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
A364285
Number T(n,k) of partitions of n with largest part k where each block of part i with multiplicity j is marked with a word of length i*j over an n-ary alphabet whose letters appear in alphabetical order and all n letters occur exactly once in the partition; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
Original entry on oeis.org
1, 0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 7, 4, 1, 0, 1, 15, 20, 5, 1, 0, 1, 31, 81, 30, 6, 1, 0, 1, 63, 287, 175, 42, 7, 1, 0, 1, 127, 952, 841, 280, 56, 8, 1, 0, 1, 255, 3025, 4545, 1638, 420, 72, 9, 1, 0, 1, 511, 9370, 23820, 10333, 2730, 600, 90, 10, 1
Offset: 0
T(4,1) = 1: 1111abcd.
T(4,2) = 7: 2ab11cd, 2ac11bd, 2ad11bc, 2bc11ad, 2bd11ac, 2cd11ab, 22abcd.
T(4,3) = 4: 3abc1d, 3abd1c, 3acd1b, 3bcd1a.
T(4,4) = 1: 4abcd.
Triangle T(n,k) begins:
1;
0, 1;
0, 1, 1;
0, 1, 3, 1;
0, 1, 7, 4, 1;
0, 1, 15, 20, 5, 1;
0, 1, 31, 81, 30, 6, 1;
0, 1, 63, 287, 175, 42, 7, 1;
0, 1, 127, 952, 841, 280, 56, 8, 1;
0, 1, 255, 3025, 4545, 1638, 420, 72, 9, 1;
...
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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:
T:= (n, k)-> b(n, k)-`if`(k=0, 0, b(n, k-1)):
seq(seq(T(n, k), k=0..n), n=0..12);
A364323
Number of partitions of 2n into n parts where each block of part i with multiplicity j is marked with a word of length i*j over a (2n)-ary alphabet whose letters appear in alphabetical order and all 2n letters occur exactly once in the partition.
Original entry on oeis.org
1, 1, 5, 76, 785, 12181, 377708, 8009002, 171155505, 4073421919, 168532394115, 6213455777530, 198071252771780, 6383569557705276, 204582355050315856, 8766238064421938746, 446196770370016437201, 20584924968627941009331, 920598569147050035793061
Offset: 0
a(2) = 5: 3abc1d, 3abd1c, 3acd1b, 3bcd1a, 22abcd.
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b:= proc(n, i) option remember; expand(`if`(n=0, 1, `if`(i<1, 0,
add(b(n-i*j, i-1)*x^j*binomial(n, i*j), j=0..n/i))))
end:
a:= n-> coeff(b(2*n$2), x, n):
seq(a(n), n=0..23);
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b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i < 1, 0,
Sum[b[n - i*j, i - 1]*x^j*Binomial[n, i*j], {j, 0, n/i}]]]];
a[n_] := Coefficient[b[2n, 2n], x, n];
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Nov 29 2023, from Maple code *)
Showing 1-3 of 3 results.
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