A286800 -id:A286800 - cp's OEIS frontend

A287029 Row sums of A286800.

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

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

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A287029 Row sums of A286800.

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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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Mathematica

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula