A000698 -id:A000698 - cp's OEIS frontend

A355721 Square table, read by antidiagonals: the g.f. for row n is given recursively by (2n-1)xR(n,x) = 1 + (2n-3)x - 1/R(n-1,x) for n >= 1 with the initial value R(0,x) = Sum_{k >= 0} A112934(k+1)x^k.

1, 1, 2, 1, 2, 6, 1, 2, 10, 26, 1, 2, 14, 74, 158, 1, 2, 18, 138, 706, 1282, 1, 2, 22, 218, 1686, 8162, 13158, 1, 2, 26, 314, 3194, 24162, 110410, 163354, 1, 2, 30, 426, 5326, 53890, 394254, 1708394, 2374078, 1, 2, 34, 554, 8178, 102722, 1019250, 7191018, 29752066, 39456386

Offset: 0

Peter Bala, Jul 15 2022

Compare with A111528, which has a similar definition.

			Square array begins
1, 2,  6,  26,   158,    1282,   13158,    163354,    2374078,     39456386, ...
1, 2, 10,  74,   706,    8162,  110410,   1708394,   29752066,    576037442, ...
1, 2, 14, 138,  1686,   24162,  394254,   7191018,  144786006,   3188449602, ...
1, 2, 18, 218,  3194,   53890, 1019250,  21256090,  483426010,  11895873410, ...
1, 2, 22, 314,  5326,  102722, 2197558,  51355514, 1297759918,  35208930050, ...
1, 2, 26, 426,  8178,  176802, 4206618, 108577674, 3011332338,  89141101506, ...
1, 2, 30, 554, 11846,  283042, 7396830, 208569034, 6288011206, 201404591042, ...
...

A. N. Stokes, Continued fraction solutions of the Riccati equation, Bull. Austral. Math. Soc. Vol. 25 (1982), 207-214.

Cf. A112934 (row 0), A000698 (row 1), A355722 (row 2), A355723 (row 3), A355724 (row 4), A355725 (row 5). Cf. A001147, A111528.

T := (n,k) -> coeff(series(hypergeom([n+1/2, 1], [], 2*x)/ hypergeom([n-1/2, 1], [], 2*x), x, 21), x, k):
# display as a sequence
seq(seq(T(n-k,k), k = 0..n), n = 0..10);
# display as a square array
seq(print(seq(T(n,k), k = 0..10)), n = 0..10);

Let d(n) = Product_{k = 1..n} 2*k-1 = A001147(n) denote the double factorial of odd numbers.
O.g.f. for row n: R(n,x) = ( Sum_{k >= 0} d(n+k)/d(n)*x^k )/( Sum_{k >= 0} d(n-1+k)/d(n-1)*x^k ).
R(n,x)/(1 - (2*n-1)*x*R(n,x)) = Sum_{k >= 0} d(n+k)/d(n)*x^k.
R(n,x) = 1/(1 + (2*n-1)*x - (2*n+1)*x/(1 + (2*n+1)*x - (2*n+3)*x/(1 + (2*n+3)*x - (2*n+5)*x/(1 + (2*n+5)*x - ... )))).
R(n,x) satisfies the Riccati differential equation 2*x^2*d/dx(R(n,x)) + (2*n-1)*x*R(n,x)^2 - (1 + (2*n-3)*x)*R(n,x) + 1 = 0 with R(n,0) = 1.
Applying Stokes 1982 gives A(x) = 1/(1 - 2*x/(1 - (2*n+1)*x/(1 - 4*x/(1 - (2*n+3)*x/(1 - 6*x/(1 - (2*n+5)*x/(1 - ... - 2*m*x/(1 - (2*n+2*m-1)*x/(1 - ... ))))))))), a continued fraction of Stieltjes type.
Row 0: A112934(n+1); Row 1; A000698(n+1).

A107716 Inverse INVERT transform of triple factorial numbers (3*n-2)!!! (A007559).

Original entry on oeis.org

1, 3, 21, 219, 2973, 49323, 964173, 21680571, 551173053, 15633866379, 489583062381, 16780438408539, 624935780160285, 25131869565110571, 1085528359404039117, 50124679063548821499, 2464153823558024331645, 128500643820213560377803, 7085182933810282490250285

Offset: 0

Paul D. Hanna, May 23 2005

Column 0 of triangle A107717.

			The triple factorials begin: {1,4,28,280,3640,58240,...}; thus the inverse INVERT transform of the triple factorials can be calculated by the g.f.s:
1/(1 + x + 4*x^2 + 28*x^3 + 280*x^4 + 3640*x^5 + 58240*x^6 +...) = (1 - x - 3*x^2 - 21*x^3 - 219*x^4 - 2973*x^5 - 49323*x^6 -...).

Alois P. Heinz, Table of n, a(n) for n = 0..380

Cf. A007559, A000698, A107717.

b:= proc(n) b(n):=  `if`(n=0, 1, b(n-1)*(3*n+1)) end:
a:= proc(n) a(n):= -`if`(n<0, 1, add(a(n-i-1)*b(i), i=0..n)) end:
seq(a(n), n=0..20);  # Alois P. Heinz, May 23 2017

m = 20; f3[n_] := Product[3k+1, {k, 0, n-1}]; A[x_] = 1-1/(1+Sum[f3[n] x^n, {n, 1, m}]); CoefficientList[A[x] + O[x]^m, x] // Rest (* Jean-François Alcover, May 01 2019 *)

a(n)=polcoeff(1-(1+sum(k=1,n+1,prod(j=0,k-1,3*j+1)*x^k)+x^2*O(x^n))^-1,n+1)

G.f.: A(x) = 1 - 1/[1 + Sum_{n>=1} (3*n-2)!!! * x^n ] where (3*n-2)!!! = Product_{k=0..n-1} (3*k+1).
a(n) = Sum_{k, 0<=k<=n} A089949(n, k)*3^k . - Philippe Deléham, Aug 15 2005
G.f.: (1 - Q(0))/x where Q(k) = 1 - x*(3*k+1)/(1 - x*(3*k+3)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 20 2013
G.f.: 1/x - 2 - 2/x/G(0), where G(k)= 1 + 1/(1 - x*(3*k+3)/(x*(3*k+4) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 25 2013
From Peter Bala, May 23 2017: (Start)
G.f. A(x) = 1/(1 + x - 4*x/(1 + 4*x - 7*x/(1 + 7*x - 10*x/(1 + 10*x - ...)))).
A(x) = 1/(1 + x - 4*x/(1 - 3*x/(1 - 7*x/(1 - 6*x/(1 - 10*x/(1 - 9*x - ...)))))). (End)

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

A005416 Vertex diagrams of order 2n.

Original entry on oeis.org

1, 1, 6, 50, 518, 6354, 89782, 1435330, 25625910, 505785122, 10944711398, 257834384850, 6572585595622, 180334118225650, 5300553714899094, 166206234856979810, 5538980473666776854, 195527829569946627138, 7288988096561232432070

Offset: 0

N. J. A. Sloane

			G.f. = 1 + x + 6*x^2 + 50*x^3 + 518*x^4 + 6354*x^5 + 89782*x^6 + 1435330*x^7 + ...

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..100
D. J. Broadhurst, Four-loop Dyson-Schwinger-Johnson anatomy, arXiv:hep-ph/9909336, 1999.
P. Cvitanovic, Asymptotic estimates and gauge invariance, Nuclear Phys. B 127 (1977), 176-188.
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. 292.

Cf. A000698, A001147, A049464.

m = 19; s[x_] = Sum[(2*n)!/(2^n*n!)*x^n, {n, 0, m}]; gf[x_] = (s[x] - 1)/(s[x]^2*x); Most[CoefficientList[Series[gf[x], {x, 0, m}], x]] (* Jean-François Alcover, Aug 31 2011, after g.f. *)

{a(n) = my(A); if( n<0, 0, A = sum( k=0, n+1, (2*k)! / k! /2^k * x^k, x^2 * O(x^n)); polcoeff( (A - 1) / (x * A^2), n))}; /* Michael Somos, Oct 11 2006 */

{a(n) = my(A); if( n<1, n==0, A = vector(n); A[1] = 1; for( k=2, n, A[k] = (2 * k - 3) * A[k-1] + sum( j=1, k-1, A[j] * A[k-j])); (2*n - 1) * A[n])}; /* Michael Somos, Jul 24 2011 */

Let s_n = (2*n)!/(2^n*n!) (A001147), S(x) = Sum_{n >= 0} s_n*x^n; sequence has g.f. A(x) satisfying 1 - 1/S(x) = x*A(x)*S(x).

a(n) = (2*n - 1) * A000698(n). [Martin and Kearney]

A140456 a(n) is the number of indecomposable involutions of length n.

Original entry on oeis.org

1, 1, 1, 3, 7, 23, 71, 255, 911, 3535, 13903, 57663, 243871, 1072031, 4812575, 22278399, 105300287, 510764095, 2527547455, 12794891007, 66012404863, 347599231103, 1863520447103, 10178746224639, 56548686860543, 319628408814847, 1835814213846271

Offset: 1

Joel B. Lewis, Jul 22 2008

nonn

An involution is a self-inverse permutation. A permutation of [n] = {1, 2, ..., n} is indecomposable if it does not fix [j] for any 0 < j < n.
From Paul Barry, Nov 26 2009: (Start)
G.f. of a(n+1) is 1/(1-x-2x^2/(1-x-3x^2/(1-x-4x^2/(1-x-5x^2/(1-...))))) (continued fraction).
a(n+1) is the binomial transform of the aeration of A000698(n+1). Hankel transform of a(n+1) is A000178(n+1). (End)
From Groux Roland, Mar 17 2011: (Start)
a(n) is the INVERTi transform of A000085(n+1)
a(n) is also the moment of order n for the density: sqrt(2/Pi^3)*exp((x-1)^2/2)/(1-(erf(I*(x-1)/sqrt(2)))^2).
More generally, if c(n)=int(x^n*rho(x),x=a..b) with rho(x) a probability density function of class C1, then the INVERTi transform of (c(1),..c(n),..) starting at n=2 gives the moments of mu(x) = rho(x) / ((s(x))^2+(Pi*rho(x))^2) with s(x) = int( rho'(t)*log(abs(1-t/x)), t=a..b) + rho(b)*log(x/(b-x)) + rho(a)*log((x-a)/x).
(End)
For n>1 sum over all Motzkin paths of length n-2 of products over all peaks p of (x_p+y_p)/y_p, where x_p and y_p are the coordinates of peak p. - Alois P. Heinz, May 24 2015

			The unique indecomposable involution of length 3 is 321. The indecomposable involutions of length 4 are 3412, 4231 and 4321.
G.f. = x + x^2 + 3*x^3 + 7*x^4 + 23*x^5 + 71*x^6 + 255*x^7 + 911*x^8 + ...

Alois P. Heinz, Table of n, a(n) for n = 1..800 (terms n = 1..50 from Joel B. Lewis)
Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.
Claudia Malvenuto and Christophe Reutenauer, Primitive Elements of the Hopf Algebras of Tableaux, arXiv:2010.06731 [math.CO], 2020.

Cf. A000085 (involutions), A000698 (indecomposable fixed-point free involutions), and A003319 (indecomposable permutations).

Cf. A001006, A258306.

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)/y, 1)
                 + b(x-1, y, false) + b(x-1, y+1, true)))
    end:
a:= n-> `if`(n=1, 1, b(n-2, 0, false)):
seq(a(n), n=1..35);  # Alois P. Heinz, May 24 2015

CoefficientList[Series[1 - 1/Total[CoefficientList[Series[E^(x + x^2/2), {x, 0, 50}], x] * Range[0, 50]! * x^Range[0, 50]], {x, 0, 50}], x]

G.f.: 1 - 1/I(x), where I(x) is the ordinary generating function for involutions (A000085).

G.f.: Q(0) +1/x, where Q(k) = 1 - 1/x - (k+1)/Q(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Sep 16 2013

A004208 a(n) = n * (2n - 1)!! - Sum_{k=0..n-1} a(k) (2n - 2k - 1)!!.

Original entry on oeis.org

1, 5, 37, 353, 4081, 55205, 854197, 14876033, 288018721, 6138913925, 142882295557, 3606682364513, 98158402127761, 2865624738913445, 89338394736560917, 2962542872271918593, 104128401379446177601, 3867079042971339087365, 151312533647578564021477

Offset: 1

N. J. A. Sloane, following a suggestion from E. W. Bowen, Aug 27 1976

nonn

a(n+1) is the moment of order n for the probability density function rho(x) = Pi^(-3/2)*sqrt(x/2)*exp(x/2)/(1-erf^2(i*sqrt(x/2))) on the interval 0..infinity, where erf is the error function and i=sqrt(-1). - Groux Roland, Nov 10 2009

E. W. Bowen, Letter to N. J. A. Sloane, Aug 27 1976.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 1..300
E. W. Bowen & N. J. A. Sloane, Correspondence, 1976
Colin K, Is it possible to make Mathematica reformulate an expression in a more numerically stable way?. see second answer by whuber.

Cf. A000698.

df := proc(n) product(2*k-1,k=1..n) end: a[1] := 1: for n from 2 to 30 do a[n] := n*df(n)-sum(a[k]*df(n-k),k=1..n-1) od;

CoefficientList[Series[D[Log[Sum[(2n-1)!!x^n,{n,0,19}]],x],{x,0,18}],x] (* Wouter Meeussen, Mar 21 2009 *)
a[ n_] := If[ n < 1, 0, n Coefficient[ Normal[ Series[ Log @ Erfc @ Sqrt @ x, {x, Infinity, n}] + x + Log[ Sqrt [Pi x]]] /. x -> -1 / 2 / x, x, n]] (* Michael Somos, May 28 2012 *)

{a(n) = if( n<1, 0, n++; polcoeff( 1 - 1 / (2 * sum( k=0, n, x^k * (2*k)! / (2^k * k!), x * O(x^n))), n))} /* Michael Somos, May 28 2012 */

a(n) = (1/2) * A000698(n+1), n > 0.
x + (5/2)*x^2 + (37/3)*x^3 + (353/4)*x^4 + (4081/5)*x^5 + (55205/6)*x^6 + ... = log(1 + x + 3*x^2 + 15*x^3 + 105*x^4 + 945*x^5 + 10395*x^6 + ...) where [1, 1, 3, 15, 105, 945, 10395, ...] = A001147(double factorials). - Philippe Deléham, Jun 20 2006
G.f.: ( 1/Q(0) - 1)/x where Q(k) = 1 - x*(2*k+1)/(1 - x*(2*k+4)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Mar 19 2013
G.f.: (2/x)/G(0) - 1/x, where G(k) = 1 + 1/(1 - 2*x*(2*k+1)/(2*x*(2*k+1) - 1 + 2*x*(2*k+4)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 31 2013
G.f.: 1/(2*x^2) - 1/(2*x) - G(0)/(2*x^2), where G(k) = 1 - x*(k+1)/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Aug 15 2013
L.g.f.: log(1/(1 - x/(1 - 2*x/(1 - 3*x/(1 - 4*x/(1 - 5*x/(1 - ...))))))) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 10 2017

Description corrected by Jeremy Magland (magland(AT)math.byu.edu), Jan 07 2000

More terms from Emeric Deutsch, Dec 21 2003

A105620 Matrix inverse square-root of triangle A105615.

Original entry on oeis.org

1, -1, 1, -2, -2, 1, -10, -4, -3, 1, -74, -20, -7, -4, 1, -706, -148, -39, -11, -5, 1, -8162, -1412, -315, -70, -16, -6, 1, -110410, -16324, -3243, -635, -116, -22, -7, 1, -1708394, -220820, -40167, -7264, -1183, -180, -29, -8, 1, -29752066, -3416788, -579159, -99191, -15065, -2049, -265, -37, -9, 1

Offset: 0

Paul D. Hanna, Apr 16 2005

Column 0 is negative A000698 (related to double factorials). Column 1 equals twice column 0 after the initial term.

			Triangle begins:
1;
-1,1;
-2,-2,1;
-10,-4,-3,1;
-74,-20,-7,-4,1;
-706,-148,-39,-11,-5,1;
-8162,-1412,-315,-70,-16,-6,1;
-110410,-16324,-3243,-635,-116,-22,-7,1;
-1708394,-220820,-40167,-7264,-1183,-180,-29,-8,1;
-29752066,-3416788,-579159,-99191,-15065,-2049,-265,-37,-9,1; ...

Cf. A105615, A105619 (matrix square), A105623 (matrix inverse), A000698 (column 0), A105621 (column 2), A105622 (row sums).

T(n,k)=local(R,M=matrix(n+1,n+1,m,j,if(m>=j,if(m==j,1,if(m==j+1,-2*j, polcoeff(1/sum(i=0,m-j,(2*i)!/i!/2^i*x^i)+O(x^m),m-j)))))); R=(M+M^0)/2;for(i=1,floor(2*log(n+2)),R=(R+M*R^(-1))/2); return(if(n

A127059 Column 2 of triangle A127058.

Original entry on oeis.org

3, 12, 108, 1332, 19908, 342252, 6583788, 139380372, 3211960068, 79950396492, 2137119431148, 61065403377012, 1858069709657028, 60006976422450732, 2050924514408985708, 73988085260209757652, 2810535115787602525188

Offset: 0

Paul D. Hanna, Jan 04 2007

nonn

Column 0 of triangle A127058 is A000698, the number of shellings of an n-cube, divided by 2^n n!. Column 1 of triangle A127058 is A115974, the number of Feynman diagrams of the proper self-energy at perturbative order n.

Vaclav Kotesovec, Table of n, a(n) for n = 0..400

Cf. A127058; other columns: A000698, A115974; A127060.

A127058[n_, k_]:= A127058[n, k] = If[k==n, n+1, Sum[A127058[j+k, k]* A127058[n-j, k+1], {j,0,n - k - 1}]]; Table[A127058[n+2, 2], {n, 0, 30}] (* G. C. Greubel, Jun 09 2019 *)

c(n)=(2*n)!/(2^n*n!);
a(n)=if(n==0, 3, (c(n+3) - 3*c(n+2) - sum(k=0, n-1, a(k)*(c(n+2-k)-c(n+1-k)) ))/2  );
vector(20, n, n--; a(n)) \\ G. C. Greubel, Jun 09 2019

@CachedFunction
def A127058(n, k):
    if (k==n): return n+1
    else: return sum(A127058(j+k, k)*A127058(n-j, k+1) for j in (0..n-k-1))
[A127058(n+2,2) for n in (0..30)] # G. C. Greubel, Jun 09 2019

a(0) = 3 and for n>0 a(n) = (1/2)*(c(n+3)-3*c(n+2)-Sum_{k=0..n-1} a(k)*(c(n+2-k)-c(n+1-k))) with c(n) = (2*n)!/(2^n*n!). - Groux Roland, Nov 14 2009
G.f.: A(x) = (1 - T(0))/x, T(k) = 1 - x*(k+3)/T(k+1) (continued fraction). - Sergei N. Gladkovskii, Dec 13 2011
G.f.: 1/x - Q(0)/x, where Q(k)= 1 - x*(2*k+3)/(1 - x*(2*k+4)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, May 21 2013
a(n) ~ 2^(n + 5/2) * n^(n+3) / exp(n). - Vaclav Kotesovec, Jan 02 2019

cp's OEIS Frontend

A355721 Square table, read by antidiagonals: the g.f. for row n is given recursively by (2*n-1)*x*R(n,x) = 1 + (2*n-3)*x - 1/R(n-1,x) for n >= 1 with the initial value R(0,x) = Sum_{k >= 0} A112934(k+1)*x^k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula

A107716 Inverse INVERT transform of triple factorial numbers (3*n-2)!!! (A007559).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula

Extensions

A005416 Vertex diagrams of order 2n.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A140456 a(n) is the number of indecomposable involutions of length n.

A355721 Square table, read by antidiagonals: the g.f. for row n is given recursively by (2n-1)xR(n,x) = 1 + (2n-3)x - 1/R(n-1,x) for n >= 1 with the initial value R(0,x) = Sum_{k >= 0} A112934(k+1)x^k.

A004208 a(n) = n * (2n - 1)!! - Sum_{k=0..n-1} a(k) (2n - 2k - 1)!!.