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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, TRows 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).

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;

[10] ...

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))))

CROSSREFS

Cf. A286781, A286782, A286783.

Sequence in context: A211175 A102469 A098886 * A257514 A089120 A286452

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 July 21 06:55 EDT 2019. Contains 325192 sequences. (Running on oeis4.)