A049294 -id:A049294 - cp's OEIS frontend

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

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

A049295 Number of subgroups of index 4 in free group of rank n+1.

Original entry on oeis.org

1, 71, 2143, 54335, 1321471, 31817471, 764217343, 18344733695, 440294408191, 10567189327871, 253613279903743, 6086723113107455, 146081381003558911, 3505953301484470271, 84142880178680889343, 2019429129941297135615, 48466299152487396933631

Offset: 0

Valery A. Liskovets

P. de la Harpe, Topics in Geometric Group Theory, Univ. Chicago Press, 2000, p. 23, N_{4,n}.
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.13(b).

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.
Index entries for linear recurrences with constant coefficients, signature (37,-368,1436,-2256,1152).

Cf. A003319, A027837, A049290-A049294.

LinearRecurrence[{37,-368,1436,-2256,1152},{1,71,2143,54335,1321471},20] (* Harvey P. Dale, Apr 14 2016 *)

a(n) = 4*24^n-4*6^n-2*4^n+4*2^n-1.

G.f.: (264*x^3+116*x^2-34*x-1) / ((x-1)*(2*x-1)*(4*x-1)*(6*x-1)*(24*x-1)). [Colin Barker, Feb 17 2013]

More terms from Carrie Westbrook (s1213407(AT)cedarville.edu)
Terms corrected by Colin Barker, May 08 2012
a(16) from Colin Barker, Feb 17 2013

A093052 Exponent of 2 in 6^n - 2^n.

Original entry on oeis.org

0, 2, 5, 4, 8, 6, 9, 8, 13, 10, 13, 12, 16, 14, 17, 16, 22, 18, 21, 20, 24, 22, 25, 24, 29, 26, 29, 28, 32, 30, 33, 32, 39, 34, 37, 36, 40, 38, 41, 40, 45, 42, 45, 44, 48, 46, 49, 48, 54, 50, 53, 52, 56, 54, 57, 56, 61, 58, 61, 60, 64, 62, 65, 64, 72, 66, 69, 68, 72

Offset: 0

Ralf Stephan, Mar 16 2004

nonn

a(n-1) is the exponent of 2 in A009168(n), A012394(n), A088991(n), A009083(n), A012036(n), A012092(n), A012395(n), A012460(n), A012465(n), A012466(n), A012467(n), (A049294(n)-1)/3.

Cf. A093050, A093051.

Join[{0},Table[IntegerExponent[6^n-2^n,2],{n,70}]] (* Harvey P. Dale, Mar 08 2012 *)

a(n)=if(n<1,0,if(n%2==0,a(n/2)+2*floor((n+2)/4)+1,n+1))

def A093052(n): return n+(~(m:=3**n-1)& m-1).bit_length() if n else 0 # Chai Wah Wu, Jul 07 2022

Recurrence: a(2n) = a(n) + [(n+1)/2] + 1, a(2n+1) = 2n+2.

a(n) = n + A007814(A024023(n)) = n + A090740(n). - Reinhard Zumkeller, Mar 27 2004

A049291 Number of subgroups of index n in free group of rank 4.

Original entry on oeis.org

1, 15, 625, 54335, 8563601, 2228419359, 893451975473, 523337983164799, 429463651385469649, 477364501208149290975, 699086688951391180496497, 1318072723102023442664430143, 3137514636520304660660007679505

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).

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, A027837, A049290, A049292, A049293, A049294, A049295.

ClearAll[a]; a[n_] := a[n] = n*n!^3 - Sum [k!^3*a[n - k], {k, 1, n - 1}]; Table[a[n], {n, 1, 13}]  (* Jean-François Alcover, Oct 08 2012, from first formula *)

{a(n)=n*polcoeff(log(sum(k=0,n,k!^3*x^k)+x*O(x^n)),n)} \\ Paul D. Hanna, Apr 13 2009

a(n) = n*n!^3 - Sum_{k=1..n-1} k!^3*a(n-k).

L.g.f.: Sum_{n>=1} a(n)*x^n/n = log( Sum_{n>=1} (n-1)!^3*x^n ). [Paul D. Hanna, Apr 13 2009]

More terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jun 17 2001

cp's OEIS Frontend

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

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A049295 Number of subgroups of index 4 in free group of rank n+1.

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A093052 Exponent of 2 in 6^n - 2^n.

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A049291 Number of subgroups of index n in free group of rank 4.

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