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
A326654
Number of colored integer partitions of n using all colors of an initial interval of the color palette such that each block of part i with multiplicity j has a pattern of i*j colors in (weakly) increasing order.
Original entry on oeis.org
1, 1, 4, 16, 70, 356, 1928, 11428, 69772, 471200, 3350320, 25067040, 195361800, 1559368544, 13165162256, 116528178688, 1074460079840, 10203335290992, 99238550358000, 979455883492672, 10002569256970848, 105957081274335392, 1164108439659208704
Offset: 0
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b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add((t->
b(n-t, min(n-t, i-1), k)*binomial(k+t-1, t))(i*j), j=0..n/i)))
end:
a:= n-> add(add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k), k=0..n):
seq(a(n), n=0..25);
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b[n_, i_, k_] := b[n, i, k] = If[n==0, 1, If[i < 1, 0, Sum[With[{t = i j}, b[n - t, Min[n - t, i - 1], k]*Binomial[k + t - 1, t]], {j, 0, n/i}]]];
a[n_] := Sum[Sum[b[n, n, k-i] (-1)^i Binomial[k, i], {i, 0, k}], {k, 0, n}];
a /@ Range[0, 25] (* Jean-François Alcover, Dec 15 2020, after Alois P. Heinz *)
A326656
Total number of colors in all colored integer partitions of n using all colors of an initial interval of the color palette such that each block of part i with multiplicity j has a pattern of i*j colors in (weakly) increasing order.
Original entry on oeis.org
0, 1, 6, 34, 191, 1208, 7840, 54152, 377396, 2868528, 22719712, 187318016, 1594593876, 13795808224, 125535871760, 1192418406800, 11747646588912, 118703814213296, 1223646182128656, 12755728151091424, 137199027931128992, 1527404635450188128, 17599899510211606336
Offset: 0
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b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add((t->
b(n-t, min(n-t, i-1), k)*binomial(k+t-1, t))(i*j), j=0..n/i)))
end:
a:= n-> add(k*add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k), k=0..n):
seq(a(n), n=0..25);
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b[n_, i_, k_] := b[n, i, k] = If[n==0, 1, If[i < 1, 0, Sum[With[{t = i j}, b[n - t, Min[n - t, i - 1], k]*Binomial[k + t - 1, t]], {j, 0, n/i}]]];
a[n_] := Sum[k Sum[b[n, n, k-i] (-1)^i Binomial[k, i], {i, 0, k}], {k, 0, n}];
a /@ Range[0, 25] (* Jean-François Alcover, Dec 15 2020, after Alois P. Heinz *)
A328158
Number of colored integer partitions of 2n using all colors of an n-set such that each block of part i with multiplicity j has a pattern of i*j colors in (weakly) increasing order.
Original entry on oeis.org
1, 2, 22, 428, 11595, 416010, 18283208, 945843148, 58252818659, 4087684096527, 317934667075551, 28164509102578546, 2781331187964705790, 294700331738309167806, 33811410232219114946609, 4297801013746798965557794, 593679426174377865941838598
Offset: 0
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b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add((t->
b(n-t, min(n-t, i-1), k)*binomial(k+t-1, t))(i*j), j=0..n/i)))
end:
a:= n-> add(b(2*n$2, n-i)*(-1)^i*binomial(n, i), i=0..n):
seq(a(n), n=0..18);
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b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[b[n - i j, Min[n - i j, i - 1], k] Binomial[k + i j - 1, i j], {j, 0, n/i}]]];
a[n_] := Sum[b[2n, 2n, n-i] (-1)^i Binomial[n, i], {i, 0, n}];
a /@ Range[0, 18] (* Jean-François Alcover, May 08 2020, after Maple *)
Showing 1-4 of 4 results.
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