A258220 -id:A258220 - cp's OEIS frontend

A258221 Row sums of A258220.

1, 2, 11, 90, 952, 12203, 182677, 3118314, 59688447, 1265193199, 29408221404, 743677646836, 20325564686926, 597051775012306, 18758388926380409, 627712133246362442, 22288938527631882996, 837033514431748421053, 33146037056721682537319, 1380365444443138768970878

Offset: 0

Alois P. Heinz, May 23 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..250

Cf. A258220, A258224.

b:= proc(x, y, t, k) option remember; `if`(y>x or y<0, 0,
      `if`(x=0, 1, b(x-1, y-1, false, k)*`if`(t, (x+k*y)/y, 1)
                 + b(x-1, y+1, true, k)  ))
    end:
A:= (n, k)-> b(2*n, 0, false, k):
T:= proc(n,k) option remember;
       add(A(n, i)*(-1)^(k-i)*binomial(k, i), i=0..k)/k!
    end:
a:= proc(n) option remember; add(T(n,k), k=0..n) end:
seq(a(n), n=0..25);

b[x_, y_, t_, k_] := b[x, y, t, k] = If[y > x || y < 0, 0,
     If[x == 0, 1, b[x - 1, y - 1, False, k]*If[t, (x + k*y)/y, 1]
                 + b[x - 1, y + 1, True, k]]];
A[n_, k_] := b[2*n, 0, False, k];
T[n_, k_] := Sum[A[n, i]*(-1)^(k - i)*Binomial[k, i], {i, 0, k}]/k!;
a[n_] := Sum[T[n, k], {k, 0, n}];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Apr 28 2022, after Alois P. Heinz *)

a(n) = Sum_{k=0..n} A258220(n,k).

A292692 a(n) = A258220(2n,n).

Original entry on oeis.org

1, 6, 217, 13997, 1283817, 151778430, 21902350480, 3726598826676, 729894290301789, 161683308639785288, 39960246577221138239, 10900687898284398341055, 3253179971490906621196740, 1054372896576358319473872858, 368810384314446516250937494104

Offset: 0

Alois P. Heinz, Sep 20 2017

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..290 (terms 0..200 from Alois P. Heinz)

Cf. A258220.

a(n) ~ c * d^n * n! / sqrt(n), where d = 25.81959672662517201... and c = 0.28310309251343879... - Vaclav Kotesovec, Dec 01 2017

Conjecture: d = (51*sqrt(17) - 107)/4. - Vaclav Kotesovec, Mar 17 2024

A005411 Number of non-vanishing Feynman diagrams of order 2n for the electron or the photon propagators in quantum electrodynamics.

Original entry on oeis.org

1, 1, 4, 25, 208, 2146, 26368, 375733, 6092032, 110769550, 2232792064, 49426061818, 1192151302144, 31123028996164, 874428204384256, 26308967412122125, 843984969276915712, 28757604639850111894, 1037239628039528906752, 39481325230750749160462

Offset: 0

N. J. A. Sloane

Cvitanovic et al. paper relates this sequence to A000698 and A005413. - Robert Munafo, Jan 24 2010
(x + 4x^2 + 25x^3 + 208x^4 + ...) = (x + 2x^2 + 7x^3 + 38x^4 + ...) * 1/(1 + x + 2x^2 + 7x^3 + 38x^4 + ...); where A094664 = (1, 1, 2, 7, 38, 286, ...). - Gary W. Adamson, Nov 16 2011.
The Martin and Kearney article has S(2,-4,1) = [1,1,4,25,...] where u_1 = u_2 = 1, u_3 = 4, u_4 =25, etc. This is almost the same as this sequence. - Michael Somos, Feb 27 2014
From Robert Coquereaux, Sep 05 2014: (Start)
Evaluation of quantum electrodynamics functional integrals in dimension 0 become usual Lebesgue integrals, their Taylor expansion around g=0 at order n give the number of Feynman diagrams.
These are graphs with two kinds of edges: a (non-oriented), f (oriented), and only one kind of vertex: aff.
Electron propagator: all the diagrams with two external edges of type f.
Photon propagator: all the diagrams with two external edges of type a.
The exponent n of g^n gives the number of vertices.
Diagrams containing loops of type f with an odd number of vertices are set to 0 (vanishing diagrams).
The coefficients of the series S(g)=Sum a(n) g^(2n) give the number of non-vanishing Feynman diagrams for the electron (or the photon) propagator.
S(g) is obtained as < 1/(1-g^2 a^2) > for the measure (E^(-(a^2/2)))/sqrt[1-g^2 a^2]da, assuming g^2 < 0, hence a formula for S(g) in terms of modified Bessel functions (setting x=g^2 gives the G.f. below).
(End)
Sum over all Dyck paths of semilength n of products over all peaks p of x_p/y_p, where x_p and y_p are the coordinates of peak p. a(3) = 3/3 +2/2*5/1 +1/1*4/2 +2/2*4/2 +1/1*3/1*5/1 = 25. - Alois P. Heinz, May 21 2015
From Sasha Kolpakov, Dec 11 2017: (Start)
Number of free index 2n subgroups in the free product Z_2*Z_2*Z_2.
Number of oriented rooted pavings (after Arques & Koch, Spehner, Lienhardt) with 2n darts.
(End)

			G.f. = 1 + x + 4*x^2 + 25*x^3 + 208*x^4 + 2146*x^5 + 26368*x^6 + 375733*x^7 + ... [Deleted g.f. restored by _N. J. A. Sloane_, Jan 30 2016]

C. Itzykson and J.-B. Zuber, Quantum Field Theory, McGraw-Hill, 1980, pages 466-467.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..400
Michael Borinsky, Renormalized asymptotic enumeration of Feynman diagrams, arXiv:1703.00840 [hep-th], 2017.
Rémi Bottinelli, Laura Ciobanu, and Alexander Kolpakov, Three-dimensional maps and subgroup growth, manuscripta math. (2021).
L. Ciobanu and A. Kolpakov, Three-dimensional maps and subgroup growth, arXiv:1712.01418 [math.GR], 2017.
P. Cvitanovic, B. Lautrup and R. B. Pearson, The number and weights of Feynman diagrams, Phys. Rev. D18, (1978), 1939-1949. DOI:10.1103/PhysRevD.18.1939
R. J. Martin and M. J. Kearney, An exactly solvable self-convolutive recurrence, arXiv:1103.4936 [math.CO], 2011.
R. J. Martin and M. J. Kearney, An exactly solvable self-convolutive recurrence, Aequat. Math., 80 (2010), 291-318. see p. 294.
A. N. Stokes, Continued fraction solutions of the Riccati equation, Bull. Austral. Math. Soc. Vol. 25 (1982), 207-214.
Wikipedia, Feynman diagram

Cf. A000108, A000698, A005412, A005413, A094664.

Column k=0 of A258219, A258220.

b:= proc(x, y, t) option remember; `if`(y>x or y<0, 0,
      `if`(x=0, 1, b(x-1, y-1, false)*`if`(t, x/y, 1) +
                   b(x-1, y+1, true)  ))
    end:
a:= n-> b(2*n, 0, false):
seq(a(n), n=0..20);  # Alois P. Heinz, May 21 2015

a[n_] := Module[{A}, A[1] = 1; A[k_] := A[k] = (2*k-4)*A[k-1]+Sum[A[j]*A[k-j], {j, 1, k-1}]; A[n]]; Table[a[n], {n, 2, 20}] (* Jean-François Alcover, Feb 27 2014, after Michael Somos *)
a[ n_] := Module[{m = n + 1, u}, If[ n < 2, Boole[n >= 0], u = Range[m]; Do[ u[[k]] = (2 k - 4) u[[k - 1]] + Sum[ u[[j]] u[[k - j]], {j, k - 1}], {k, 2, m}]; u[[m]]]]; (* Michael Somos, Feb 27 2014 *)
a[n_]:=SeriesCoefficient[(1-BesselK[1,-(1/(4 g^2))]/BesselK[0,-(1/(4 g^2))])/(2 g^2),{g,0,2*n}]; (* Robert Coquereaux, Sep 05 2014 *)

{a(n) = my(A); if( n<1, n==0, n++; A = vector(n); A[1] = 1; for( k=2, n, A[k] = (2 * k - 4) * A[k-1] + sum( j=1, k-1, A[j] * A[k-j])); A[n])}; /* Michael Somos, Jul 24 2011 */

From Peter Bala, Mar 07 2011: (Start)
Given the o.g.f. A(x), the function F(x) := A(x^2) satisfies the differential equation F(x) = 1 + x^3*d/dx(F(x)) + x^2*F(x)^2 (equation 3.53, P. Cvitanovic et al.).
Conjectural o.g.f. A(x) as a continued fraction:
1 + x/(1 - 4*x - 3^2*x^2/(1 - 8*x - 5^2*x^2/(1 - 12*x - 7^2*x^2/(1 - 16*x - ...)))).
Asymptotics: a(n) ~ 1/Pi*2^(n+1)*n!*(1 - 1/(2*n) - 3/(8*n^2)). (End)
Given u(1) = 1, u(n) = (2*n - 4) * u(n-1) + Sum_{k=1..n-1} u(k) * u(n-k) when n>1, then a(n) = u(n+1) if n>0. - Michael Somos, Jul 24 2011
G.f.: 1/Q(0) where Q(k) = 1 - x*(2*k+1)/(1 - x*(2*k+3)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 19 2013
G.f.: 1/x^2 - 1/x - Q(0)/x^2, where Q(k) = 1 - x*(2*k+1)/(1 - x*(2*k+1)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, May 20 2013
G.f.: 1/x^2 - 1/x - G(0)/(2*x^2), where G(k) = 1 + 1/(1 - 2*x*(2*k+1)/(2*x*(2*k+1) - 1 + 2*x*(2*k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 29 2013
G.f.: W(0)/x - 1/x, where W(k) = 1 - x*(2*k+1)/( x*(2*k+1) - 1/(1 - x*(2*k+3)/( x*(2*k+3) - 1/W(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Aug 26 2013
G.f.: G(0)/x -1/x, where G(k) = 1 - x*(2*k+1)/(x - 1/G(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Jan 21 2014
G.f.: 1/(2*x) - BesselK(1,-1/(4*x))/(2*x*BesselK(0,-1/(4*x))) where BesselK[p,z] denotes the modified Bessel function of the second kind (order p, argument z). This is a small improvement of a result obtained in 1980 book "Quantum Field Theory". - Robert Coquereaux, Sep 05 2014
Asymptotics: a(n) ~ 2*(2/Pi)^(1/2)*(2/e)^n*n^(n+1/2), cf. Ciobanu and Kolpakov in Links. - Sasha Kolpakov, Dec 11 2017
From Peter Bala, Jun 27 2022: (Start)
O.g.f. as a continued fraction of Stieltjes type: A(x) = 1/(1 - x/(1 - 3*x/(1 - 3*x/(1 - 5*x/(1 - 5*x/(1 - 7*x/(1 - 7*x/(1 - ...)))))))) follows by applying the result of Stokes to the Riccati differential equation 2*x^2*A'(x) = -1 + A(x) - x*A^2(x).
The even part of the continued fraction gives A(x) = 1/(1 - x - 3*x^2/(1 - 6*x - 15*x^2/(1 - 10*x - 35*x^2/(1 - 14*x - 63*x^2/(1 - 18*x - ... - (4*n^2-1)*x^2/(1 - (4*n+2)*x -...)))))), a continued fraction of Jacobi type (a J-fraction). (End)

Name corrected by Charles R Greathouse IV, Jan 24 2014
Name clarified by Robert Coquereaux, Sep 05 2014
a(0)=1 prepended, programs and formulas edited by Alois P. Heinz, Jun 22 2015

A258219 A(n,k) is the sum over all Dyck paths of semilength n of products over all peaks p of (x_p+k*y_p)/y_p, where x_p and y_p are the coordinates of peak p; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 4, 1, 3, 10, 25, 1, 4, 18, 74, 208, 1, 5, 28, 153, 706, 2146, 1, 6, 40, 268, 1638, 8162, 26368, 1, 7, 54, 425, 3172, 20898, 110410, 375733, 1, 8, 70, 630, 5500, 44164, 307908, 1708394, 6092032, 1, 9, 88, 889, 8838, 82850, 702844, 5134293, 29752066, 110769550

Offset: 0

Alois P. Heinz, May 23 2015

A Dyck path of semilength n is a (x,y)-lattice path from (0,0) to (2n,0) that does not go below the x-axis and consists of steps U=(1,1) and D=(1,-1). A peak of a Dyck path is any lattice point visited between two consecutive steps UD.

Conjecture: the g.f. G(k,x) for the k-th column satisfies the Riccati differential equation 2*x^2*d/dx(G(k,x)) + 1 + (k*x - 1)*G(k,x) + x*G^2(k,x) = 0 and hence, by Stokes 1982, has the continued fraction representation G(k,x) = 1/(1 - (k+1)*x/(1 - 3*x/(1 - (k+3)*x/(1 - 5*x/(1 - (k+5)*x/(1 - 7*x/(1 - ...))))))) of Stieltjes type. - Peter Bala, Jul 28 2022

			Square array A(n,k) begins:
     1,    1,     1,     1,     1,      1, ...
     1,    2,     3,     4,     5,      6, ...
     4,   10,    18,    28,    40,     54, ...
    25,   74,   153,   268,   425,    630, ...
   208,  706,  1638,  3172,  5500,   8838, ...
  2146, 8162, 20898, 44164, 82850, 143046, ...
  ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
A. N. Stokes, Continued fraction solutions of the Riccati equation, Bull. Austral. Math. Soc. Vol. 25 (1982), 207-214.
Wikipedia, Feynman diagram
Wikipedia, Lattice path

Columns k=0-2 give: A005411 (for n>0), A000698(n+1), A005412(n+1).
Rows n=0-2 give: A000012, A000027(k+1), A028552(k+1).
Main diagonal gives A292693.
Cf. A258220, A258222.

b:= proc(x, y, t, k) option remember; `if`(y>x or y<0, 0,
      `if`(x=0, 1, b(x-1, y-1, false, k)*`if`(t, (x+k*y)/y, 1)
                 + b(x-1, y+1, true, k)  ))
    end:
A:= (n,k)-> b(2*n, 0, false, k):
seq(seq(A(n, d-n), n=0..d), d=0..12);

b[x_, y_, t_, k_] := b[x, y, t, k] = If[y>x || y<0, 0, If[x==0, 1, b[x-1, y -1, False, k]*If[t, (x+k*y)/y, 1] + b[x-1, y+1, True, k]]]; A[n_, k_] := b[2*n, 0, False, k]; Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Jan 09 2016, after Alois P. Heinz *)

A(n,k) = Sum_{i=0..min(n,k)} C(k,i) * i! * A258220(n,i).

A258223 T(n,k) = 1/k! * Sum_{i=0..k} (-1)^(k-i) C(k,i) A258222(n,i); triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 1, 1, 2, 8, 3, 5, 69, 77, 15, 14, 692, 1749, 890, 105, 42, 8120, 41998, 41909, 12039, 945, 132, 110278, 1114808, 1944225, 1018865, 186594, 10395, 429, 1707965, 33058519, 94833341, 80595226, 25798856, 3260067, 135135, 1430, 29750636, 1093994697, 4979407614, 6439957299, 3201618970, 687652446, 63390060, 2027025

Offset: 0

Alois P. Heinz, May 23 2015

			Triangle T(n,k) begins:
:   1;
:   1,      1;
:   2,      8,       3;
:   5,     69,      77,      15;
:  14,    692,    1749,     890,     105;
:  42,   8120,   41998,   41909,   12039,    945;
: 132, 110278, 1114808, 1944225, 1018865, 186594, 10395;

Alois P. Heinz, Rows n = 0..140, flattened

Column k=0 gives A000108.
Main diagonal gives A001147.
Row sums give A258224.
T(2n,n) gives A292695.
Cf. A258220, A258222.

b:= proc(x, y, t, k) option remember; `if`(y>x or y<0, 0,
      `if`(x=0, 1, b(x-1, y-1, false, k)*`if`(t, (k*x+y)/y, 1)
                 + b(x-1, y+1, true, k)  ))
    end:
A:= (n, k)-> b(2*n, 0, false, k):
T:= (n, k)-> add(A(n, i)*(-1)^(k-i)*binomial(k, i), i=0..k)/k!:
seq(seq(T(n, k), k=0..n), n=0..10);

b[x_, y_, t_, k_] := b[x, y, t, k] = If[y > x || y < 0, 0, If[x == 0, 1, b[x-1, y-1, False, k]*If[t, (k*x + y)/y, 1] + b[x-1, y+1, True, k]]];
A[n_, k_] := b[2*n, 0, False, k];
T[n_, k_] := Sum[A[n, i]*(-1)^(k - i)*Binomial[k, i], {i, 0, k}]/k!;
Table[T[n, k], {n, 0, 10}, { k, 0, n}] // Flatten (* Jean-François Alcover, Jun 06 2018, from Maple *)

T(n,k) = 1/k! * Sum_{i=0..k} (-1)^(k-i) *C(k,i) * A258222(n,i).

cp's OEIS Frontend

A258221 Row sums of A258220.

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A292692 a(n) = A258220(2n,n).

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A005411 Number of non-vanishing Feynman diagrams of order 2n for the electron or the photon propagators in quantum electrodynamics.

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A258219 A(n,k) is the sum over all Dyck paths of semilength n of products over all peaks p of (x_p+k*y_p)/y_p, where x_p and y_p are the coordinates of peak p; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A258223 T(n,k) = 1/k! * Sum_{i=0..k} (-1)^(k-i) *C(k,i) * A258222(n,i); triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A258223 T(n,k) = 1/k! * Sum_{i=0..k} (-1)^(k-i) C(k,i) A258222(n,i); triangle T(n,k), n>=0, 0<=k<=n, read by rows.