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A286784 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.

%I A286784 #34 Sep 08 2022 08:46:19
%S A286784 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 A286784 1100,330,36,1,429,3003,7007,7007,3185,637,49,1,1430,11440,32032,
%U A286784 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 A286784 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.
%C A286784 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 A286784 Gheorghe Coserea, <a href="/A286784/b286784.txt">Rows n=0..122, flattened</a>
%H A286784 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 A286784 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 A286784 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 A286784 T(n,m) = C(2*n,n+m)*C(n+1,m)/(n+1). - _Vladimir Kruchinin_, Sep 23 2018
%e A286784 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 A286784 Triangle starts:
%e A286784 n\k  [0]   [1]    [2]     [3]     [4]     [5]    [6]    [7]   [8] [9]
%e A286784 [0]  1;
%e A286784 [1]  1,    1;
%e A286784 [2]  2,    4,     1;
%e A286784 [3]  5,    15,    9,      1;
%e A286784 [4]  14,   56,    56,     16,     1;
%e A286784 [5]  42,   210,   300,    150,    25,     1;
%e A286784 [6]  132,  792,   1485,   1100,   330,    36,    1;
%e A286784 [7]  429,  3003,  7007,   7007,   3185,   637,   49,    1;
%e A286784 [8]  1430, 11440, 32032,  40768,  25480,  7840,  1120,  64,   1;
%e A286784 [9]  4862, 43758, 143208, 222768, 179928, 77112, 17136, 1836, 81, 1;
%e A286784 [10] ...
%t A286784 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 A286784 (PARI)
%o A286784 A286784_ser(N,t='t) = my(x='x+O('x^N)); serreverse(Ser(x*(1-x)^2/(1+(t-1)*x)))/x;
%o A286784 concat(apply(p->Vecrev(p), Vec(A286784_ser(12))))
%o A286784 \\ test: y=A286784_ser(50); y*(1-x*y)^2 == 1 + ('t-1)*x*y
%o A286784 (Maxima)
%o A286784 T(n,m):=(binomial(2*n,n+m)*binomial(n+1,m))/(n+1); /* _Vladimir Kruchinin_, Sep 23 2018 */
%o A286784 (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 A286784 Cf. A286781, A286782, A286783.
%K A286784 nonn,tabl
%O A286784 0,4
%A A286784 _Gheorghe Coserea_, May 14 2017