A113775
Number of sets of lists (cf. A000262) whose list sizes are not a multiple of 3.
Original entry on oeis.org
1, 1, 3, 7, 49, 321, 2131, 19783, 195777, 2101249, 25721731, 340358151, 4902173233, 75688032577, 1253701725459, 22347046050631, 418439924732161, 8318748086461953, 175769214730290307, 3871849719998940679, 89734800330818444721, 2187944831367633226561
Offset: 0
-
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 *)
A115275
Number of partitions of {1,...,n} into blocks such that no block size is repeated more than 3 times.
Original entry on oeis.org
1, 1, 2, 5, 14, 51, 187, 820, 3670, 18191, 97917, 554500, 3334465, 20871592, 138440031, 972083845, 6985171390, 52194795327, 412903730293, 3313067916192, 28017395030419, 241504438776956, 2189375704925081, 19771679215526507, 187677937412341677
Offset: 0
-
with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1,
`if`(i<1, 0, add(multinomial(n, n-i*j, i$j)/j!*
b(n-i*j, i-1), j=0..min(3, n/i))))
end:
a:= n-> b(n$2):
seq(a(n), n=0..25); # Alois P. Heinz, Sep 17 2015
-
multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[multinomial[n, Join[{n-i*j}, Array[i&, j]]]/j!*b[n - i*j, i-1], {j, 0, Min[3, n/i]}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Oct 29 2015, after Alois P. Heinz *)
A115277
Number of partitions of {1,...,n} into blocks such that no even sized block is repeated.
Original entry on oeis.org
1, 1, 2, 5, 12, 37, 143, 562, 2320, 10941, 54865, 283890, 1604155, 9558226, 58668223, 384572975, 2631778832, 18576630237, 137919691717, 1060303298138, 8415786131309, 69538205444478, 591734670548037, 5194542789203877, 47127033586211659, 438972204436025198
Offset: 0
-
with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
multinomial(n, n-i*j, i$j)/j!*b(n-i*j, i-1), j=0..min(
`if`(irem(i, 2)=0, 1, n), n/i))))
end:
a:= n-> b(n$2):
seq(a(n), n=0..30); # Alois P. Heinz, Mar 08 2015
-
multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[multinomial[n, Join[{n-i*j}, Array[i&, j]]]/j! * b[n-i*j, i-1], {j, 0, Min[If[Mod[i, 2]==0, 1, n], n/i]}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Oct 25 2015, after Alois P. Heinz *)
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
-
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 *)
Showing 1-4 of 4 results.