A005412 -id:A005412 - cp's OEIS frontend

A258179 Sum over all Dyck paths of semilength n of products over all peaks p of y_p^2, where y_p is the y-coordinate of peak p.

1, 1, 5, 34, 312, 3649, 52161, 889843, 17796555, 411120395, 10838039407, 322752018060, 10762432731362, 398802951148255, 16312276452291935, 732189190349581890, 35876807697443520000, 1910107567584518883891, 110035833179472385285367, 6832792252684597270659486

Offset: 0

Alois P. Heinz, May 22 2015

nonn

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.

Alois P. Heinz, Table of n, a(n) for n = 0..300
Wikipedia, Lattice path

Cf. A000108, A000698, A005411, A005412, A258172, A258173, A258174, A258175, A258176, A258177, A258178, A258180, A258181.

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, y^2, 1) +
                   b(x-1, y+1, true)  ))
    end:
a:= n-> b(2*n, 0, false):
seq(a(n), n=0..20);

nmax = 20; Clear[g]; g[nmax+1] = 1; g[k_] := g[k] = 1 - x/( (k+2)^2*x - 1/g[k+1]); CoefficientList[Series[g[0], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 20 2015, after Sergei N. Gladkovskii *)

G.f.: T(0), where T(k) = 1 - x/( (k+2)^2*x - 1/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Aug 20 2015

A258180 Sum over all Dyck paths of semilength n of products over all peaks p of C(x_p,y_p), where x_p and y_p are the coordinates of peak p.

Original entry on oeis.org

1, 1, 4, 33, 517, 15326, 852912, 91023697, 19716262702, 8794395041567, 8016790849841585, 15556074485786226848, 64891787190080888991273, 561815453349204340865790817, 10402242033224422585780623039909, 423787530114579490372987256671625678

Offset: 0

Alois P. Heinz, May 22 2015

nonn

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.

Alois P. Heinz, Table of n, a(n) for n = 0..75
Wikipedia, Lattice path

Cf. A000108, A000698, A005411, A005412, A258172, A258173, A258174, A258175, A258176, A258177, A258178, A258179, A258181.

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, binomial(x, y), 1) +
                   b(x-1, y+1, true)  ))
    end:
a:= n-> b(2*n, 0, false):
seq(a(n), n=0..20);

b[x_, y_, t_] := b[x, y, t] = If[y > x || y < 0, 0, If[x == 0, 1, b[x - 1, y - 1, False]*If[t, Binomial[x, y], 1] + b[x - 1, y + 1, True]]];
a[n_] := b[2*n, 0, False];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Apr 23 2016, translated from Maple *)

A258181 Sum over all Dyck paths of semilength n of products over all peaks p of 2^(x_p-y_p), where x_p and y_p are the coordinates of peak p.

Original entry on oeis.org

1, 1, 5, 89, 5933, 1540161, 1584150165, 6497470064169, 106497075348688637, 6980195689972655145233, 1829876050804408046228327525, 1918781572083632396857805205324025, 8047973452254281276702044410544321359565, 135022681866797995009325363468217320506328688097

Offset: 0

Alois P. Heinz, May 22 2015

nonn

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.

Alois P. Heinz, Table of n, a(n) for n = 0..55
Wikipedia, Lattice path

Cf. A000108, A000698, A005411, A005412, A258172, A258173, A258174, A258175, A258176, A258177, A258178, A258179, A258180.

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, 2^(x-y), 1) +
                   b(x-1, y+1, true)  ))
    end:
a:= n-> b(2*n, 0, false):
seq(a(n), n=0..15);

b[x_, y_, t_] := b[x, y, t] = If[y > x || y < 0, 0, If[x == 0, 1, b[x - 1, y - 1, False]*If[t, 2^(x - y), 1] + b[x - 1, y + 1, True]]];
a[n_] := b[2*n, 0, False];
Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Apr 23 2016, translated from Maple *)

a(n) ~ c * 2^(n*(n-1)), where c = 1.47818066525747143617276638534... . - Vaclav Kotesovec, Jun 01 2015

A111546 Column 2 of triangle A111544.

Original entry on oeis.org

1, 3, 15, 99, 783, 7083, 71415, 789939, 9485343, 122721723, 1701224775, 25156450179, 395362560303, 6583219735563, 115817825451735, 2147443419579219, 41868118883289663, 856527397513863003, 18350158259899381095, 410942059850878349859, 9603217302778609785423

Offset: 0

Paul D. Hanna, Aug 07 2005

nonn

Also forms the columns of triangle A111548, which is the matrix inverse of triangle A111544.

Reinhard Zumkeller, Table of n, a(n) for n = 0..250
Glenn Bruda, Asymptotic expansions for the reciprocal Hardy-Littlewood logarithmic integrals, arXiv:2412.19866 [math.CO], 2024. See page 4.

Cf. A111544, A111530, A111548, A230253.

Cf. A005412.

a111546 n = a111546_list !! n
a111546_list = 1 : f 2 [1] where
   f v ws@(w:_) = y : f (v + 1) (y : ws) where
                  y = v * w + (sum $ zipWith (*) ws $ reverse ws)
-- Reinhard Zumkeller, Jan 24 2014

{a(n)=if(n<0,0,(matrix(n+3,n+3,m,j,if(m==j,1,if(m==j+1,-m+1, -(m-j-1)*polcoeff(log(sum(i=0,m,(i+2)!/2!*x^i)),m-j-1))))^-1)[n+3,3])}

a(n)=(1/2)*((n+3)!-3*(n+2)!-sum(k=0,n-2,(n+1-k)!*a(k+1))) \\ Formula by Groux Roland, implemented & checked to conform to given terms by M. F. Hasler, Dec 12 2010

G.f.: log(Sum_{n>=0} (n+2)!/2!*x^n) = Sum_{n>=1} a(n)*x^n/n. a(n) = 3*A111530(n) = -A111548(n+1, 0) for n>0.
a(n+1) = (1/2)*((n+4)!-3*(n+3)!-Sum_{k=0..n-1} (n+2-k)!*a(k+1)).
a(n+1) is the moment of order n for the measure of density: 2*x^2*exp(-x)/((x^2*exp(-x)*Ei(x)-x-1)^2+Pi^2*x^4*exp(-2*x)), on the interval 0..infinity. - Groux Roland, Dec 10 2010
a(n) = Sum_{k=0..n} A200659(n,k)*2^k. - Philippe Deléham, Nov 21 2011
G.f.: 1/(1-3x/(1-2x/(1-4x/(1-3x/(1-5x/(1-4x/(1-...(continued fraction). - Philippe Deléham, Nov 21 2011
G.f. 2 - U(0) where U(k)= 1 - x*(k+1)/(1 - x*(k+3)/U(k+1)); (continued fraction, 2-step). - Sergei N. Gladkovskii, Jun 29 2012
G.f. -1/G(0) where G(k) = x - 1 - k*x - x*(k+2)/G(k+1); (continued fraction, Euler's 1st kind, 1-step). - Sergei N. Gladkovskii, Aug 13 2012
G.f.: A(x) = 1/(G(0) - x) where G(k) = 1 + (k+1)*x - x*(k+3)/G(k+1) ; (recursively defined continued fraction). - Sergei N. Gladkovskii, Dec 26 2012
G.f.: 1/Q(0), where Q(k)= 1 - x + k*x - x*(k+2)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 03 2013
G.f.: 1/x -2 -2/(x*G(0)), where G(k)= 1 + 1/(1 - x*(k+3)/(x*(k+3) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 06 2013
G.f.: 1/x - 2 - 1/(x*W(0)), where W(k) = 1 - x*(k+3)/( x*(k+3) - 1/(1 - x*(k+1)/( x*(k+1) - 1/W(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Aug 25 2013
G.f.: W(0), where W(k) = 1 - x*(k+3)/( x*(k+3) - 1/(1 - x*(k+2)/( x*(k+2) - 1/W(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Aug 26 2013
a(n) ~ n! * n^3 / 2. - Vaclav Kotesovec, May 24 2025

A115974 Number of Feynman diagrams (vanishing and non-vanishing) of order 2n for the proper self-energy function of quantum electrodynamics (QED).

Original entry on oeis.org

1, 2, 6, 42, 414, 5058, 72486, 1182762, 21573054, 434358018, 9565348806, 228740050602, 5904853053534, 163728751178178, 4855046674314726, 153367360732387242, 5143219420761900414, 182530741698302811138, 6835913695777897799046, 269455018264860747728682, 11152465473005099074500894, 483617145128737549802831298

Offset: 0

R. J. Mathar, Mar 15 2006

nonn

The number of diagrams of A000698 left if the connected improper diagrams are removed: a(n)<=A000698(n+1). G.f. is essentially the inversion of the G.f. of A000698.
From Groux Roland, Mar 22 2011: (Start)
a(n) is the INVERTi transform of A001147(n+2), starting at n=2.
Let rho(x)=sqrt(x)*exp(-x/2)/sqrt(2*Pi); s(x)=integral(rho'(t)*log(abs(1-t/x)),t=0..infinity), and mu(x)=rho(x)/((s(x))^2+Pi^2*(rho(x))^2), then a(n+1) is the moment of order n for the measure of density mu(x) over the interval 0..infinity.
(End)
Vanishing diagrams: QED diagrams containing electron loops with an odd number of vertices are set to 0 (Furry theorem). See comments in A000698. This sequence (which is twice A167872(n-1) for n>=1) counts all the diagrams (vanishing and non-vanishing) for the self-energy function of QED. The sequence A005412 gives the number of non-vanishing diagrams for the self-energy. - Robert Coquereaux, Sep 12 2014

			There are A000698(3)=10 self-energy diagrams of order 4, (n=2). Four of them are chained diagrams of order 2, (n=1) (of two kinds) which are simply connected, which leaves 10-4=6=a(2) proper diagrams.

A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems, McGraw-Hill, 1971.

Alois P. Heinz, Table of n, a(n) for n = 0..400
P. Cvitanovic, B. Lautrup and R. B. Pearson, The number and weights of Feynman diagrams, Phys. Rev. D 18 (1978), 1939-1949.
R. J. Mathar, Table of Third and Fourth Order Feynman Diagrams of the Interacting Fermion Green's Function, Int. J. Quantum. Chem. 107 (10) (2007) 1975-1984.
Adrian Ocneanu, On the inner structure of a permutation: bicolored partitions and Eulerians, trees and primitives; arXiv preprint arXiv:1304.1263 [math.CO], 2013.
Wikipedia, Feynman diagram

Cf. A000698, A005411, A005412, A167872.

A000698 := proc(n::integer) local resul,fac,pows,c,c1,p,i ; if n = 0 then RETURN(1) ; else pows := combinat[partition](n) ; resul := 0 ; for p from 1 to nops(pows) do c := combinat[permute](op(p,pows)) ; c1 := op(1,c) ; fac := nops(c) ; for i from 1 to nops(c1) do fac := fac*doublefactorial(2*op(i,c1)-1) ; od ; resul := resul-(-1)^nops(c1)*fac ; od : fi ; RETURN(resul) ; end:
A115974 := proc(n::integer) local resul,m ; resul := A000698(n+1) ; for m from 1 to n-1 do resul := resul-A115974(m)*A000698(n+1-m) ; od: RETURN(resul) ; end:
for n from 1 to 20 do printf("%a,",A115974(n)) ; od ; # R. J. Mathar, Apr 24 2006

(* b = A000698 *) b[n_] := b[n] = (2n-1)!! - Sum[b[n-k]*(2k-1)!!, {k, n-1}]; a[0] = 1; a[n_] := a[n] = b[n+1] - Sum[a[m]*b[n+1-m], {m, n-1}]; Array[a, 22, 0] (* Jean-François Alcover, Jul 10 2017 *)

a(n) = A000698(n+1) - Sum_{m=1..n-1} a(m)*A000698(n+1-m).
1-Sum_{n>=1} a(n)*x^n = 1/(1+Sum_{n>=1} A000698(n+1)*x^n) (G.f.)
G.f. 2 - Q(0) where Q(k) = 1 - (k+2)*x/Q(k+1); (continued fraction, 1-step). - Sergei N. Gladkovskii, Aug 20 2012
G.f. 2 - x - x/(Q(0)-1) where Q(k) = 1 + (4*k+1)*x/(1 - (4*k+3)*x/((4*k+3)*x + 1/Q(k+1))); (continued fraction, 3rd kind, 3-step). - Sergei N. Gladkovskii, Sep 12 2012
G.f.: 2 + x/(G(0)-1) where G(k) = 1 - x*(k+1)/G(k+1); (recursively defined continued fraction). - Sergei N. Gladkovskii, Dec 10 2012
G.f.: 2 - G(0) where G(k) = 1 + (2*k+1)*x - x*(2*k+3)/G(k+1); (recursively defined continued fraction). - Sergei N. Gladkovskii, Dec 26 2012
G.f.: 2 - x - Q(0), where Q(k) = 1 - x*(2*k+3)/(1 - x*(2*k+2)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, May 21 2013
Call Sf the G.f. for the sequence 1, 2, 10, 74, ..., i.e., A000698 with first term (equal to 1) dropped. Call Sigmaf the G.f. for the sequence 0, 2, 6, 42, ..., i.e., this sequence A115974 with a first term of order 0 (equal to 0) added. Then Sf = 1/(1-Sigmaf). - Robert Coquereaux, Sep 14 2014
a(n) ~ 2^(n + 3/2) * n^(n+1) / exp(n). - Vaclav Kotesovec, Jan 02 2019

More terms from R. J. Mathar, Apr 24 2006, Nov 07 2006
Name and definition clarified by Robert Coquereaux, Sep 14 2014
a(0)=1 prepended 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).

A005413 Number of non-vanishing Feynman diagrams of order 2n+1 for the electron-electron-photon proper vertex function in quantum electrodynamics (QED).

Original entry on oeis.org

1, 1, 7, 72, 891, 12672, 202770, 3602880, 70425747, 1503484416, 34845294582, 872193147840, 23469399408510, 676090493459712, 20771911997290116, 678287622406488192, 23466105907996232835, 857623856612704266240

Offset: 0

N. J. A. Sloane

From Robert Coquereaux, Sep 12 2014: (Start)
QED diagrams are graphs with two kinds of edges (lines): a (non-oriented), f (oriented), and only one kind of (internal) vertex: aff.
They may have internal and external (i.e., pendant) lines.
QED diagrams containing loops of type f with an odd number of vertices are set to 0 (vanishing diagrams).
Proper diagrams also called one-particle-irreducible diagrams (1PI) are connected diagrams that remain connected when an arbitrary internal line is cut.
The proper vertex function of QED is described by proper (1PI) diagrams with one external line of type a (photon) and two external lines of type f (electron). Non-vanishing diagrams only exist if the number of vertices is odd.
The number of non-vanishing Feynman diagrams for the proper vertex function is obtained from  g*Gamma(g) = g (1 + 1 g^2 + 7 g^4 + 72 g^6 + ...) where the exponent p of g^p gives the number of (internal) vertices, p is called the order of the diagram.
Write g*Gamma(g) = g (1 + x + 7 x2 + 72 x3 + ...) with x = g^2.
The sequence a(n) gives the coefficient of x^n.
Relation with A005411: Gamma (g) = (S(g) - 1)/(g^2 S(g)^3) where S(g) = 1 + g^2 + 4 g^4 + 25 g^6 + ... is sum A005411(n) g^(2n), hence the g.f. in terms of modified Bessel functions.
(End)

			G.f. = 1 + x + 7*x^2 + 72*x^3 + 891*x^4 + 12672*x^5 + 202770*x^6 + 3602880*x^7 + ...

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

Robert Coquereaux, Table of n, a(n) for n = 0..250
P. Cvitanovic, B. Lautrup and R. B. Pearson, The number and weights of Feynman diagrams, Phys. Rev. D18, pp. 1939-1949 (1978).
Kevin Hartnett, Physicists uncover strange numbers in particle collisions, Quanta Magazine, November 15 2016.
R. J. Martin and M. J. Kearney, An exactly solvable self-convolutive recurrence, Aequat. Math., 80 (2010), 291-318. see p. 310.
R. J. Martin and M. J. Kearney, An exactly solvable self-convolutive recurrence, arXiv:1103.4936 [math.CO], 2011.
Wikipedia, Feynman diagram

Cvitanovic et al. relate this sequence to A000698 and A005411. - Robert Munafo, Jan 24 2010

a005413 n = a005413_list !! (n-1)
a005413_list = 1 : zipWith (*) [1 ..]
                           (zipWith (+) (tail a005412_list)
                           (zipWith (*) [4, 6 ..] a005413_list))
-- Reinhard Zumkeller, Jan 24 2014

a[n_]:= SeriesCoefficient[(4*x*(-2*x + (1 - BesselK[1, -(1/(4*x))]/BesselK[0, -(1/(4*x))])))/ (1 - BesselK[1, -(1/(4*x))]/BesselK[0, -(1/(4*x))])^3, {x,0,n}] (* Robert Coquereaux, Sep 12 2014 *)

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

See recurrence in Martin-Kearney paper.
a(n) = (n-1)*(A005412(n) + 2*n*A005412(n-1)) if n > 1.
From Robert Coquereaux, Sep 12 2014: (Start)
The g.f. for this sequence is (U - 1)/(U^3 x) where U is the g.f. for A005411.
G.f.: (4*x*(-2*x + (1 - K(1, -(1/(4*x))) / K(0, -(1/(4*x))))))/
   (1 - K(1, -(1/(4*x))) / K(0, -(1/(4*x))))^3
where K(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 the 1980 book "Quantum Field Theory".
(End)

Name clarified and reference added by Robert Coquereaux, Sep 12 2014

A172455 The case S(6,-4,-1) of the family of self-convolutive recurrences studied by Martin and Kearney.

Original entry on oeis.org

1, 7, 84, 1463, 33936, 990542, 34938624, 1445713003, 68639375616, 3676366634402, 219208706540544, 14397191399702118, 1032543050697424896, 80280469685284582812, 6725557192852592984064, 603931579625379293509683

Offset: 1

N. J. A. Sloane, Nov 20 2010

nonn

			G.f. = x + 7*x^2 + 84*x^3 + 1463*x^4 + 33936*x^5 + 990542*x^6 + 34938624*x^7 + ...
a(2) = 7 since (6*2 - 4) * a(2-1) - (a(1) * a(2-1)) = 7.

Vincenzo Librandi, Table of n, a(n) for n = 1..200
R. J. Martin and M. J. Kearney, An exactly solvable self-convolutive recurrence, Aequat. Math., 80 (2010), 291-318. see p. 307.
R. J. Martin and M. J. Kearney, An exactly solvable self-convolutive recurrence, arXiv:1103.4936 [math.CO], 2011.
NIST Digital Library of Mathematical Functions, Airy Functions.
A. N. Stokes, Continued fraction solutions of the Riccati equation, Bull. Austral. Math. Soc. Vol. 25 (1982), 207-214.
Eric Weisstein's World of Mathematics, Airy Functions, contains the definitions of Ai(x), Bi(x).

Cf. A000079 S(1,1,-1), A000108 S(0,0,1), A000142 S(1,-1,0), A000244 S(2,1,-2), A000351 S(4,1,-4), A000400 S(5,1,-5), A000420 S(6,1,-6), A000698 S(2,-3,1), A001710 S(1,1,0), A001715 S(1,2,0), A001720 S(1,3,0), A001725 S(1,4,0), A001730 S(1,5,0), A003319 S(1,-2,1), A005411 S(2,-4,1), A005412 S(2,-2,1), A006012 S(-1,2,2), A006318 S(0,1,1), A047891 S(0,2,1), A049388 S(1,6,0), A051604 S(3,1,0), A051605 S(3,2,0), A051606 S(3,3,0), A051607 S(3,4,0), A051608 S(3,5,0), A051609 S(3,6,0), A051617 S(4,1,0), A051618 S(4,2,0), A051619 S(4,3,0), A051620 S(4,4,0), A051621 S(4,5,0), A051622 S(4,6,0), A051687 S(5,1,0), A051688 S(5,2,0), A051689 S(5,3,0), A051690 S(5,4,0), A051691 S(5,5,0), A053100 S(6,1,0), A053101 S(6,2,0), A053102 S(6,3,0), A053103 S(6,4,0), A053104 S(7,1,0), A053105 S(7,2,0), A053106 S(7,3,0), A062980 S(6,-8,1), A082298 S(0,3,1), A082301 S(0,4,1), A082302 S(0,5,1), A082305 S(0,6,1), A082366 S(0,7,1), A082367 S(0,8,1), A105523 S(0,-2,1), A107716 S(3,-4,1), A111529 S(1,-3,2), A111530 S(1,-4,3), A111531 S(1,-5,4), A111532 S(1,-6,5), A111533 S(1,-7,6), A111546 S(1,0,1), A111556 S(1,1,1), A143749 S(0,10,1), A146559 S(1,1,-2), A167872 S(2,-3,2), A172450 S(2,0,-1), A172485 S(-1,-2,3), A177354 S(1,2,1), A292186 S(4,-6,1), A292187 S(3, -5, 1).

a[1] = 1; a[n_]:= a[n] = (6*n-4)*a[n-1] - Sum[a[k]*a[n-k], {k, 1, n-1}]; Table[a[n], {n, 1, 20}] (* Vaclav Kotesovec, Jan 19 2015 *)

{a(n) = local(A); if( n<1, 0, A = vector(n); A[1] = 1; for( k=2, n, A[k] = (6 * k - 4) * A[k-1] - sum( j=1, k-1, A[j] * A[k-j])); A[n])} /* Michael Somos, Jul 24 2011 */

S(v1, v2, v3, N=16) = {
  my(a = vector(N)); a[1] = 1;
  for (n = 2, N, a[n] = (v1*n+v2)*a[n-1] + v3*sum(j=1,n-1,a[j]*a[n-j])); a;
};
S(6,-4,-1)
\\ test: y = x*Ser(S(6,-4,-1,201)); 6*x^2*y' == y^2 - (2*x-1)*y - x
\\ Gheorghe Coserea, May 12 2017

a(n) = (6*n - 4) * a(n-1) - Sum_{k=1..n-1} a(k) * a(n-k) if n>1. - Michael Somos, Jul 24 2011
G.f.: x / (1 - 7*x / (1 - 5*x / (1 - 13*x / (1 - 11*x / (1 - 19*x / (1 - 17*x / ... )))))). - Michael Somos, Jan 03 2013
a(n) = 3/(2*Pi^2)*int((4*x)^((3*n-1)/2)/(Ai'(x)^2+Bi'(x)^2), x=0..inf), where Ai'(x), Bi'(x) are the derivatives of the Airy functions. [Vladimir Reshetnikov, Sep 24 2013]
a(n) ~ 6^n * (n-1)! / (2*Pi) [Martin + Kearney, 2011, p.16]. - Vaclav Kotesovec, Jan 19 2015
6*x^2*y' = y^2 - (2*x-1)*y - x, where y(x) = Sum_{n>=1} a(n)*x^n. - Gheorghe Coserea, May 12 2017
G.f.: x/(1 - 2*x - 5*x/(1 - 7*x/(1 - 11*x/(1 - 13*x/(1 - ... - (6*n - 1)*x/(1 - (6*n + 1)*x/(1 - .... Cf. A062980. - Peter Bala, May 21 2017

A005414 Feynman diagrams of order 2n with vertex skeletons.

Original entry on oeis.org

1, 1, 13, 93, 1245, 18093, 308605, 5887453, 124221373, 2864305277, 71589605885, 1927010749181, 55572839581437, 1709604517055229, 55893262628149245, 1935654236127347709, 70799043456576835581, 2727771901780930132989, 110438840436968476274685, 4688223534904569925386237

Offset: 1

N. J. A. Sloane

nonn

P. Cvitanovic, B. Lautrup and R. B. Pearson, Number and weights of Feynman diagrams, Phys. Rev. D 18 (1978), 1939-1949.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Gheorghe Coserea, Table of n, a(n) for n = 1..202
Michael Borinsky, Renormalized asymptotic enumeration of Feynman diagrams, arXiv:1703.00840 [hep-th], 2017.
P. Cvitanovic, B. Lautrup and R. B. Pearson, The number and weights of Feynman diagrams, Phys. Rev. D18, 1939 (1978).

Cf. A001147, A000698, A005411, A005412, A005413, A005416, A049464.

seq[nn_] := Module[{x, y=0, y1=0, n=1}, While[n++; True, y1 = x^2 + x^4 + 2x^6 - 3x^2 y + x^4 (-y + x D[y, x]/2) - x^6 (8y + x D[y, x]/2) + y^2 + x y D[y, x] + (x^2 - x^4)(3y^2 + 3/2 x y D[y, x]) + x^6 (12y^2 + 3/2 x y D[y, x]) - x^2 (y^3 + 3/2 x y^2 D[y, x]) + x^4 (5y^3 + 3/2 x y^2 D[y, x]) - x^6 (8y^3 + 3/2 x y^2 D[y, x]) + (-x^4 + x^6)(2y^4 + 1/2 x y^3 D[y, x]) + O[x]^(2nn+1); If[y1 == y, Break[]]; y = y1]; CoefficientList[y, x^2]] // Rest;
seq[20] (* Jean-François Alcover, Oct 05 2018, after Gheorghe Coserea *)

seq(N) = {
  my(x='x+O('x^(2*N+1)), y=0, y1=0, n=1);
  while (n++,
  y1 = x^2 + x^4 + 2*x^6 - 3*x^2*y + x^4*(-y + 1/2*x*y') +
       -x^6*(8*y + 1/2*x*y') + y^2 + x*y*y' +
       (x^2 - x^4)*(3*y^2 + 3/2*x*y*y') + x^6*(12*y^2 + 3/2*x*y*y') +
       -x^2*(y^3 + 1/2*x*3*y^2*y') + x^4*(5*y^3 + 1/2*x*3*y^2*y') +
       -x^6*(8*y^3 + 1/2*x*3*y^2*y') + (-x^4+x^6)*(2*y^4 + 1/8*x*4*y^3*y');
  if (y1 == y, break); y=y1);
  vector(N, n, polcoeff(y, 2*n));
};
seq(20) \\ Gheorghe Coserea, Oct 17 2017

a(n) ~ 4*exp(-5/2)/Pi * n * 2^n * n! * (1 - 9/(4*n) - 67/(32*n^2) + O(1/n^3)). (see Borinsky link) - Gheorghe Coserea, Oct 19 2017

More terms from Gheorghe Coserea, Oct 17 2017

A268163 Number of labeled binary-ternary rooted non-planar trees, indexed by number of leaves.

Original entry on oeis.org

0, 1, 1, 4, 25, 220, 2485, 34300, 559405, 10525900, 224449225, 5348843500, 140880765025, 4063875715900, 127418482316125, 4314607214417500, 156920190449147125, 6100643259005795500, 252476539015516440625, 11081983532721088487500, 514215436341672155715625

Offset: 0

Murray R. Bremner, Jan 27 2016

nonn

This can also be interpreted as the number of multilinear monomials of degree n in a nonassociative algebra with an (anti)commutative binary operation and a completely (skew-)symmetric ternary operation; the number of variables in the monomial corresponds to the number of leaves in the tree.

This sequence also enumerates a certain class of Feynman diagrams; see the references, links, and crossrefs below.

			For n = 4 and using the monomial interpretation, the 25 multilinear monomials of degree 4 are as follows, where [-,-] is the binary operation and (-,-,-) is the ternary operation:
[[[a,b],c],d], [[[a,b],d],c], [[[a,c],b],d], [[[a,c],d],b], [[[a,d],b],c], [[[a,d],c],b], [[[b,c],a],d], [[[b,c],d],a], [[[b,d],a],c], [[[b,d],c],a], [[[c,d],a],b], [[[c,d],b],a], [[a,b],[c,d]], [[a,c],[b,d]], [[a,d],[b,c]], [(a,b,c),d], [(a,b,d),c], [(a,c,d),b], [(b,c,d),a], ([a,b],c,d), ([a,c],b,d), ([a,d],b,c), ([b,c],a,d), ([b,d],a,c), ([c,d],a,b).

J. Bedford, On Perturbative Field Theory and Twistor String Theory, Ph.D. Thesis, 2007, Queen Mary, University of London.
B. Feng and M. Luo, An introduction to on-shell recursion relations, Review Article, Frontiers of Physics, October 2012, Volume 7, Issue 5, pp. 533-575.
K. Kampf, A new look at the nonlinear sigma model, 17th International Conference in Quantum Chromodynamics (QCD 14), Nuclear and Particle Physics Proceedings, Volumes 258-259, January-February 2015, pp. 86-89.
M. L. Mangano and S. J. Parke, Multi-parton amplitudes in gauge theories, Physics Reports, Volume 200, Issue 6, February 1991, pp. 301-367.

Alois P. Heinz, Table of n, a(n) for n = 0..400.
J. Bedford, On Perturbative Field Theory and Twistor String Theory, arXiv:0709.3478 [hep-th], 2007, (see page 23).
D. Carmi, TeV scale strings and scattering amplitudes at the LHC, arXiv:1109.5161 [hep-th], 2011-2015, (see page 23).
B. Feng, M. Luo, An introduction to on-shell recursion relations, arXiv:1111.5759 [hep-th], 2011-2012, (see page 2).
K. Kampf, A new look at the nonlinear sigma model (see pages 7 and 13).
M. L. Mangano, S. J. Parke, Multi-parton amplitudes in gauge theories, arXiv:hep-th/0509223, 2005, (see page 5).
G. Travaglini, Harmony of (super) form factors (see page 3).
A. Volovich, Yang-Mills amplitudes and twistor string theory (see bottom of page 1).
James Christopher Whitehead, The Production of Pairs of Isolated Photons at Higher Orders in QCD, Ph. D. Thesis, Durham University (UK, 2021).
Wikipedia, Feynman diagram

Cf. A001147. The number of labeled binary rooted non-planar trees.
Cf. A025035. The number of labeled ternary rooted non-planar trees.
Cf. A268172. The corresponding number of unlabelled trees.
Cf. A005411. Number of non-vanishing Feynman diagrams of order 2n for the electron or the photon propagators in quantum electrodynamics.
Cf. A005412. Number of non-vanishing Feynman diagrams of order 2n for the vacuum polarization (the proper two-point function of the photon) and for the self-energy (the proper two-point function of the electron) in quantum electrodynamics (QED).
Cf. A005413. Number of non-vanishing Feynman diagrams of order 2n+1 for the electron-electron-photon proper vertex function in quantum electrodynamics (QED).
Cf. A005414. Feynman diagrams of order 2n with vertex skeletons.
Other sequences related to Feynman diagrams: A115974, A122023, A167872, A214298, A214299.
Cf. A000311.

with(combinat):
b:= proc(n, i, v) option remember; `if`(n=0,
      `if`(v=0, 1, 0), `if`(i<1 or v<1 or nAlois P. Heinz, Jan 28 2016
# second Maple program:
a:= proc(n) option remember; `if`(n<3, [0, 1$2][n+1],
       ((24*n-36)*a(n-1)+(3*n-5)*(3*n-7)*a(n-2))/11)
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Jan 28 2016

a[0]=0; a[1]=1; a[2]=1; a[n_]:=a[n]=(12(2n-3)a[n-1]+(3n-5)(3n-7)a[n-2])/11; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Feb 24 2016, after Alois P. Heinz *)

a(n) = ((24*n-36)*a(n-1)+(3*n-5)*(3*n-7)*a(n-2))/11 for n>2. - Alois P. Heinz, Jan 28 2016
Because of Koszul duality for operads, the exponential generating function is the compositional inverse of the power series x-x^2/2-x^3/6 (email of Vladimir Dotsenko to Murray R. Bremner, Jan 28 2016).
a(n) ~ sqrt(9-4*sqrt(3)) * ((12+9*sqrt(3))/11)^n * n^(n-1) / (3 * exp(n)). - Vaclav Kotesovec, Feb 24 2016

cp's OEIS Frontend

A258179 Sum over all Dyck paths of semilength n of products over all peaks p of y_p^2, where y_p is the y-coordinate of peak p.

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A258180 Sum over all Dyck paths of semilength n of products over all peaks p of C(x_p,y_p), where x_p and y_p are the coordinates of peak p.

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A258181 Sum over all Dyck paths of semilength n of products over all peaks p of 2^(x_p-y_p), where x_p and y_p are the coordinates of peak p.

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A111546 Column 2 of triangle A111544.

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Haskell

PARI

PARI

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A115974 Number of Feynman diagrams (vanishing and non-vanishing) of order 2n for the proper self-energy function of quantum electrodynamics (QED).

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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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A005413 Number of non-vanishing Feynman diagrams of order 2n+1 for the electron-electron-photon proper vertex function in quantum electrodynamics (QED).

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