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

1, 1, 1, 2, 4, 1, 5, 15, 9, 1, 14, 56, 56, 16, 1, 42, 210, 300, 150, 25, 1, 132, 792, 1485, 1100, 330, 36, 1, 429, 3003, 7007, 7007, 3185, 637, 49, 1, 1430, 11440, 32032, 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

Offset

0,4

Comments

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).

Links

Gheorghe Coserea, Rows n=0..122, flattened

Luca G. Molinari, Hedin's equations and enumeration of Feynman's diagrams, arXiv:cond-mat/0401500 [cond-mat.str-el], 2005.

Formula

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.

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).

T(n,m) = C(2*n,n+m)*C(n+1,m)/(n+1). - Vladimir Kruchinin, Sep 23 2018

Example

A(x;t) = 1 + (1 + t)*x + (2 + 4*t + t^2)*x^2 + (5 + 15*t + 9*t^2 + t^3)*x^3 + ...
Triangle starts:
n\k  [0]   [1]    [2]     [3]     [4]     [5]    [6]    [7]   [8] [9]
[0]  1;
[1]  1,    1;
[2]  2,    4,     1;
[3]  5,    15,    9,      1;
[4]  14,   56,    56,     16,     1;
[5]  42,   210,   300,    150,    25,     1;
[6]  132,  792,   1485,   1100,   330,    36,    1;
[7]  429,  3003,  7007,   7007,   3185,   637,   49,    1;
[8]  1430, 11440, 32032,  40768,  25480,  7840,  1120,  64,   1;
[9]  4862, 43758, 143208, 222768, 179928, 77112, 17136, 1836, 81, 1;
[10] ...

Mathematica

Flatten@Table[Binomial[2 n, n + m] Binomial[n + 1, m] / (n + 1), {n, 0, 10}, {m, 0, n}] (* Vincenzo Librandi, Sep 23 2018 *)

Prog

(PARI)
A286784_ser(N, t='t) = my(x='x+O('x^N)); serreverse(Ser(x*(1-x)^2/(1+(t-1)*x)))/x;
concat(apply(p->Vecrev(p), Vec(A286784_ser(12))))
\\ test: y=A286784_ser(50); y*(1-x*y)^2 == 1 + ('t-1)*x*y
(Maxima)
T(n, m):=(binomial(2*n, n+m)*binomial(n+1, m))/(n+1); /* Vladimir Kruchinin, Sep 23 2018 */
(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

Crossrefs

Cf. A286781, A286782, A286783.

Sequence in context: A124037 A126126 A090285 * A047908 A125847 A078886

Adjacent sequences: A286781 A286782 A286783 * A286785 A286786 A286787

Keyword

nonn,tabl

Author

Gheorghe Coserea, May 14 2017

Status

approved