A286781 -id:A286781 - cp's OEIS frontend

A286786 Column 1 of A286781.

1, 9, 91, 1063, 14193, 213953, 3602891, 67168527, 1375636129, 30741614905, 745133551611, 19485223248311, 547092691302545, 16422216867929457, 524970306508659691, 17809453107819266335, 639153386976421052481, 24196474723945543441769, 963736849031626750711451, 40289411871795861783689799

Offset: 1

Gheorghe Coserea, May 16 2017

nonn

Gheorghe Coserea, Table of n, a(n) for n = 1..210

Cf. A286781.

A286781_ser(N, t='t) = {
  my(x='x+O('x^N), y0=1+O('x^N), y1=0, n=1);
  while(n++,
    y1 = (1 + x*y0 + 2*x^2*y0')*(1 - x*y0*(1-t))/(1-x*y0)^2;
    if (y1 == y0, break()); y0 = y1;);
  y0;
};
Kol(K,N=20) = {
  my(s = A286781_ser(N+K, 't+O('t^(K+1))));
  vector(N, n, polcoeff(polcoeff(s, K+n-1), K));
};
Kol(1)

A286787 Column 2 of A286781.

Original entry on oeis.org

1, 23, 416, 7344, 134613, 2620379, 54636792, 1223392968, 29409134545, 757686550455, 20870680635528, 612964613117960, 19140704352872949, 633710701752022635, 22185391759982205904, 819180275431111135536, 31826528430233802890049, 1298154677953792936043447, 55473817874169725876166480

Offset: 2

Gheorghe Coserea, May 16 2017

nonn

Gheorghe Coserea, Table of n, a(n) for n = 2..211

Cf. A286781.

A286781_ser(N, t='t) = {
  my(x='x+O('x^N), y0=1+O('x^N), y1=0, n=1);
  while(n++,
    y1 = (1 + x*y0 + 2*x^2*y0')*(1 - x*y0*(1-t))/(1-x*y0)^2;
    if (y1 == y0, break()); y0 = y1;);
  y0;
};
Kol(K,N=20) = {
  my(s = A286781_ser(N+K, 't+O('t^(K+1))));
  vector(N, n, polcoeff(polcoeff(s, K+n-1), K));
};
Kol(2)

A286788 Column 3 of A286781.

Original entry on oeis.org

1, 46, 1350, 34362, 842751, 20862684, 533394516, 14251251012, 400275122533, 11849888387786, 370032953752618, 12183464636804638, 422529226131247475, 15413172354966378040, 590459747844232859976, 23715530712277152761928, 997006212536288359365609, 43800374567467950834916070

Offset: 3

Gheorghe Coserea, May 16 2017

nonn

Gheorghe Coserea, Table of n, a(n) for n = 3..212

Cf. A286781, A286786, A286787.

A286789 Column 4 of A286781.

Original entry on oeis.org

1, 80, 3550, 125195, 4009832, 124266346, 3854188670, 121943460540, 3979525634005, 134806148044750, 4756728585598416, 175132764223880035, 6731931366383154760, 270129431174068396380, 11308903054417009540644, 493544598057755406560688, 22431785171561428611320865

Offset: 4

Gheorghe Coserea, May 16 2017

nonn

Gheorghe Coserea, Table of n, a(n) for n = 4..213

Cf. A286781, A286786, A286787.

A286790 Column 5 of A286781.

Original entry on oeis.org

1, 127, 8085, 382358, 15653598, 598415692, 22280470762, 828351390146, 31225895068199, 1204894853397543, 47870402081423775, 1965205314867907972, 83531844407610101388, 3679988322044604002256, 168093873703636990132596, 7960504783782783371445468

Offset: 5

Gheorghe Coserea, May 16 2017

nonn

Gheorghe Coserea, Table of n, a(n) for n = 5..214

Cf. A286781, A286786, A286787.

A286791 Column 6 of A286781.

Original entry on oeis.org

1, 189, 16576, 1023340, 52563182, 2445890678, 108248782988, 4693939818672, 203214066766123, 8891156436472127, 396276632116578640, 18084968846773765272, 847923332620865268252, 40927316768759272656876, 2036183873275911481501656, 104482592642320957223810448

Offset: 6

Gheorghe Coserea, May 16 2017

nonn

Gheorghe Coserea, Table of n, a(n) for n = 6..215

Cf. A286781, A286786, A286787.

A286792 Column 7 of A286781.

Original entry on oeis.org

1, 268, 31356, 2471008, 156688950, 8769288752, 457164358716, 22959079494036, 1135195898868631, 56066461542293032, 2793059847097230456, 141275485011659499936, 7287917154352619470284, 384587934464798917826784, 20802129691800205763744280, 1154767942465769017202459544

Offset: 7

Gheorghe Coserea, May 16 2017

nonn

Gheorghe Coserea, Table of n, a(n) for n = 7..216

Cf. A286781, A286786, A286787.

A286793 Column 8 of A286781.

Original entry on oeis.org

1, 366, 55650, 5493004, 423998498, 28236562692, 1719625254528, 99364697625603, 5581585635090040, 309848573192308938, 17191725056094651066, 960908576231662848594, 54403314773991195041214, 3132006013784614723699158, 183841318769012678798846646, 11022935912713058281592718756

Offset: 8

Gheorghe Coserea, May 16 2017

nonn

Gheorghe Coserea, Table of n, a(n) for n = 8..217

Cf. A286781, A286786, A286787.

A286794 Row sums of A286781.

Original entry on oeis.org

1, 3, 20, 189, 2232, 31130, 497016, 8907885, 176829104, 3849436062, 91187523000, 2335691914050, 64344487654800, 1897619527612692, 59667237154623280, 1993022006345620605, 70488571028815935072, 2631925423768158446390, 103469607286411235941944, 4272438866376100717458486

Offset: 0

Gheorghe Coserea, May 16 2017

nonn

			A(x) = 1 + 3*x + 20*x^2 + 189*x^3 + 2232*x^4 + 31130*x^5 + ...

Gheorghe Coserea, Table of n, a(n) for n = 0..301
Michael Borinsky, Renormalized asymptotic enumeration of Feynman diagrams, arXiv:1703.00840 [hep-th], 2017.
E. Z. Kuchinskii and M. V. Sadovskii, Combinatorics of Feynman diagrams for the problems with gaussian random field, arXiv:cond-mat/9706062 [cond-mat.dis-nn], 1997.
Luca G. Molinari, Hedin's equations and enumeration of Feynman's diagrams, arXiv:cond-mat/0401500 [cond-mat.str-el], 2005.

Cf. A208975, A286781.

max = 22; (* B(x) is A000699(x) *) B[_] = 0;
Do[B[x_] = x + x^2 D[B[x]^2/x, x] + O[x]^max // Normal, max];
A[x_] = (1 - x/B[x])/x + O[x]^max;
Drop[CoefficientList[A[x], x], -2] (* Jean-François Alcover, Oct 25 2018 *)

A286781_ser(N, t='t) = {
  my(x='x+O('x^N), y0=1+O('x^N), y1=0, n=1);
  while(n++,
    y1 = (1 + x*y0 + 2*x^2*y0')*(1 - x*y0*(1-t))/(1-x*y0)^2;
    if (y1 == y0, break()); y0 = y1;);
  y0;
};
Vec(A286781_ser(20,1))

A000699_seq(N) = {
  my(a = vector(N)); a[1] = 1;
  for (n=2, N, a[n] = sum(k=1, n-1, (2*k-1)*a[k]*a[n-k])); a;
};
A286794_seq(N) = Vec((1-1/Ser(A000699_seq(N+1)))/x);
A286794_seq(20)

a(n) = Sum_{k=0..n} A286781(n,k).
A(x) = (1-x/A000699(x))/x, A208975(x) = 1 + x*A(-x).
a(n) ~ 4*exp(-1)/sqrt(Pi) * n^(3/2) * 2^n * n! * (1 - 3/(8*n) - 215/(128*n^2) + O(1/n^3)). (see Borinsky link) - Gheorghe Coserea, Oct 23 2017

A286782 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.

Original entry on oeis.org

1, 1, 6, 3, 50, 45, 5, 518, 637, 161, 7, 6354, 9567, 3744, 414, 9, 89782, 156123, 80784, 14850, 880, 11, 1435330, 2781389, 1749969, 446706, 46150, 1651, 13, 25625910, 54043365, 39305685, 12641265, 1877925, 121275, 2835, 15, 505785122, 1141864959, 928825464, 354665628, 68167144, 6500086, 281792, 4556, 17, 10944711398, 26137086451, 23244466392, 10134495804, 2361060574, 297418362, 19443460, 595764, 6954, 19

Offset: 0

Gheorghe Coserea, May 14 2017

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 vertex function in a many-body theory of fermions with two-body interaction (see Molinari link).

			A(x;t) = 1 + x + (6 + 3*t)*x^2 + (50 + 45*t + 5*t^2)*x^3 + ...
Triangle starts:
n\k  [0]       [1]       [2]       [3]       [4]      [5]     [6]    [7]
[0]  1;
[1]  1;
[2]  6,        3;
[3]  50,       45,       5;
[4]  518,      637,      161,      7;
[5]  6354,     9567,     3744,     414,      9;
[6]  89782,    156123,   80784,    14850,    880,     11;
[7]  1435330,  2781389,  1749969,  446706,   46150,   1651,   13;
[8]  25625910, 54043365, 39305685, 12641265, 1877925, 121275, 2835,  15;
[9]  ...

Gheorghe Coserea, Rows n=0..123, flattened
Luca G. Molinari, Hedin's equations and enumeration of Feynman's diagrams, arXiv:cond-mat/0401500 [cond-mat.str-el], 2005.

max = 10; y0[x_, t_] = 1; y1[x_, t_] = 0; For[n = 1, n <= max, n++, y1[x_, t_] = (1 + x*y0[x, t] + 2*x^2*D[y0[x, t], x])*(1 - x*y0[x, t]*(1 - t))/(1 - x*y0[x, t])^2 + O[x]^n // Normal; y0[x_, t_] = y1[x, t]];
row[n_] := (2n+1) CoefficientList[Coefficient[y0[x, t], x, n], t];
T[0, 0] = 1; T[n_, k_] := row[n-1][[k+1]];
Table[T[n, k], {n, 0, max}, {k, 0, If[n == 0, 0, n-1]}] // Flatten (* Jean-François Alcover, May 19 2017, adapted from PARI *)

A286781_ser(N,t='t) = {
  my(x='x+O('x^N), y0=1+O('x^N), y1=0, n=1);
  while(n++,
    y1 = (1 + x*y0 + 2*x^2*y0')*(1 - x*y0*(1-t))/(1-x*y0)^2;
    if (y1 == y0, break()); y0 = y1;);
  y0;
};
A286782_ser(N,t='t) = my(s=A286781_ser(N,t)); 1 + x*s + 2*x^2 * deriv(s,'x);
concat(apply(p->Vecrev(p), Vec(A286782_ser(10))))

A(x;t) = Sum_{n>=0} P_n(t)*x^n = 1 + x*s + 2*x^2 * deriv(s,x), where s(x;t) = A286781(x;t) and P_n(t) = Sum_{k=0..n-1} T(n,k)*t^k for n>0.

T(n+1,k) = (2*n+1)*A286781(n,k), A005416(n)=T(n,0), A088218(n)=P_n(-1).

cp's OEIS Frontend

A286786 Column 1 of A286781.

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A286787 Column 2 of A286781.

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A286788 Column 3 of A286781.

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A286789 Column 4 of A286781.

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A286790 Column 5 of A286781.

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A286791 Column 6 of A286781.

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A286792 Column 7 of A286781.

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A286793 Column 8 of A286781.

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A286794 Row sums of A286781.

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A286782 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.

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