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

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

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