A000684 -id:A000684 - cp's OEIS frontend

A000686 Number of 4-colored labeled graphs on n nodes, divided by 4.

1, 7, 85, 1777, 63601, 3882817, 403308865, 71139019777, 21276992674561, 10778161937857537, 9238819435213784065, 13390649605615389843457, 32796747486424209782108161, 135669064080920007649863745537, 947468281528010179181982467702785, 11166618111585805201637975219611631617

Offset: 1

N. J. A. Sloane

Sequence represents 1/4 of the number of 4-colored labeled graphs on n nodes. Indeed, on p. 413 of the Read paper, column 4 is 4, 28, 340, 7108, ... - Emeric Deutsch, May 06 2004

R. C. Read, personal communication.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

S. R. Finch, Bipartite, k-colorable and k-colored graphs. (4*{a(n)})
S. R. Finch, Bipartite, k-colorable and k-colored graphs, June 5, 2003. [Cached copy, with permission of the author]
R. C. Read, The number of k-colored graphs on labelled nodes, Canad. J. Math., 12 (1960), 410-414.
R. C. Read, Letter to N. J. A. Sloane, Oct. 29, 1976.

Cf. A000683, A000685, A223887.

b[n_] := Sum[ 2^((i-j)*j + i*(n-i))*Binomial[n, i]*Binomial[i, j], {i, 0, n}, {j, 0, i}]; a[n_] := 1/4*Sum[ Binomial[n, k]*2^(k*(n-k))*b[k], {k, 0, n}]; Table[a[n], {n, 1, 14}] (* Jean-François Alcover, Dec 07 2011, after Emeric Deutsch *)

N=66;  x='x+O('x^N);
E=sum(n=0, N, x^n/(n!*2^binomial(n,2)) );
tgf=E^4-1;  v=Vec(tgf);
v=vector(#v, n, v[n] * n! * 2^(n*(n-1)/2) ) / 4
/* Joerg Arndt, Apr 10 2013 */

a(n) = (1/4)*Sum_{k=0..n} binomial(n, k)*2^(k(n-k))*b(k), where b(0)=1 and b(k) = 3*A000685(k) for k > 0. - Emeric Deutsch, May 06 2004
From Peter Bala, Apr 12 2013: (Start)
a(n) = (1/4)*A223887(n).
a(n) = (1/4)*Sum_{k = 0..n} binomial(n,k)*2^(k*(n-k))*b(k)*b(n-k), where b(n) := Sum_{k = 0..n} binomial(n,k)*2^(k*(n-k)).
Let E(x) = Sum_{n >= 0} x^n/(n!*2^C(n,2)). Then a generating function for this sequence is (1/4)*(E(x)^4 - 1) = Sum_{n >= 0} a(n)*x^n/(n!*2^C(n,2)) = x + 7*x^2/(2!*2) + 85*x^3/(3!*2^3) + .... (End)

More terms from Pab Ter (pabrlos(AT)yahoo.com) and Emeric Deutsch, May 05 2004

A001226 Lerch's function q_2(n) = (2^{phi(t)} - 1)/t where t = 2*n - 1.

Original entry on oeis.org

1, 1, 3, 9, 7, 93, 315, 17, 3855, 13797, 195, 182361, 41943, 9709, 9256395, 34636833, 31775, 479349, 1857283155, 430185, 26817356775, 102280151421, 372827, 1497207322929, 89756051247, 84215045, 84973577874915, 19991120505, 1205604855, 4885260612740877

Offset: 1

N. J. A. Sloane

nonn

Lerch defines q_u = (u^phi(p) - 1)/p for any odd integer p and for u prime to p. - Altug Alkan, Dec 19 2017

L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 1, p. 109.

T. D. Noe, Table of n, a(n) for n = 1..200
R. C. Read, Letter to N. J. A. Sloane, Oct. 29, 1976

Cf. A000010.

Table[(2^EulerPhi[n] - 1)/n, {n, 1, 40, 2}] (* T. D. Noe, Jun 26 2012 *)

a(n) = (2^eulerphi(2*n-1) - 1)/(2*n-1) \\ Andrew Howroyd, Dec 18 2017

A193198 G.f.: A(x) = Sum_{n>=0} x^n/(1 - 3^n*x)^n.

Original entry on oeis.org

1, 1, 4, 28, 352, 7696, 296704, 19845568, 2325071872, 472050401536, 167325747134464, 102717666720160768, 109887628080679616512, 203517277347030338768896, 656102983404750860283019264, 3660938644168893995628877692928

Offset: 0

Paul D. Hanna, Jul 17 2011

nonn

			G.f.: A(x) = 1 + x + 4*x^2 + 28*x^3 + 352*x^4 + 7696*x^5 +...
where:
A(x) = 1 + x/(1-3*x) + x^2/(1-9*x)^2 + x^3/(1-27*x)^3 + x^4/(1-81*x)^4 +...

Seiichi Manyama, Table of n, a(n) for n = 0..90

Cf. A000684, A135079, A193199.

{a(n)=local(A=1);A=1+sum(m=1,n,x^m/(1-3^m*x +x*O(x^n))^m);polcoeff(A,n)}

{a(n)=if(n==0,1,sum(k=0,n-1,binomial(n-1,k)*3^(k*(n-k))))}

a(n) = Sum_{k=0..n-1} binomial(n-1,k)*3^(k*(n-k)) for n>0 with a(0)=1.

A000222 Coefficients of ménage hit polynomials.

Original entry on oeis.org

0, 0, 1, 3, 6, 38, 213, 1479, 11692, 104364, 1036809, 11344859, 135548466, 1755739218, 24504637741, 366596136399, 5852040379224, 99283915922264, 1783921946910417, 33840669046326579, 675849838112277598, 14174636583759324798

Offset: 0

N. J. A. Sloane, Simon Plouffe

nonn

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 198.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
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..150
R. C. Read, Letter to N. J. A. Sloane, Oct. 29, 1976

A diagonal of A058057.

max = 30; f[x_, y_] := Sum[n!*((x*y)^n/(1+x*(y-1))^(2*n+1)), {n, 0, max}]; t = MapIndexed[Take[#1, First[#2]]&, CoefficientList[Series[f[x, y], {x, 0, max}, {y, 0, max}], {x, y}]] ; a[0] = a[1] = 0; a[n_] := t[[n+1, n-1]]; Table[a[n], {n, 0, max}] (* Jean-François Alcover, Mar 11 2014, after Vladeta Jovovic *)

a(n) ~ 2/exp(2) * n!. - Vaclav Kotesovec, Sep 06 2014

a(n)+2*a(n+p)+a(n+2*p) is divisible by p for any prime p. - Mark van Hoeij, Jun 13 2019

A002537 a(2n) = a(2n-1) + 3a(2n-2), a(2n+1) = 2a(2n) + 3a(2n-1).

Original entry on oeis.org

1, 1, 4, 11, 23, 79, 148, 533, 977, 3553, 6484, 23627, 43079, 157039, 286276, 1043669, 1902497, 6936001, 12643492, 46094987, 84025463, 306335887, 558412276, 2035832213, 3711069041, 13529634721, 24662841844, 89914587851

Offset: 0

N. J. A. Sloane

nonn

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
A. Tarn, Approximations to certain square roots and the series of numbers connected therewith, Mathematical Questions and Solutions from the Educational Times, 1 (1916), 8-12.

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
R. C. Read, Letter to N. J. A. Sloane, Oct. 29, 1976
Albert Tarn, Approximations to certain square roots and the series of numbers connected therewith [Annotated scanned copy]
Index entries for linear recurrences with constant coefficients, signature (0, 8, 0, -9).

A002537:=(1+z-4*z**2+3*z**3)/(1-8*z**2+9*z**4); # conjectured by Simon Plouffe in his 1992 dissertation

LinearRecurrence[{0,8,0,-9},{1,1,4,11},40] (* Harvey P. Dale, Jul 24 2012 *)

a(n)=8a(n-2)-9a(n-4). - Mario Catalani (mario.catalani(AT)unito.it), Apr 24 2003
G.f.: (1+x-4x^2+3x^3)/(1-8x^2+9x^4). a(n)/A002536(n) converges to sqrt(7). - Mario Catalani (mario.catalani(AT)unito.it), Apr 24 2003
a(n+1) = x^n + (-1)^n*(x-2)^n where x = (1+sqrt(7)) and the term is divided by 2 for a(2) and a(3), 4 for a(4) and a(5)... 2^n for a(2n) and a(2n+1). - Ben Paul Thurston, Aug 30 2006

More terms from James Sellers, Sep 08 2000

A000450 Coefficients of ménage hit polynomials.

Original entry on oeis.org

1, 10, 50, 385, 3130, 28764, 291900, 3249210, 39367395, 515874470, 7270929806, 109691447395, 1763782644020, 30114243100760, 544123405603800, 10373304279494964, 208092476689208805, 4381794437494059810, 96635643877147885450, 2227562535558247510885

Offset: 4

N. J. A. Sloane, Simon Plouffe

nonn

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 198.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. C. Read, Letter to N. J. A. Sloane, Oct. 29, 1976

A diagonal of A058057.

max = 23; f[x_, y_] := Sum[n!*((x*y)^n/(1 + x*(y-1))^(2*n+1)), {n, 0, max} ]; t = MapIndexed[Take[#1, #2[[1]]]&, CoefficientList[Series[f[x, y], {x, 0, max}, {y, 0, max}], {x, y}]]; a[n_] := t[[n+1, n-3]]; Table[a[n], {n, 4, max}] (* Jean-François Alcover, Feb 08 2016, after Vladeta Jovovic in A058057 *)

A360704 Expansion of Sum_{k>=0} (x * (1 + 2^k * x))^k.

Original entry on oeis.org

1, 1, 3, 9, 41, 257, 2209, 27009, 455553, 10831873, 360452609, 16786663425, 1102243190785, 101146710556673, 13109796072955905, 2379217548538511361, 609386444958743363585, 219178211386515281412097, 111098724276069341895720961, 79284929294467154275606200321

Offset: 0

Seiichi Manyama, Feb 17 2023

nonn

Cf. A117402, A360592, A360696, A360705.

Cf. A000684.

my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (x*(1+2^k*x))^k))

a(n) = sum(k=0, n\2, 2^(k*(n-k))*binomial(n-k, k));

a(n) = Sum_{k=0..floor(n/2)} 2^(k*(n-k)) * binomial(n-k,k).

A006536 Switching classes of digraphs.

Original entry on oeis.org

2, 6, 38, 684, 50224, 13946352, 14061131152, 50947324188128, 666754350781269760, 31763991166456912086656, 5550343429517595744902558208, 3581353054755185129132443153728000

Offset: 1

N. J. A. Sloane

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

R. C. Read, Letter to N. J. A. Sloane, Oct. 29, 1976

Cf. A002854, A049313.

More terms from Vladeta Jovovic, Jul 22 2000

cp's OEIS Frontend

A000686 Number of 4-colored labeled graphs on n nodes, divided by 4.

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A001226 Lerch's function q_2(n) = (2^{phi(t)} - 1)/t where t = 2*n - 1.

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A193198 G.f.: A(x) = Sum_{n>=0} x^n/(1 - 3^n*x)^n.

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A000222 Coefficients of ménage hit polynomials.

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A002537 a(2n) = a(2n-1) + 3a(2n-2), a(2n+1) = 2a(2n) + 3a(2n-1).

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A000450 Coefficients of ménage hit polynomials.

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A360704 Expansion of Sum_{k>=0} (x * (1 + 2^k * x))^k.

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PARI

PARI

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A006536 Switching classes of digraphs.

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