OFFSET
0,2
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 GW approximation of the polarization function in a many-body theory of fermions with two-body interaction (see Molinari link).
LINKS
Gheorghe Coserea, Rows n = 0..123, flattened
FORMULA
y(x;t) = Sum_{n>=0} P_n(t)*x^n = 1/(1-x*s)^2, where s(x;t) = A286784(x;t) and P_n(t) = Sum_{k=0..n-1} T(n,k)*t^k for n>0.
A000108(n+1) = T(n,0), A002058(n+3) = T(n,1), A014106(n-1) = T(n,n-2), A006013(n) = P_n(1), A211789(n+1) = P_n(2).
T(n,k) = C(n-1,k)*C(2*n+2,n-k)/(n+1). - Vladimir Kruchinin, Jan 14 2022
EXAMPLE
A(x;t) = 1 + 2*x + (5 + 2*t)*x^2 + (14 + 14*t + 2*t^2)*x^3 + ...
Triangle starts:
n\k | 0 1 2 3 4 5 6 7 8
-----+-----------------------------------------------------------
0 | 1;
1 | 2;
2 | 5, 2;
3 | 14, 14, 2;
4 | 42, 72, 27, 2;
5 | 132, 330, 220, 44, 2;
6 | 429, 1430, 1430, 520, 65, 2;
7 | 1430, 6006, 8190, 4550, 1050, 90, 2;
8 | 4862, 24752, 43316, 33320, 11900, 1904, 119, 2;
9 | 16796, 100776, 217056, 217056, 108528, 27132, 3192, 152, 2;
PROG
(PARI)
A286784_ser(N, t='t) = my(x='x+O('x^N)); serreverse(Ser(x*(1-x)^2/(1+(t-1)*x)))/x;
concat(apply(p->Vecrev(p), Vec(A286785_ser(12))))
(Maxima)
T(n, k):=(binomial(n-1, k)*binomial(2*(n+1), n-k))/(n+1); /* Vladimir Kruchinin, Jan 14 2022 */
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Gheorghe Coserea, May 15 2017
STATUS
approved