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A286784 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section. 7
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 (list; table; graph; refs; listen; history; text; internal format)
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

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Last modified July 22 05:47 EDT 2019. Contains 325213 sequences. (Running on oeis4.)