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A286782 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section. 17
1, 1, 6, 3, 50, 45, 5, 518, 637, 161, 7, 6354, 9567, 3744, 414, 9, 89782, 156123, 80784, 14850, 880, 11, 1435330, 2781389, 1749969, 446706, 46150, 1651, 13, 25625910, 54043365, 39305685, 12641265, 1877925, 121275, 2835, 15, 505785122, 1141864959, 928825464, 354665628, 68167144, 6500086, 281792, 4556, 17, 10944711398, 26137086451, 23244466392, 10134495804, 2361060574, 297418362, 19443460, 595764, 6954, 19 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
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 vertex function in a many-body theory of fermions with two-body interaction (see Molinari link).
LINKS
Gheorghe Coserea, Rows n=0..123, flattened
Luca G. Molinari, Hedin's equations and enumeration of Feynman's diagrams, arXiv:cond-mat/0401500 [cond-mat.str-el], 2005.
FORMULA
A(x;t) = Sum_{n>=0} P_n(t)*x^n = 1 + x*s + 2*x^2 * deriv(s,x), where s(x;t) = A286781(x;t) and P_n(t) = Sum_{k=0..n-1} T(n,k)*t^k for n>0.
T(n+1,k) = (2*n+1)*A286781(n,k), A005416(n)=T(n,0), A088218(n)=P_n(-1).
EXAMPLE
A(x;t) = 1 + x + (6 + 3*t)*x^2 + (50 + 45*t + 5*t^2)*x^3 + ...
Triangle starts:
n\k [0] [1] [2] [3] [4] [5] [6] [7]
[0] 1;
[1] 1;
[2] 6, 3;
[3] 50, 45, 5;
[4] 518, 637, 161, 7;
[5] 6354, 9567, 3744, 414, 9;
[6] 89782, 156123, 80784, 14850, 880, 11;
[7] 1435330, 2781389, 1749969, 446706, 46150, 1651, 13;
[8] 25625910, 54043365, 39305685, 12641265, 1877925, 121275, 2835, 15;
[9] ...
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_] := (2n+1) CoefficientList[Coefficient[y0[x, t], x, n], t];
T[0, 0] = 1; T[n_, k_] := row[n-1][[k+1]];
Table[T[n, k], {n, 0, max}, {k, 0, If[n == 0, 0, n-1]}] // Flatten (* Jean-François Alcover, May 19 2017, adapted from PARI *)
PROG
(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;
};
A286782_ser(N, t='t) = my(s=A286781_ser(N, t)); 1 + x*s + 2*x^2 * deriv(s, 'x);
concat(apply(p->Vecrev(p), Vec(A286782_ser(10))))
CROSSREFS
Sequence in context: A229130 A349492 A088390 * A054380 A348171 A348142
KEYWORD
nonn,tabf
AUTHOR
Gheorghe Coserea, May 14 2017
STATUS
approved

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Last modified June 17 18:36 EDT 2024. Contains 373463 sequences. (Running on oeis4.)