of n, a(n) for n = 0..200
Luca G. Molinari, Nicola Manini, Enumeration of many-body skeleton diagrams, arXiv:cond-mat/0512342 [cond-mat.str-el], 2006.

Cf. A286796.

max = 20; 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 // Simplify; y0[x_, t_] = y1[x, t]];
a[n_] := CoefficientList[SeriesCoefficient[y0[x, t]/(1 - x*t*y0[x, t]), {x, 0, n}], t] // Total;
Table[a[n], {n, 0, max-1}] (* Jean-François Alcover, May 24 2017, adapted from 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);
Vec(A286796_ser(20,1))

A049464_ser(N) = {  \\ for A049464(0)=0
  my(s=Ser(concat(1, vector(N+1, n, (2*n)!/(2^n*n!)))), g=(1/s - 1/s^2)/x);
  1 - 1/subst(g, 'x, serreverse(x*g^2*s^2));
};
A286797_ser(N) = my(q=A049464_ser(N)); q/(x-x*q);
Vec(A286797_ser(20))

a(n) = Sum_{k=0..n} A286796(n,k).

a(n) ~ 2^(n + 5/2) * n^(n+2) / exp(n+2). - Vaclav Kotesovec, Mar 08 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

Gheorghe Coserea, May 21 2017

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)

			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] ...

Gheorghe Coserea, Rows n=0..123, flattened
Luca G. Molinari, Nicola Manini, Enumeration of many-body skeleton diagrams, arXiv:cond-mat/0512342 [cond-mat.str-el], 2006.

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

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 *)

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'

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).

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

Gheorghe Coserea, May 22 2017

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

			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] ...

Gheorghe Coserea, Rows n=1..123, flattened
Luca G. Molinari, Nicola Manini, Enumeration of many-body skeleton diagrams, arXiv:cond-mat/0512342 [cond-mat.str-el], 2006.

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

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 *)

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)

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).

A287029 Row sums of A286800.

Original entry on oeis.org

1, 3, 13, 147, 1965, 30979, 559357, 11289219, 250794109, 6066778627, 158533572861, 4447703062787, 133309656009469, 4251322261512195, 143749952968507389, 5137921526511802371, 193589838004887201789, 7670544451820808601603, 318892867844484240154621, 13881730766388536085356547

Offset: 1

Gheorghe Coserea, May 22 2017

nonn

			A(x) = x + 3*x^2 + 13*x^3 + 147*x^4 + 1965*x^5 + 30979*x^6 + ...

Gheorghe Coserea, Table of n, a(n) for n = 1..200
Michael Borinsky, Renormalized asymptotic enumeration of Feynman diagrams, arXiv:1703.00840 [hep-th], 2017.
Luca G. Molinari, Nicola Manini, Enumeration of many-body skeleton diagrams, arXiv:cond-mat/0512342 [cond-mat.str-el], 2006.

Cf. A049464, A286799, A286800, A287039.

terms = 20; y[, ] = 0; Do[y[x_, t_] = (1/(-1 + y[x, t])) x (-1 - y[x, t]^2 - 2 y[x, t] (-1 + D[y[x, t], x]) + t x (-1 + y[x, t]) (2 (-1 + y[x, t])^2 + (x (-1 + y[x, t]) + y[x, t]) D[y[x, t], x])) + O[x]^n // Normal // Simplify, {n, terms+1}];
Total[CoefficientList[#, t]]& /@ CoefficientList[y[x, t], x] // Rest

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));
};
A287029_ser(N) = A286800_ser(N+1, 1);
Vec(A287029_ser(20))

a(n) = Sum_{k=0..floor((2*n-1)/3)} A286800(n,k) for n>=1.

a(n) ~ 4*exp(-7/2)/sqrt(Pi) * n^(3/2) * 2^n * n! * (1 - 15/(8*n) - 503/(128*n^2) + O(1/n^3)). (see Borinsky link) - Gheorghe Coserea, Oct 21 2017

A287039 Row sums of A286782.

Original entry on oeis.org

1, 1, 9, 100, 1323, 20088, 342430, 6461208, 133618275, 3006094768, 73139285178, 1914937983000, 53720914023150, 1608612191370000, 51235727245542684, 1730349877484075120, 61783682196714238755, 2326122843950925857376, 92117389831885545623650, 3828375469597215729851928

Offset: 0

Gheorghe Coserea, May 18 2017

nonn

Gheorghe Coserea, Table of n, a(n) for n = 0..302
Michael Borinsky, Renormalized asymptotic enumeration of Feynman diagrams, arXiv:1703.00840 [hep-th], 2017.
Luca G. Molinari, Hedin's equations and enumeration of Feynman's diagrams, arXiv:cond-mat/0401500 [cond-mat.str-el], 2005.
Luca G. Molinari, Nicola Manini, Enumeration of many-body skeleton diagrams, arXiv:cond-mat/0512342 [cond-mat.str-el], 2006.

Cf. A000699, A049464, A286782, A287029.

max = 21; (* B(x) is A000699(x) *) B[_] = 0;
Do[B[x_] = x + x^2 D[B[x]^2/x, x] + O[x]^max // Normal, max];
Join[{1}, Drop[CoefficientList[(1-x/B[x])/x + O[x]^max, x], -2] Table[2n-1, {n, max-2}]] (* Jean-François Alcover, Oct 25 2018, from PARI *)

A000699_seq(N) = {
  my(a = vector(N)); a[1] = 1;
  for (n=2, N, a[n] = sum(k=1, n-1, (2*k-1)*a[k]*a[n-k])); a;
};
A286794_seq(N) = Vec((1-1/Ser(A000699_seq(N+1)))/x);
A287039_seq(N) = {
  my(s = A286794_seq(N));
  concat(1, vector(#s, n, (2*n-1)*s[n]));
};
A287039_seq(19)

a(n) ~ 4*exp(-1)/sqrt(Pi) * n^(3/2) * 2^n * n! * (1 - 19/(8*n) - 23/(128*n^2) + O(1/n^3)). (see Borinsky link) - Gheorghe Coserea, Oct 21 2017

A005414 Feynman diagrams of order 2n with vertex skeletons.

Original entry on oeis.org

1, 1, 13, 93, 1245, 18093, 308605, 5887453, 124221373, 2864305277, 71589605885, 1927010749181, 55572839581437, 1709604517055229, 55893262628149245, 1935654236127347709, 70799043456576835581, 2727771901780930132989, 110438840436968476274685, 4688223534904569925386237

Offset: 1

N. J. A. Sloane

nonn

P. Cvitanovic, B. Lautrup and R. B. Pearson, Number and weights of Feynman diagrams, Phys. Rev. D 18 (1978), 1939-1949.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Gheorghe Coserea, Table of n, a(n) for n = 1..202
Michael Borinsky, Renormalized asymptotic enumeration of Feynman diagrams, arXiv:1703.00840 [hep-th], 2017.
P. Cvitanovic, B. Lautrup and R. B. Pearson, The number and weights of Feynman diagrams, Phys. Rev. D18, 1939 (1978).

Cf. A001147, A000698, A005411, A005412, A005413, A005416, A049464.

seq[nn_] := Module[{x, y=0, y1=0, n=1}, While[n++; True, y1 = x^2 + x^4 + 2x^6 - 3x^2 y + x^4 (-y + x D[y, x]/2) - x^6 (8y + x D[y, x]/2) + y^2 + x y D[y, x] + (x^2 - x^4)(3y^2 + 3/2 x y D[y, x]) + x^6 (12y^2 + 3/2 x y D[y, x]) - x^2 (y^3 + 3/2 x y^2 D[y, x]) + x^4 (5y^3 + 3/2 x y^2 D[y, x]) - x^6 (8y^3 + 3/2 x y^2 D[y, x]) + (-x^4 + x^6)(2y^4 + 1/2 x y^3 D[y, x]) + O[x]^(2nn+1); If[y1 == y, Break[]]; y = y1]; CoefficientList[y, x^2]] // Rest;
seq[20] (* Jean-François Alcover, Oct 05 2018, after Gheorghe Coserea *)

seq(N) = {
  my(x='x+O('x^(2*N+1)), y=0, y1=0, n=1);
  while (n++,
  y1 = x^2 + x^4 + 2*x^6 - 3*x^2*y + x^4*(-y + 1/2*x*y') +
       -x^6*(8*y + 1/2*x*y') + y^2 + x*y*y' +
       (x^2 - x^4)*(3*y^2 + 3/2*x*y*y') + x^6*(12*y^2 + 3/2*x*y*y') +
       -x^2*(y^3 + 1/2*x*3*y^2*y') + x^4*(5*y^3 + 1/2*x*3*y^2*y') +
       -x^6*(8*y^3 + 1/2*x*3*y^2*y') + (-x^4+x^6)*(2*y^4 + 1/8*x*4*y^3*y');
  if (y1 == y, break); y=y1);
  vector(N, n, polcoeff(y, 2*n));
};
seq(20) \\ Gheorghe Coserea, Oct 17 2017

a(n) ~ 4*exp(-5/2)/Pi * n * 2^n * n! * (1 - 9/(4*n) - 67/(32*n^2) + O(1/n^3)). (see Borinsky link) - Gheorghe Coserea, Oct 19 2017

More terms from Gheorghe Coserea, Oct 17 2017

A294158 Row sums of A291844.

Original entry on oeis.org

1, 1, 6, 52, 602, 8223, 128917, 2273716, 44509914, 957408649, 22449011336, 570032756328, 15587503694363, 456793916757139, 14284890417759141, 474896318288651220, 16726743380843538668, 622282429409944248297, 24385251974172090147514, 1004017088910699487855180

Offset: 0

Gheorghe Coserea, Oct 24 2017

nonn

Gheorghe Coserea, Table of n, a(n) for n = 0..202
Luca G. Molinari, Nicola Manini, Enumeration of many-body skeleton diagrams, arXiv:cond-mat/0512342 [cond-mat.str-el], 2006.

Cf. A049464(y), A287039(x), A286799(z), A287029(u), A291844.

A291843_ser(N, t='t) = {
  my(x='x+O('x^N), y=1, y1=0, n=1,
  dn = 1/(-2*t^2*x^4 - (2*t^2+3*t)*x^3 - (2*t+1)*x^2 + (2*t-1)*x + 1));
  while (n++,
   y1 = (2*x^2*y'*((-t^2 + t)*x + (-t + 1) + (t^2*x^2 + (t^2 + t)*x + t)*y) +
        (t*x^2 + t*x)*y^2 - (2*t^2*x^3 + 3*t*x^2 + (-t + 1)*x - 1))*dn;
   if (y1 == y, break); y = y1;); y;
};
A291844_ser(N, t='t) = {
  my(z = A291843_ser(N+1,t));
  ((1+x)*z - 1)*(1 + t*x)/((1-t + t*(1+x)*z)*x*z^2);
};
Vec(A291844_ser(20,t=1))

a(n) = Sum_{k=0..floor((2*n-1)/3)} A291844(n,k), n > 0.

cp's OEIS Frontend

A005416 Vertex diagrams of order 2n.

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A286797 Row sums of A286796.

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

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

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A287029 Row sums of A286800.

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A287039 Row sums of A286782.

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A005414 Feynman diagrams of order 2n with vertex skeletons.

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