A102736 -id:A102736 - cp's OEIS frontend

A113775 Number of sets of lists (cf. A000262) whose list sizes are not a multiple of 3.

1, 1, 3, 7, 49, 321, 2131, 19783, 195777, 2101249, 25721731, 340358151, 4902173233, 75688032577, 1253701725459, 22347046050631, 418439924732161, 8318748086461953, 175769214730290307, 3871849719998940679, 89734800330818444721, 2187944831367633226561

Offset: 0

Vladeta Jovovic, Jan 19 2006

Alois P. Heinz, Table of n, a(n) for n = 0..445

Cf. A000726, A003724, A115276, A000246, A102736, A088009, A001590.

nmax := 30: B := x*(1+x)/(1-x^3) : egf := 0 : for i from 0 to nmax do egf := convert(egf+taylor(B^i,x=0,nmax+1)/i!,polynom) : od: for i from 0 to nmax do printf("%d ", i!*coeftayl(egf,x=0,i)) ; od: # R. J. Mathar, Feb 06 2008
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add(`if`(0=
      irem(j, 3), 0, a(n-j)*j!*binomial(n-1, j-1)), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, May 10 2016

CoefficientList[Series[E^(x*(1+x)/(1-x^3)), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Sep 25 2013 *)

E.g.f.: exp(x*(1+x)/(1-x^3)).
a(n) = a(n-1) + 2*(n-1)*a(n-2) + 2*(n-3)*(n-2)*(n-1)*a(n-3) + 2*(n-3)*(n-2)*(n-1)*a(n-4) + (n-4)*(n-3)*(n-2)*(n-1)*a(n-5) - (n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*a(n-6). - Vaclav Kotesovec, Sep 25 2013
a(n) ~ 6^(-1/4) * n^(n-1/4) * exp(2/3*sqrt(6*n)-n) * (1 - 43/(48*sqrt(6*n))). - Vaclav Kotesovec, Sep 25 2013

2 more terms from R. J. Mathar, Feb 06 2008

A247007 Number of permutations on [n] admitting a ninth root.

Original entry on oeis.org

1, 1, 2, 4, 16, 80, 400, 2800, 22400, 179200, 1792000, 19712000, 216832000, 2818816000, 39463424000, 552487936000, 8839806976000, 150276718592000, 2554704216064000, 48539380105216000, 970787602104320000, 19415752042086400000, 427146544925900800000, 9824370533295718400000, 225960522265801523200000, 5649013056645038080000000, 146874339472770990080000000, 3818732827816549381939200000

Offset: 0

Alois P. Heinz, Sep 09 2014

nonn

Differs from A102736 first at n=27.

Alois P. Heinz, Table of n, a(n) for n = 0..300
H. S. Wilf, Generatingfunctionology, 2nd edn., Academic Press, NY, 1994, Theorem 4.8.2.

Column k=9 of A247005.

with(combinat): with(numtheory): with(padic):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      `if`(irem(j, mul(p^ordp(9, p), p=factorset(i)))=0, (i-1)!^j*
      multinomial(n, n-i*j, i$j)/j!*b(n-i*j, i-1), 0), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..27);

A113774 Number of partitions of {1,...,n} into block sizes not a multiple of 3.

Original entry on oeis.org

1, 1, 2, 4, 11, 32, 112, 415, 1732, 7678, 37115, 190016, 1039546, 5996083, 36528196, 233492044, 1564012751, 10940385668, 79762304116, 604791685063, 4760047233424, 38825234812882, 327641201731475, 2856835856307428, 25702896025566886, 238331921722835203

Offset: 0

Vladeta Jovovic, Jan 19 2006

Alois P. Heinz, Table of n, a(n) for n = 0..500

Cf. A003724, A115276, A000246, A102736, A088009, A001590.

nmax := 30: B := add(op(1+(i mod 3),[0,1,1])*x^i/i!,i=0..nmax) : egf := 0 : for i from 0 to nmax do egf := convert(egf+taylor(B^i,x=0,nmax+1)/i!,polynom) : od: for i from 0 to nmax do printf("%d ", i!*coeftayl(egf,x=0,i)) ; od: # R. J. Mathar, Feb 06 2008
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add(`if`(
      irem(j, 3)=0, 0, binomial(n-1, j-1)*a(n-j)), j=1..n))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Mar 17 2015

a=Sum[x^(3i)/(3i)!,{i,1,20}]; Range[0, 20]! CoefficientList[Series[Exp[Exp[x] - 1 - a], {x, 0, 20}], x] (* Geoffrey Critzer, Jan 02 2011 *)

E.g.f.: exp(B(x)), where B(x) is e.g.f. of A011655.

More terms from R. J. Mathar, Feb 06 2008

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A113775 Number of sets of lists (cf. A000262) whose list sizes are not a multiple of 3.

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A247007 Number of permutations on [n] admitting a ninth root.

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Maple

A113774 Number of partitions of {1,...,n} into block sizes not a multiple of 3.

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Maple

Mathematica

Formula

Extensions