A286795 -id:A286795 - cp's OEIS frontend

A049464 Number of n-photon quenched skeletons.

1, 1, 1, 7, 63, 729, 10113, 161935, 2923135, 58547761, 1286468225, 30747331223, 793992877247, 22031281255689, 653827064820993, 20670172958564127, 693662602935500031, 24632233419065156193, 922938914156271368961

Offset: 0

N. J. A. Sloane

Gheorghe Coserea, Table of n, a(n) for n = 0..202
Michael Borinsky, Renormalized asymptotic enumeration of Feynman diagrams, arXiv:1703.00840 [hep-th], 2017.
D. J. Broadhurst, Four-loop Dyson-Schwinger-Johnson anatomy, arXiv:hep-ph/9909336, 1999.
Ali Assem Mahmoud, An Asymptotic Expansion for the Number of 2-Connected Chord Diagrams, arXiv:2009.12688 [math.CO], 2020.
Ali Assem Mahmoud, Chord Diagrams and the Asymptotic Analysis of QED-type Theories, arXiv:2011.04291 [hep-th], 2020.
Ali Assem Mahmoud, An asymptotic expansion for the number of two-connected chord diagrams, J. Math. Phys. (2023) Vol. 64, 122301. See Section V.
Luca G. Molinari and Nicola Manini, Enumeration of many-body skeleton diagrams, arXiv:cond-mat/0512342 [cond-mat.str-el], 2006.

Cf. A001147, A005416, A286795.

terms = 19; y[] = 0; Do[y[x] = (x + (1 + x)*y[x]^2 + 2*x*y[x]*y'[x])/(1 + 2*x) + O[x]^terms // Normal, terms]; CoefficientList[1 + y[x], x] (* Jean-François Alcover, Aug 14 2013, updated Jan 12 2018 *)

seq(N) = {
  my(s=Ser(concat(1, vector(N, n, (2*n)!/(2^n*n!)))), g=(1/s - 1/s^2)/x);
  Vec(1 - 1/subst(g, 'x, serreverse(x*g^2*s^2)));
};
concat(1, seq(19))
\\ test: y='x*Ser(seq(200)); 0==2*x*y*y' + (1+x)*y^2 - (2*x+1)*y + x
\\ Gheorghe Coserea, Oct 12 2017

Reference gives recurrence.
From Gheorghe Coserea, Oct 22 2017: (Start)
a(n) ~ 2*exp(-2)/sqrt(Pi) * n^(1/2) * 2^n * n! * (1 - 21/(8*n) - 87/(128*n^2) + O(1/n^3)). (see Borinsky link)
For n > 0 we have a(n) == 1 (mod 8) if n mod 8 in {1,2,5,6}, otherwise a(n) == 7 (mod 8).
G.f. y(x) satisfies (with a(0)=0): g = 1 + g*y(x*g^2*s^2), where s = A001147(x) and g = A005416(x). (eqn. (7) in Broadhurst link)
0 = 2*x*y*deriv(y,x) + (1+x)*y^2 - (2*x+1)*y + x.
(End)

A291844 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, 29, 23, 274, 292, 36, 3145, 4068, 994, 16, 42294, 62861, 22250, 1512, 651227, 1075562, 484840, 61027, 1060, 11295242, 20275944, 10867381, 1977879, 93188, 280, 217954807, 418724047, 255929070, 59896915, 4823178, 80632, 4632600152, 9418874022, 6387031115, 1798212190, 204846125, 7410676, 37056, 107572674851, 229535650138, 169414005231, 55017177704, 8022471066, 463514918, 7255380, 7040

Offset: 0

Gheorghe Coserea, Oct 24 2017

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

			A(x;t) = 1 + x + (4 + 2*t)*x^2 + (29 + 23*t)*x^3 + (274 + 292*t + 36*t^2)*x^4 + ...
Triangle starts:
n\k  [0]        [1]        [2]        [3]       [4]      [5]
[0]  1;
[1]  1;
[2]  4,         2;
[3]  29,        23;
[4]  274,       292,       36;
[5]  3145,      4068,      994,       16;
[6]  42294,     62861,     22250,     1512;
[7]  651227,    1075562,   484840,    61027,    1060;
[8]  11295242,  20275944,  10867381,  1977879,  93188,   280;
[9]  217954807, 418724047, 255929070, 59896915, 4823178, 80632;
[10] ...

Gheorghe Coserea, Rows n = 0..124, flattened
Luca G. Molinari, Nicola Manini, Enumeration of many-body skeleton diagrams, arXiv:cond-mat/0512342 [cond-mat.str-el], 2006.

Cf. A286795, A286798, A286800, A291843.

Columns k=0..5 give A294160 (k=0), A294161 (k=1), A294162 (k=2), A294163 (k=3), A294164 (k=4), A294165 (k=5).

m = maxExponent = 13; Z[_] = 0;
Do[Z[t_] = -(((1 - l + l (1+t) Z[t]) (-((t Z[t])/(1 + l t)) - (1 - t - 2 l t^2)/(1 - l + l (1+t) Z[t]) - 2 t^2 Z'[t]))/((1+t) (1 - t - 2 l t^2))) + O[t]^m // Normal // Simplify, {m}];
gamma[t_] = ((1 + l t)(-1 + Z[t] + t Z[t]))/(Z[t]^2 (t + l t (-1 + Z[t] + t Z[t]))) + O[t]^m // Normal // Simplify;
CoefficientList[# + O[l]^m, l]& /@ Most @ CoefficientList[gamma[t], t] // Flatten (* Jean-François Alcover, Nov 17 2018 *)

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);
};
concat(apply(p->Vecrev(p), Vec(A291844_ser(12))))

y(x;t) = Sum_{n>=0} P_n(t)*x^n satisfies y = ((1+x)*z - 1) * (1 + t*x)/((1-t + t*(1+x)*z)*x*z^2), where z = A291843(x;t) and P_n(t) = Sum_{k=0..floor((2*n-1)/3)} T(n,k)*t^k for n > 0.

A294158(n) = P_n(1), A294159(n)=P_n(-1), A294160(n)=P_n(0).

A291843 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.

Original entry on oeis.org

1, 0, 1, 5, 3, 36, 33, 2, 329, 388, 72, 3655, 5101, 1545, 64, 47844, 75444, 30700, 3023, 20, 721315, 1248911, 621937, 97200, 3134, 12310199, 22964112, 13269140, 2793713, 180936, 1656, 234615096, 465344235, 301698501, 78495574, 7733807, 205620, 352, 4939227215, 10316541393, 7336995966, 2239771686, 293933437, 13977294, 140660

Offset: 0

Gheorghe Coserea, Oct 23 2017

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

			A(x;t) = 1 + x^2 + (5 + 3*t)*x^3 + (36 + 33*t + 2*t^2)*x^4 + ...
Triangle starts:
n\k  [0]        [1]        [2]        [3]       [4]      [5]     [6]
[0]  1;
[1]  0;
[2]  1;
[3]  5,         3;
[4]  36,        33,        2;
[5]  329,       388,       72;
[6]  3655,      5101,      1545,      64;
[7]  47844,     75444,     30700,     3023,     20;
[8]  721315,    1248911,   621937,    97200,    3134;
[9]  12310199,  22964112,  13269140,  2793713,  180936,  1656;
[10] 234615096, 465344235, 301698501, 78495574, 7733807, 205620, 352;
[11] ...

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. A082582, A267827, A278990, A286795, A286798, A286800.

nmax = 11; Clear[Z, Zp]; Z[_] = 0;
Do[
Zp[t_] = Z'[t] + O[t]^n // Normal;
Z[t_] = (-(1/(2L t (1+t)))) (-1 + t - 2L t + 2L^2 t^4 (1 + Zp[t]) + t^2 (1 + 2L + 2L Zp[t]) + L t^3 (3 + 2L + 2(1+L) Zp[t]) + Sqrt[4L t (1+t) (1 + L t)(-1 + t + 2L t^2 + 2(-1 + L) t^2 Zp[t]) + (-1 + t (1 + t + L (-2 + t (2 + t (3 + 2L (1+t))))) + 2L t^2 (1+t)(1 + L t) Zp[t])^2]) + O[t]^n // Normal // Simplify,
{n, nmax+1}];
CoefficientList[#, L]& /@ CoefficientList[Z[t], t] /. {} -> {0} // Flatten (* Jean-François Alcover, Oct 23 2018 *)

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;
};
concat(apply(p->if(p === Pol(0,'t), [0], Vecrev(p)), Vec(A291843_ser(12))))
\\ test: y=A291843_ser(56); 2*x^2*deriv(y,x) == (1-x-2*t*x^2)*((1+x)*y-1)/(1-t + t*(1+x)*y) - y*x/(1+t*x)

y(x;t) = Sum_{n>=0} P_n(t)*x^n satisfies 2*x^2*deriv(y,x) = (1-x-2*t*x^2)*((1+x)*y-1)/(1-t + t*(1+x)*y) - y*x/(1+t*x), with y(0;t)=1, where P_n(t) = Sum_{k=0..floor((2*n-2)/3)} T(n,k)*t^k for n > 0. (see eqn. (24) in Molinari link)
A278990(n) = P_n(0), A294166(n) = P_n(1), A082582(n) = P_n(-1) for n > 1.
A267827(n) = T(3*n+1, 2*n), n > 0. - Danny Rorabaugh, Nov 10 2017

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

Gheorghe Coserea, May 21 2017

			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]  ...

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_] = 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 *)

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))))

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.

cp's OEIS Frontend

A049464 Number of n-photon quenched skeletons.

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A291844 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.

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A291843 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula