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A286785 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.

%I A286785 #32 Jan 26 2022 08:53:41
%S A286785 1,2,5,2,14,14,2,42,72,27,2,132,330,220,44,2,429,1430,1430,520,65,2,
%T A286785 1430,6006,8190,4550,1050,90,2,4862,24752,43316,33320,11900,1904,119,
%U A286785 2,16796,100776,217056,217056,108528,27132,3192,152,2,58786,406980,1046520,1302336,854658,301644,55860,5040,189,2,208012,1634380,4903140,7354710,6056820,2826516,743820,106260,7590,230,2
%N A286785 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.
%C A286785 Row n>0 contains n terms.
%C A286785 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).
%H A286785 Gheorghe Coserea, <a href="/A286785/b286785.txt">Rows n = 0..123, flattened</a>
%F A286785 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.
%F A286785 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).
%F A286785 T(n,k) = C(n-1,k)*C(2*n+2,n-k)/(n+1). - _Vladimir Kruchinin_, Jan 14 2022
%e A286785 A(x;t) = 1 + 2*x + (5 + 2*t)*x^2 + (14 + 14*t + 2*t^2)*x^3 + ...
%e A286785 Triangle starts:
%e A286785    n\k |     0       1       2       3       4      5     6    7  8
%e A286785   -----+-----------------------------------------------------------
%e A286785    0   |     1;
%e A286785    1   |     2;
%e A286785    2   |     5,      2;
%e A286785    3   |    14,     14,      2;
%e A286785    4   |    42,     72,     27,      2;
%e A286785    5   |   132,    330,    220,     44,      2;
%e A286785    6   |   429,   1430,   1430,    520,     65,     2;
%e A286785    7   |  1430,   6006,   8190,   4550,   1050,    90,    2;
%e A286785    8   |  4862,  24752,  43316,  33320,  11900,  1904,  119,   2;
%e A286785    9   | 16796, 100776, 217056, 217056, 108528, 27132, 3192, 152, 2;
%o A286785 (PARI)
%o A286785 A286784_ser(N,t='t) = my(x='x+O('x^N)); serreverse(Ser(x*(1-x)^2/(1+(t-1)*x)))/x;
%o A286785 A286785_ser(N,t='t) = 1/(1-x*A286784_ser(N,t))^2;
%o A286785 concat(apply(p->Vecrev(p), Vec(A286785_ser(12))))
%o A286785 (Maxima)
%o A286785 T(n,k):=(binomial(n-1,k)*binomial(2*(n+1),n-k))/(n+1); /* _Vladimir Kruchinin_, Jan 14 2022 */
%Y A286785 Cf. A000108, A286781, A286782, A286783.
%K A286785 nonn,tabf
%O A286785 0,2
%A A286785 _Gheorghe Coserea_, May 15 2017