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 A286785 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section. 6
 1, 2, 5, 2, 14, 14, 2, 42, 72, 27, 2, 132, 330, 220, 44, 2, 429, 1430, 1430, 520, 65, 2, 1430, 6006, 8190, 4550, 1050, 90, 2, 4862, 24752, 43316, 33320, 11900, 1904, 119, 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 (list; graph; refs; listen; history; text; internal format)
 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; A286785_ser(N, t='t) = 1/(1-x*A286784_ser(N, t))^2; 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 Cf. A000108, A286781, A286782, A286783. Sequence in context: A211175 A102469 A098886 * A340043 A257514 A089120 Adjacent sequences: A286782 A286783 A286784 * A286786 A286787 A286788 KEYWORD nonn,tabf AUTHOR Gheorghe Coserea, May 15 2017 STATUS approved

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Last modified August 8 15:25 EDT 2024. Contains 375022 sequences. (Running on oeis4.)