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, 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.
-
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)
Original entry on oeis.org
3, 45, 637, 9567, 156123, 2781389, 54043365, 1141864959, 26137086451, 645573913005, 17138071687053, 487130581207775, 14771502665168715, 476244289169954253, 16274079501768450421, 587711952558035789055, 22370368544174736836835, 895269564785985107345453, 37585737112233443277746589
Offset: 2
-
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;
};
A286782_ser(N, t='t) = my(s=A286781_ser(N, t)); 1 + x*s + 2*x^2 * deriv(s,'x);
Kol(K, N=20) = {
my(v = A286782_ser(N+K+1, 't+O('t^(K+1))));
vector(N, n, polcoeff(polcoeff(v, K+n), K));
};
Kol(1)
Original entry on oeis.org
5, 161, 3744, 80784, 1749969, 39305685, 928825464, 23244466392, 617591825445, 17426790660465, 521767015888200, 16550044554184920, 555080426233315521, 19645031754312701685, 732117928079412794832, 28671309640088889743760, 1177581551918650706931813, 50628032440197924505694433
Offset: 3
Original entry on oeis.org
7, 414, 14850, 446706, 12641265, 354665628, 10134495804, 299276271252, 9206327818259, 296247209694650, 9990889751320686, 353320474467334502, 13098406010068671725, 508634687713890475320, 20666091174548150099160, 877474636354254652191336, 38883242288915246015258751
Offset: 4
Original entry on oeis.org
9, 880, 46150, 1877925, 68167144, 2361060574, 80937962070, 2804699592420, 99488140850125, 3639765997208250, 137945128982354064, 5429115690940281085, 222153735090644107080, 9454530091092393873300, 418429413013429353003828, 19248239324252460855866832
Offset: 5
Original entry on oeis.org
11, 1651, 121275, 6500086, 297418362, 12566729532, 512450827526, 20708784753650, 843099166841373, 34941950748528747, 1483982464524137025, 64851775390640963076, 2923614554266353548580, 136159567915650348083472, 6555661074441842615171244, 326380696135094118229264188
Offset: 6
Original entry on oeis.org
13, 2835, 281792, 19443460, 1103826822, 56255485594, 2706219574700, 126736375104144, 5893207936217567, 275625849530635937, 13077128859847095120, 632973909637081784520, 31373163306972014925324, 1596165353981611633618164, 83483538804312370741567896
Offset: 7
Original entry on oeis.org
15, 4556, 595764, 51891168, 3603845850, 219232218800, 12343437685332, 665813305327044, 35191072864927561, 1850193230895670056, 97757094648403065960, 5227192945431401497632, 284228769019752159341076, 15768105313056755630898144, 894491576747408847841004040
Offset: 8
Original entry on oeis.org
17, 6954, 1168650, 126339092, 10599962450, 762387192684, 49869132381312, 3080305626393693, 184192325957971320, 10844700061730812830, 636093827075502089442, 37475434473034851095166, 2230535905733638996689774, 134676258592738433119063794, 8272859344605570545948099070
Offset: 9
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
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] ...
For vertex and polarization functions see
A286782 and
A286783. For GWA of the self-energy and polarization functions see
A286784 and
A286785.
-
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 *)
-
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))
Showing 1-10 of 16 results.
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