A003319 -id:A003319 - cp's OEIS frontend

A111533 Row 6 of table A111528.

1, 1, 8, 78, 876, 10956, 149472, 2195208, 34398288, 571525200, 10022997888, 184897670112, 3578224662720, 72486450479808, 1534267158087168, 33877135427154048, 779208751651730688, 18645519786163266816

Offset: 0

Paul D. Hanna, Aug 06 2005

			(1/6)*(log(1 + 6*x + 42*x^2 + 336*x^3 + ... + (n+5)!/5!)*x^n + ...)
= x + 8/2*x^2 + 78/3*x^3 + 876/4*x^4 + 10956/5*x^5 + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..440
A. N. Stokes, Continued fraction solutions of the Riccati equation, Bull. Austral. Math. Soc. Vol. 25 (1982), 207-214.

Cf: A111528 (table), A003319 (row 1), A111529 (row 2), A111530 (row 3), A111531 (row 4), A111532 (row 5), A111534 (diagonal).

Cf. A001725, A001730.

m = 18; (-1/(6x)) ContinuedFractionK[-i x, 1 + i x, {i, 6, m+5}] + O[x]^m // CoefficientList[#, x]& (* Jean-François Alcover, Nov 02 2019 *)

{a(n)=if(n<0,0,if(n==0,1, (n/6)*polcoeff(log(sum(m=0,n,(m+5)!/5!*x^m) + x*O(x^n)),n)))} \\ fixed by Vaclav Kotesovec, Jul 27 2015

G.f.: (1/6)*log(Sum_{n>=0} (n+5)!/5!*x^n) = Sum_{n>=1} a(n)*x^n/n.
G.f.: 1/(1 + 6*x - 7*x/(1 + 7*x - 8*x/(1 + 8*x -... (continued fraction).
a(n) = Sum_{k=0..n} 6^(n-k)*A089949(n,k). - Philippe Deléham, Oct 16 2006
G.f.: (5 + 1/Q(0))/6, where Q(k) = 1 - 4*x + k*x - x*(k+2)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 04 2013
G.f.: G(0)/2, where G(k) = 1 + 1/(1 - x*(k+1)/(x*(k-2) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 05 2013
a(n) ~ n! * n^6/6! * (1 + 9/n + 19/n^2 - 69/n^3 - 704/n^4 - 5880/n^5 - 65736/n^6 - 896832/n^7 - 14068080/n^8 - 246800304/n^9 - 4760585136/n^10). - Vaclav Kotesovec, Jul 27 2015
From Peter Bala, May 25 2017: (Start)
O.g.f.: A(x) = ( Sum_{n >= 0} (n+6)!/6!*x^n ) / ( Sum_{n >= 0} (n+5)!/5!*x^n ).
1/(1 - 6*x*A(x)) = Sum_{n >= 0} (n+5)!/5!*x^n. Cf. A001725.
A(x)/(1 - 6*x*A(x)) = Sum_{n >= 0} (n+6)!/6!*x^n. Cf. A001730.
A(x) satisfies the Riccati equation x^2*A'(x) + 6*x*A^2(x) - (1 + 5*x)*A(x) + 1 = 0.
G.f. as an S-fraction: A(x) = 1/(1 - x/(1 - 7*x/(1 - 2*x/(1 - 8*x/(1 - 3*x/(1 - 9*x/(1 - ... - n*x/(1 - (n+6)*x/(1 - ... ))))))))), by Stokes 1982.
A(x) = 1/(1 + 6*x - 7*x/(1 - x/(1 - 8*x/(1 - 2*x/(1 - 9*x/(1 - 3*x/(1 - ... - (n + 6)*x/(1 - n*x/(1 - ... ))))))))). (End)

A113869 Coefficients in asymptotic expansion of probability that a random pair of elements from the alternating group A_k generates all of A_k.

Original entry on oeis.org

1, -1, -1, -4, -23, -171, -1542, -16241, -194973, -2622610, -39027573, -636225591, -11272598680, -215668335091, -4431191311809, -97316894892644, -2275184746472827, -56421527472282127, -1479397224086870294, -40897073524132164189, -1188896226524012279617

Offset: 0

N. J. A. Sloane, Jan 26 2006

Vaclav Kotesovec, Table of n, a(n) for n = 0..420
L. Babai, The probability of generating the symmetric group, J. Combin. Theory, A52 (1989), 148-153.
J. Bovey and A. Williamson, The probability of generating the symmetric group, Bull. London Math. Soc. 10 (1978) 91-96.
J. D. Dixon, The probability of generating the symmetric group, Math. Z. 110 (1969) 199-205.
J. D. Dixon, Asymptotics of Generating the Symmetric and Alternating Groups, Electronic Journal of Combinatorics, vol 11(2), R56.
Thibault Godin, An analogue to Dixon's theorem for automaton groups, arXiv preprint arXiv:1610.03301 [math.GR], 2016.
Richard J. Martin, and Michael J. Kearney, Integral representation of certain combinatorial recurrences, Combinatorica: 35:3 (2015), 309-315.

Cf. A003319, A112354, A113871, A114038, A158094, A168246, A306153, A316084.

A003319[n_] := A003319[n] = n! - Sum[ k!*A003319[n-k], {k, 1, n-1}]; a[n_] := -Sum[ A003319[i]*StirlingS2[n-1, i-1], {i, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Dec 11 2012, after N. J. A. Sloane *)

The probability that a random pair of elements from the alternating group A_k generates all of A_k is P_k ~ 1-1/k-1/k^2-4/k^3-23/k^4-171/k^5-... = Sum_{n >= 0} a(n)/k^n.
Furthermore, P_k ~ 1 - Sum_{n >= 1} A003319(n)/[k]n, where [k]_n = k(k-1)(k-2)...(k-n+1). Therefore for n >= 2, a(n) = - Sum{i=1..n} A003319(i)*Stirling_2(n-1, i-1). - N. J. A. Sloane.
a(n) ~ -n! / (4 * (log(2))^(n+1)). - Vaclav Kotesovec, Jul 28 2015

A059332 Determinant of n X n matrix A defined by A[i,j] = (i+j-1)! for 1 <= i,j <= n.

Original entry on oeis.org

1, 1, 2, 24, 3456, 9953280, 859963392000, 3120635156889600000, 634153008009974906880000000, 9278496603801318870491332608000000000, 12218100099725239100847669366019325952000000000000, 1769792823810713244721831122736499011207487815680000000000000000

Offset: 0

Noam Katz (noamkj(AT)hotmail.com), Jan 26 2001

nonn

Hankel transform of n! (A000142(n)) and of A003319. - Paul Barry, Oct 07 2008
Hankel transform of A000255. - Paul Barry, Apr 22 2009
Monotonic magmas of size n, i.e., magmas with elements labeled 1..n where product(i,j) >= max(i,j). - Chad Brewbaker, Nov 03 2013
Also called the bouncing factorial function. - Alexander Goebel, Apr 08 2020

			a(4) = 3456 because the relevant matrix is {1 2 6 24 / 2 6 24 120 / 6 24 120 720 / 24 120 720 5040 } and the determinant is 3456.

Alois P. Heinz, Table of n, a(n) for n = 0..32
Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.
Googology Wiki, Bouncing Factorial

Cf. A010790, A385867.

Cf. A162014 and A055209. - Johannes W. Meijer, Jun 27 2009

with(linalg): Digits := 500: A059332 := proc(n) local A, i, j: A := array(1..n,1..n): for i from 1 to n do for j from 1 to n do A[i,j] := (i+j-1)! od: od: RETURN(det(A)) end: for n from 1 to 20 do printf(`%d,`, A059332(n)) od;
# second Maple program:
a:= proc(n) option remember;
      `if`(n=0, 1, a(n-1)*n!^2/n)
    end:
seq(a(n), n=0..12);  # Alois P. Heinz, Apr 29 2020

Table[n! BarnesG[n+1]^2, {n, 1, 10}] (* Jean-François Alcover, Sep 19 2016 *)

A059332(n)=matdet(matrix(n,n,i,j,(i+j-1)!)) \\ M. F. Hasler, Nov 03 2013

a(n) = 2^binomial(n,2)*prod(k=1,n-1, binomial(k+2,2)^(n-1-k)) \\ Ralf Stephan, Nov 04 2013

def mono_choices(a,b,n)
    n - [a,b].max
end
def all_mono_choices(n)
    accum =1
    0.upto(n-1) do |i|
        0.upto(n-1) do |j|
            accum = accum * mono_choices(i,j,n)
        end
    end
    accum
end
1.upto(12) do |k|
puts all_mono_choices(k)
end # Chad Brewbaker, Nov 03 2013

a(n) = a(n-1)*(n!)*(n-1)! for n >= 2 so a(n) = product k=1, 2, ..., n k!*(k-1)!.
a(n) = 2^C(n,2)*Product_{k=1..(n-1), C(k+2,2)^(n-1-k)}. - Paul Barry, Jan 15 2009
a(n) = n!*product(k!, k=0..n-1)^2. - Johannes W. Meijer, Jun 27 2009
a(n) ~ (2*Pi)^(n+1/2) * exp(1/6 - n - 3*n^2/2) * n^(n^2 + n + 1/3) / A^2, where A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Aug 01 2015

More terms from James Sellers, Jan 29 2001
Offset corrected. Comment and formula aligned with new offset by Johannes W. Meijer, Jun 24 2009
a(0)=1 prepended by Alois P. Heinz, Apr 08 2020

A089949 Triangle T(n,k), read by rows, given by [0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...] DELTA [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, ...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 1, 6, 6, 0, 1, 12, 34, 24, 0, 1, 20, 110, 210, 120, 0, 1, 30, 270, 974, 1452, 720, 0, 1, 42, 560, 3248, 8946, 11256, 5040, 0, 1, 56, 1036, 8792, 38338, 87504, 97296, 40320, 0, 1, 72, 1764, 20580, 129834, 463050, 920184, 930960, 362880

Offset: 0

Philippe Deléham, Jan 11 2004

Row reverse appears to be A111184. - Peter Bala, Feb 17 2017

			Triangle begins:
  1;
  0, 1;
  0, 1,  2;
  0, 1,  6,   6;
  0, 1, 12,  34,  24;
  0, 1, 20, 110, 210,  120;
  0, 1, 30, 270, 974, 1452, 720; ...

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

Cf. A084938, A111184.
Diagonals: A000007, A000012, A002378, A000142.
Row sums: A003319.

m = 10;
gf = (1/x)*(1-1/(1+Sum[Product[(1+k*y), {k, 0, n-1}]*x^n, {n, 1, m}]));
CoefficientList[#, y]& /@ CoefficientList[gf + O[x]^m, x] // Flatten (* Jean-François Alcover, May 11 2019 *)

PARI
```
T(n,k)=if(nPaul D. Hanna, Aug 16 2005
```

Sum_{k=0..n} x^(n-k)*T(n,k) = A111528(x, n); see A000142, A003319, A111529, A111530, A111531, A111532, A111533 for x = 0, 1, 2, 3, 4, 5, 6. - Philippe Deléham, Aug 09 2005
Sum_{k=0..n} T(n,k)*3^k = A107716(n). - Philippe Deléham, Aug 15 2005
Sum_{k=0..n} T(n,k)*2^k = A000698(n+1). - Philippe Deléham, Aug 15 2005
G.f.: A(x, y) = (1/x)*(1 - 1/(1 + Sum_{n>=1} [Product_{k=0..n-1}(1+k*y)]*x^n )). - Paul D. Hanna, Aug 16 2005

A027837 Number of subgroups of index n in free group of rank 3.

Original entry on oeis.org

1, 7, 97, 2143, 68641, 3011263, 173773153, 12785668351, 1169623688353, 130305512589247, 17376934722756577, 2733655173624167551, 501034099176714373921, 105847486567006696384831

Offset: 1

Valery A. Liskovets

P. de la Harpe, Topics in Geometric Group Theory, Univ. Chicago Press, 2000, p. 23.
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.13(b).

Reinhard Zumkeller, Table of n, a(n) for n = 1..250
M. Hall, Subgroups of finite index in free groups, Canad. J. Math., 1 (1949), 187-190.
V. A. Liskovets and A. Mednykh, Enumeration of subgroups in the fundamental groups of orientable circle bundles over surfaces, Commun. in Algebra, 28, No. 4 (2000), 1717-1738.

Cf. A003319, A049290-A049295.

Cf. A001044.

a027837 n = a027837_list !! (n-1)
a027837_list = f 1 [] where
   f x ys = y : f (x + 1) (y : ys) where
            y = a001044 x * x - sum (zipWith (*) ys $ tail a001044_list)
-- Reinhard Zumkeller, Sep 05 2015

a[n_] := a[n] = n*n!^2 - Sum [k!^2*a[n-k], {k, 1, n-1}]; Table[ a[n], {n, 1, 14}] (* Jean-François Alcover, Dec 13 2011, after formula *)

{a(n)=n*polcoeff(log(sum(k=0,n,k!^2*x^k)+x*O(x^n)),n)} \\ Paul D. Hanna, Apr 13 2009

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

L.g.f.: Sum_{n>=1} a(n)*x^n/n = log( Sum_{n>=1} (n-1)!^2*x^n ). - Paul D. Hanna, Apr 13 2009

More terms from Larry Reeves (larryr(AT)acm.org), Oct 05 2000

Further terms from Naohiro Nomoto, Jun 18 2001

A077607 Convolutory inverse of the factorial sequence.

Original entry on oeis.org

1, -2, -2, -8, -44, -296, -2312, -20384, -199376, -2138336, -24936416, -314142848, -4252773824, -61594847360, -950757812864, -15586971531776, -270569513970944, -4959071121374720, -95721139472072192, -1941212789888952320, -41271304403571227648

Offset: 1

Clark Kimberling, Nov 11 2002

sign

|a(n)| is the number of permutations on [n] for which no proper initial interval of [n] is mapped to an interval. - David Callan, Nov 11 2003

			a(4)= -8 = -24*1-6*(-2)-2*(-2). (a(1),a(2),...,a(n))(*)(1,2,3!,...,n!)=(1,0,0,...,0), where (*) denotes convolution.

Alois P. Heinz, Table of n, a(n) for n = 1..449
Jean-Christophe Aval, Jean-Christophe Novelli, Jean-Yves Thibon, The # product in combinatorial Hopf algebras, dmtcs:2892 - Discrete Mathematics & Theoretical Computer Science, January 1, 2011, DMTCS Proceedings vol. AO, 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011).
Richard J. Martin, and Michael J. Kearney, Integral representation of certain combinatorial recurrences, Combinatorica: 35:3 (2015), 309-315.
Ioannis Michos, Christina Savvidou, Enumeration of super-strong Wilf equivalence classes of permutations, arXiv:1803.08818 [math.CO], 2018.
Vincent Pilaud, V. Pons, Permutrees, arXiv preprint arXiv:1606.09643 [math.CO], 2016 (Unsigned version).

Cf. A000142, A003319, A111529, A260491.

a:= proc(n) option remember; `if`(n=1, 1,
      -add((n-i+1)!*a(i), i=1..n-1))
    end:
seq(a(n), n=1..25);  # Alois P. Heinz, Dec 20 2017

Clear[a]; a[1]=1; a[n_]:=a[n]=Sum[-(n-j+1)!*a[j],{j,1,n-1}]; Table[a[n],{n,1,20}] (* Vaclav Kotesovec, Jul 27 2015 *)
terms=21; 1/Sum[(k+1)!*x^k, {k, 0, terms}]+O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Dec 20 2017, after Vladeta Jovovic *)

def A077607_list(len):
    R, C = [1], [1]+[0]*(len-1)
    for n in (1..len-1):
        for k in range(n, 0, -1):
            C[k] = C[k-1] * (k+1)
        C[0] = -sum(C[k] for k in (1..n))
        R.append(C[0])
    return R
print(A077607_list(21)) # Peter Luschny, Feb 28 2016

a(n) = -n!*a(1)-(n-1)!*a(2)-...-2!*a(n-1), with a(1)=1.
G.f.: 1/Sum_{k>=0} (k+1)!*x^k. - Vladeta Jovovic, May 04 2003
From Sergei N. Gladkovskii, Aug 15 2012 - Nov 07 2013: (Start)
Continued fractions:
G.f.: U(0) - x where U(k) = 1-x*(k+1)/(1-x*(k+2)/U(k+1)).
G.f.: A(x) = G(0) - x where G(k) = 1 + (k+1)*x - x*(k+2)/G(k+1).
G.f.: G(0) where G(k) = 1 - x*(k+2)/(1 - x*(k+1)/G(k+1)).
G.f.: (x-x^(2/3))/(Q(0)-1), where Q(k) = 1-(k+1)*x^(2/3)/(1-x^(1/3)/(x^(1/3) - 1/Q(k+1))).
G.f.: 1 - x - x/Q(0), where Q(k)= 1 + k*x - x*(k+2)/Q(k+1).
G.f.: 2/G(0) where G(k)= 1 + 1/(1 - x*(k+2)/(x*(k+2) + 1/G(k+1))).
G.f.: 1/W(0) where W(k) = 1-x*(k+2)/(x*(k+2)-1/(1 - x*(k+1)/(x*(k+1) - 1/W(k+1)))).
G.f.: x/(1- Q(0)) - x, where Q(k) = 1  - (k+1)*x/(1  - (k+1)*x/Q(k+1)).
G.f.: 1-x-x*T(0), where T(k) = 1-x*(k+2)/(x*(k+2)-(1+k*x)*(1+x+k*x)/T(k+1)). (End)
a(n) ~ -n! * (1 - 4/n - 8/n^3 - 76/n^4 - 752/n^5 - 8460/n^6 - 107520/n^7 - 1522124/n^8 - 23717424/n^9 - 402941324/n^10), for coefficients see A260491. - Vaclav Kotesovec, Jul 27 2015
a(n) = -2*A111529(n-2), for n>=2. - Vaclav Kotesovec, Jul 29 2015

More terms from Vaclav Kotesovec, Jul 29 2015

A049290 Array T(n,k) = number of subgroups of index k in free group of rank n, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 7, 13, 1, 1, 15, 97, 71, 1, 1, 31, 625, 2143, 461, 1, 1, 63, 3841, 54335, 68641, 3447, 1, 1, 127, 23233, 1321471, 8563601, 3011263, 29093, 1, 1, 255, 139777, 31817471, 1035045121, 2228419359, 173773153, 273343, 1, 1, 511, 839425

Offset: 1

N. J. A. Sloane, Sep 09 2000

			Array T(n,k) (n >= 1, k >= 1) begins:
1,  1,   1,     1,       1, ...
1,  3,  13,    71,     461, ...
1,  7,  97,  2143,   68641, ...
1, 15, 625, 54335, 8563601, ...

P. de la Harpe, Topics in Geometric Group Theory, Univ. Chicago Press, 2000, p. 23.
J. H. Kwak and J. Lee, Enumeration of graph coverings, surface branched coverings and related group theory, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161.
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.13(b).

Alois P. Heinz, Antidiagonals n = 1..37, flattened
J. H. Kwak and J. Lee, Enumeration of graph coverings and surface branched coverings, Lecture Note Series 1 (2001), Com^2MaC-KOSEF, Korea. See chapter 3. [Broken link?]
V. A. Liskovets and A. Mednykh, Enumeration of subgroups in the fundamental groups of orientable circle bundles over surfaces, Commun. in Algebra, 28, No. 4 (2000), 1717-1738.

Rows give A003319, A027837, A049291.
Columns give A000225, A049294, A049295.
Main diagonal is A057014.

T:= proc(n,k) option remember; k* k!^(n-1) -add(j!^(n-1) *T(n, k-j), j=1..k-1) end: seq(seq(T(d+1-k, k), k=1..d), d=1..10); # Alois P. Heinz, Oct 29 2009

nmax = 10; t[n_, k_] := t[n, k] = k*k!^(n-1) - Sum[j!^(n-1)*t[n, k-j], {j, 1, k-1}]; Flatten[ Table[ t[n-k+1, k], {n, 1, nmax}, {k, 1, n}]] (* Jean-François Alcover, Nov 09 2011, after Maple *)

More terms from Alois P. Heinz, Oct 29 2009

A273526 Number of 123-avoiding indecomposable permutations.

Original entry on oeis.org

1, 1, 3, 11, 38, 127, 423, 1423, 4854, 16787, 58776, 208001, 742888, 2674427, 9694831, 35357655, 129644774, 477638683, 1767263172, 6564120401, 24466267000, 91482563619, 343059613628, 1289904147301, 4861946401428, 18367353072127, 69533550915978, 263747951750333, 1002242216651340, 3814986502092275

Offset: 1

Alice L.L. Gao and Philip B. Zhang, May 24 2016

nonn

Indranil Ghosh, Table of n, a(n) for n = 1..1000
A. L. L. Gao, S. Kitaev, P. B. Zhang. On pattern avoiding indecomposable permutations, arXiv:1605.05490 [math.CO], 2016.
Murray Tannock, Equivalence classes of mesh patterns with a dominating pattern, MSc Thesis, Reykjavik Univ., May 2016.

Cf. A000108, A003319.

a[n_] := CatalanNumber[n] - (n - 1)
Table[a[n], {n, 1, 30}] (* Alice L.L. Gao, May 24 2016 *)

a(n) = binomial(2*n,n)/(n+1) - n + 1; \\ Indranil Ghosh, Mar 04 2017

import math
f=math.factorial
def C(n,r): return f(n)/f(r)/f(n-r)
def Catalan(n): return C(2*n,n)/(n+1)
def A273526(n): return Catalan(n)-n+1 # Indranil Ghosh, Mar 04 2017

a(n) = C(n) - n + 1, where C(n) is the Catalan number.

A321710 Triangle read by rows: T(n,k) is the number of rooted hypermaps of genus k with n darts.

Original entry on oeis.org

1, 3, 12, 1, 56, 15, 288, 165, 8, 1584, 1611, 252, 9152, 14805, 4956, 180, 54912, 131307, 77992, 9132, 339456, 1138261, 1074564, 268980, 8064, 2149888, 9713835, 13545216, 6010220, 579744, 13891584, 81968469, 160174960, 112868844, 23235300, 604800, 91287552, 685888171, 1805010948, 1877530740, 684173164, 57170880, 608583680, 5702382933, 19588944336, 28540603884, 16497874380, 2936606400, 68428800, 4107939840, 47168678571, 206254571236, 404562365316, 344901105444, 108502598960, 8099018496

Offset: 1

Gheorghe Coserea, Nov 17 2018

Row n contains floor((n+1)/2) = A008619(n-1) terms.

			Triangle starts:
n\k  [0]       [1]        [2]         [3]         [4]        [5]
[1]  1;
[2]  3;
[3]  12,       1;
[4]  56,       15;
[5]  288,      165,       8;
[6]  1584,     1611,      252;
[7]  9152,     14805,     4956,       180;
[8]  54912,    131307,    77992,      9132;
[9]  339456,   1138261,   1074564,    268980,     8064;
[10] 2149888,  9713835,   13545216,   6010220,    579744;
[11] 13891584, 81968469,  160174960,  112868844,  23235300,  604800;
[12] 91287552, 685888171, 1805010948, 1877530740, 684173164, 57170880;
[13] ...

Gheorghe Coserea, Rows n = 1..42, flattened
Alain Giorgetti and Timothy R. S. Walsh, Enumeration of hypermaps of a given genus, Ars Math. Contemp. 15 (2018) 225-266.
Timothy R. Walsh, Space-efficient generation of nonisomorphic maps and hypermaps
T. R. Walsh, Space-Efficient Generation of Nonisomorphic Maps and Hypermaps, J. Int. Seq. 18 (2015) # 15.4.3.
P. G. Zograf, Enumeration of Grothendieck's Dessins and KP Hierarchy, International Mathematics Research Notices, Volume 2015, Issue 24, 1 January 2015, 13533-13544.
Peter Zograf, Enumeration of Grothendieck's Dessins and KP Hierarchy, arXiv:1312.2538 [math.CO], 2014.

Columns k=0..9 give: A000257 (k=0), A118093 (k=1), A214817 (k=2), A214818 (k=3), A318104 (k=4), A321705 (k=5), A321706 (k=6), A321707 (k=7), A321708 (k=8), A321709 (k=9).
Row sums give A003319(n+1).
Cf. A008619, A060593.

l1[f_,n_] := Sum[(i-1)t[i]D[f,t[i-1]], {i,2,n}];
m1[f_,n_] := Sum[(i-1)t[j]t[i-j]D[f,t[i-1]] + j(i-j)t[i+1]D[f,t[j],t[i-j]], {i,2,n},{j,i-1}];
ff[1] = x^2 t[1];
ff[n_] := ff[n] = Simplify@(2x l1[ff[n-1],n] + m1[ff[n-1],n] + Sum[t[i+1]j(i-j)D[ff[k],t[j]]D[ff[n-1-k],t[i-j]], {i,2,n-1},{j,i-1},{k,n-2}]) / n;
row[n_]:=Reverse[CoefficientList[n ff[n] /. {t[_]->x}, x]][[;;;;2]][[;;Quotient[n+1,2]]];
Table[row[n], {n,14}] (* Andrei Zabolotskii, Jun 27 2025, after the PARI code *)

L1(f, N) = sum(i=2, N, (i-1)*t[i]*deriv(f, t[i-1]));
M1(f, N) = {
  sum(i=2, N, sum(j=1, i-1, (i-1)*t[j]*t[i-j]*deriv(f, t[i-1]) +
      j*(i-j)*t[i+1]*deriv(deriv(f, t[j]), t[i-j])));
};
F(N) = {
  my(u='x, v='x, f=vector(N)); t=vector(N+1, n, eval(Str("t", n)));
  f[1] = u*v*t[1];
  for (n=2, N, f[n] = (u + v)*L1(f[n-1], n) + M1(f[n-1], n) +
    sum(i=2, n-1, t[i+1]*sum(j=1, i-1,
    j*(i-j)*sum(k=1, n-2, deriv(f[k], t[j])*deriv(f[n-1-k], t[i-j]))));
    f[n] /= n);
  f;
};
seq(N) = {
  my(f=F(N), v=substvec(f, t, vector(#t, n, 'x)),
     g=vector(#v, n, Polrev(Vec(n * v[n]))));
  apply(p->Vecrev(substpol(p, 'x^2, 'x)), g);
};
concat(seq(14))

A000257(n)=T(n,0), A118093(n)=T(n,1), A214817(n)=T(n,2), A214818(n)=T(n,3), A060593(n)=T(2*n+1,n)=(2*n)!/(n+1), A003319(n+1)=Sum_{k=0..floor((n-1)/2)} T(n,k).

A084519 Number of indecomposable ground-state 3-ball juggling sequences of period n.

Original entry on oeis.org

1, 1, 3, 13, 47, 173, 639, 2357, 8695, 32077, 118335, 436549, 1610471, 5941181, 21917583, 80856053, 298285687, 1100404333, 4059496479, 14975869477, 55247410055, 203812962077, 751885445295, 2773777080149, 10232728055191

Offset: 1

Antti Karttunen, Jun 02 2003

nonn

This sequence counts the length n asynchronic site swaps given in A084511/A084512.
First differences of A084518. INVERTi transform of A084509. Cf. also A084529, A003319.
Equals left border of triangle A145463. - Gary W. Adamson, Oct 11 2008

Carsten Elsner, Dominic Klyve and Erik R. Tou, A zeta function for juggling sequences, Journal of Combinatorics and Number Theory, Volume 4, Issue 1, 2012, pp. 1-13; ISSN 1942-5600

Fan Chung, R. L. Graham, Primitive juggling sequences, Am. Math. Monthly 115 (3) (2008) 185-194
Index entries for sequences related to juggling
Index entries for linear recurrences with constant coefficients, signature (3, 2, 2).

Cf. A145463. - Gary W. Adamson, Oct 11 2008

INVERTi([seq(A084509(n),n=1..80)]);
with(combinat); A084519 := proc(n) option remember; local c,i,k; A084509(n)-add(add(mul(A084519(i),i=c),c=composition(n,k)),k=2..n); end;

LinearRecurrence[{3,2,2},{1,1,3},30] (* Harvey P. Dale, Jul 20 2013 *)

a(n) seems to satisfy the recurrence: a(1) = a(2) = 1, a(3) = 3 and a(n) = 3*a(n-1)+2*a(n-2)+2*a(n-3). If so, a(n) = floor(A*B^n+1/2) where B = 3.6890953... is the real positive root of x^3-3x^2-2x-2 = 0 and A = 0.0687059... is the real positive root of 118*x^3+118*x^2+35*x-3 = 0. - Benoit Cloitre, Jun 14 2003 [This conjecture is established in the Chung-Graham paper.]

G.f.: x*(1-2*x-2*x^2)/(1-3*x-2*x^2-2*x^3). - Colin Barker, Jan 14 2012

cp's OEIS Frontend

A111533 Row 6 of table A111528.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A113869 Coefficients in asymptotic expansion of probability that a random pair of elements from the alternating group A_k generates all of A_k.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A059332 Determinant of n X n matrix A defined by A[i,j] = (i+j-1)! for 1 <= i,j <= n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Ruby

Formula

Extensions

A089949 Triangle T(n,k), read by rows, given by [0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...] DELTA [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, ...] where DELTA is the operator defined in A084938.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A027837 Number of subgroups of index n in free group of rank 3.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula

Extensions

A077607 Convolutory inverse of the factorial sequence.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Sage

Formula

Extensions

A049290 Array T(n,k) = number of subgroups of index k in free group of rank n, read by antidiagonals.

Views

Author

Keywords