A178682
The number of functions f:{1,2,...,n}->{1,2,...,n} such that the number of elements that are mapped to m is divisible by m.
Original entry on oeis.org
1, 1, 2, 5, 13, 42, 150, 576, 2266, 9966, 47466, 237019, 1224703, 6429152, 35842344, 212946552, 1325810173, 8488092454, 55276544436, 362961569008, 2465240278980, 17538501945077, 130454679958312, 1002493810175093, 7838007702606372, 61789072382062638
Offset: 0
a(3) = 5 because there are 5 such functions: (1,1,1), (1,2,2), (2,1,2), (2,2,1), (3,3,3).
G.f. = 1 + x + 2*x^2 + 5*x^3 + 13*x^4 + 42*x^5 + 150*x^6 + 576*x^7 + ...
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m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( (&*[(&+[x^(k*j)/Factorial(k*j): k in [0..m]]): j in [1..m]]) )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Jan 26 2019
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b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(b(n-i*j, i-1)*binomial(n, i*j), j=0..n/i)))
end:
a:= n-> b(n$2):
seq(a(n), n=0..30); # Alois P. Heinz, Aug 30 2015
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Range[0,20]! CoefficientList[Series[Product[Sum[x^(j i)/(j i)!,{i,0,20}],{j,1,20}],{x,0,20}],x]
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m=30; my(x='x+O('x^m)); Vec(serlaplace(prod(j=1, m, sum(k=0,m, x^(k*j)/(k*j)!)))) \\ G. C. Greubel, Jan 26 2019
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m = 30; T = taylor(product(sum(x^(k*j)/factorial(k*j) for k in (0..m)) for j in (1..m)), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, Jan 26 2019
A261777
Number of compositions of n where the (possibly scattered) maximal subsequence of part i with multiplicity j is marked with i words of length j over an n-ary alphabet whose letters appear in alphabetical order and all n letters occur exactly once in the composition.
Original entry on oeis.org
1, 1, 3, 19, 115, 951, 10281, 116313, 1436499, 20203795, 338834053, 5824666893, 108142092169, 2118605140237, 44375797806315, 1039641056342619, 25413053107195539, 646983321301050147, 17311013062443870681, 481282277347815404745, 13913039361920333694165
Offset: 0
a(3) = 19: 3a|b|c, 3a|c|b, 3b|a|c, 3b|c|a, 3c|a|b, 3c|b|a, 2a|b1c, 2b|a1c, 2a|c1b, 2c|a1b, 2b|c1a, 2c|b1a, 1a2b|c, 1a2c|b, 1b2a|c, 1b2c|a, 1c2a|b, 1c2b|a, 111abc.
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with(combinat):
b:= proc(n, i, p) option remember; `if`(n=0, p!, `if`(i<1, 0, add(
b(n-i*j, i-1, p+j)/j!*multinomial(n, n-i*j, j$i), j=0..n/i)))
end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..25);
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multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_, p_] := b[n, i, p] = If[n == 0, p!, If[i<1, 0, Sum[b[n - i*j, i-1, p + j]/j!*multinomial[n, Join[{n - i*j}, Table[j, {i}]]], {j, 0, n/i}]]];
a[n_] := b[n, n, 0];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Mar 04 2022, after Alois P. Heinz *)
A327677
Number of colored compositions of n using all colors of an n-set such that any part i has a color pattern of i (distinct) colors in increasing order.
Original entry on oeis.org
1, 1, 3, 13, 71, 481, 3861, 35743, 373591, 4347103, 55671713, 777540523, 11754153869, 191114449579, 3324296885339, 61575268263193, 1209681079172663, 25116819005925409, 549458325556099551, 12629191765880480035, 304232436498153748441, 7663883684722855430077
Offset: 0
a(3) = 13: 3abc, 2ab1c, 2ac1b, 2bc1a, 1a2bc, 1b2ac, 1c2ab, 1a1b1c, 1a1c1b, 1b1a1c, 1b1c1a, 1c1a1b, 1c1b1a.
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b:= proc(n, i, p) option remember; `if`(n=0, p!, `if`(i<1, 0,
add(b(n-i*j, i-1, p+j)*binomial(n, i*j), j=0..n/i)))
end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..23);
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b[n_, i_, p_] := b[n, i, p] = If[n == 0, p!, If[i < 1, 0,
Sum[b[n - i*j, i - 1, p + j]*Binomial[n, i*j], {j, 0, n/i}]]];
a[n_] := b[n, n, 0];
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Aug 01 2021, after Alois P. Heinz *)
Showing 1-3 of 3 results.
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