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%I #34 Sep 08 2022 08:46:19
%S 1,1,1,2,4,1,5,15,9,1,14,56,56,16,1,42,210,300,150,25,1,132,792,1485,
%T 1100,330,36,1,429,3003,7007,7007,3185,637,49,1,1430,11440,32032,
%U 40768,25480,7840,1120,64,1,4862,43758,143208,222768,179928,77112,17136,1836,81,1,16796,167960,629850,1162800,1162800,651168,203490,34200,2850,100,1
%N Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.
%C 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 self-energy function in a many-body theory of fermions with two-body interaction (see Molinari link).
%H Gheorghe Coserea, <a href="/A286784/b286784.txt">Rows n=0..122, flattened</a>
%H Luca G. Molinari, <a href="https://arxiv.org/abs/cond-mat/0401500">Hedin's equations and enumeration of Feynman's diagrams</a>, arXiv:cond-mat/0401500 [cond-mat.str-el], 2005.
%F y(x;t) = Sum_{n>=0} P_n(t)*x^n satisfies y*(1-x*y)^2 = 1 + (t-1)*x*y, where P_n(t) = Sum_{k=0..n} T(n,k)*t^k.
%F A000108(n) = T(n,0), A001791(n) = T(n,1), A002055(n+3) = T(n,2), A000290(n) = T(n,n-1), A006013(n) = P_n(1), A003169(n+1) = P_n(2).
%F T(n,m) = C(2*n,n+m)*C(n+1,m)/(n+1). - _Vladimir Kruchinin_, Sep 23 2018
%e A(x;t) = 1 + (1 + t)*x + (2 + 4*t + t^2)*x^2 + (5 + 15*t + 9*t^2 + t^3)*x^3 + ...
%e Triangle starts:
%e n\k [0] [1] [2] [3] [4] [5] [6] [7] [8] [9]
%e [0] 1;
%e [1] 1, 1;
%e [2] 2, 4, 1;
%e [3] 5, 15, 9, 1;
%e [4] 14, 56, 56, 16, 1;
%e [5] 42, 210, 300, 150, 25, 1;
%e [6] 132, 792, 1485, 1100, 330, 36, 1;
%e [7] 429, 3003, 7007, 7007, 3185, 637, 49, 1;
%e [8] 1430, 11440, 32032, 40768, 25480, 7840, 1120, 64, 1;
%e [9] 4862, 43758, 143208, 222768, 179928, 77112, 17136, 1836, 81, 1;
%e [10] ...
%t Flatten@Table[Binomial[2 n, n + m] Binomial[n + 1, m] / (n + 1), {n, 0, 10}, {m, 0, n}] (* _Vincenzo Librandi_, Sep 23 2018 *)
%o (PARI)
%o A286784_ser(N,t='t) = my(x='x+O('x^N)); serreverse(Ser(x*(1-x)^2/(1+(t-1)*x)))/x;
%o concat(apply(p->Vecrev(p), Vec(A286784_ser(12))))
%o \\ test: y=A286784_ser(50); y*(1-x*y)^2 == 1 + ('t-1)*x*y
%o (Maxima)
%o T(n,m):=(binomial(2*n,n+m)*binomial(n+1,m))/(n+1); /* _Vladimir Kruchinin_, Sep 23 2018 */
%o (Magma) /* As triangle */ [[(Binomial(2*n, n+m)*Binomial(n+1, m))/(n+1): m in [0..n]]: n in [0.. 15]]; // _Vincenzo Librandi_, Sep 23 2018
%Y Cf. A286781, A286782, A286783.
%K nonn,tabl
%O 0,4
%A _Gheorghe Coserea_, May 14 2017
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