A242896
Number T(n,k) of compositions of n into k parts with distinct multiplicities, where parts are counted without multiplicities; triangle T(n,k), n>=0, 0<=k<=max{i:A000292(i)<=n}, read by rows.
Original entry on oeis.org
1, 0, 1, 0, 2, 0, 2, 0, 3, 3, 0, 2, 10, 0, 4, 12, 0, 2, 38, 0, 4, 56, 0, 3, 79, 0, 4, 152, 60, 0, 2, 251, 285, 0, 6, 284, 498, 0, 2, 594, 1438, 0, 4, 920, 2816, 0, 4, 1108, 5208, 0, 5, 2136, 11195, 0, 2, 3402, 24094, 0, 6, 4407, 38523, 0, 2, 8350, 85182
Offset: 0
T(5,1) = 2: [1,1,1,1,1], [5].
T(5,2) = 10: [1,1,1,2], [1,1,2,1], [1,2,1,1], [2,1,1,1], [1,2,2], [2,1,2], [2,2,1], [1,1,3], [1,3,1], [3,1,1].
Triangle T(n,k) begins:
1;
0, 1;
0, 2;
0, 2;
0, 3, 3;
0, 2, 10;
0, 4, 12;
0, 2, 38;
0, 4, 56;
0, 3, 79;
0, 4, 152, 60;
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b:= proc(n, i, s) option remember; `if`(n=0, add(j, j=s)!,
`if`(i<1, 0, expand(add(`if`(j>0 and j in s, 0, `if`(j=0, 1, x)*
b(n-i*j, i-1, `if`(j=0, s, s union {j}))/j!), j=0..n/i))))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2, {})):
seq(T(n), n=0..16);
-
b[n_, i_, s_List] := b[n, i, s] = If[n == 0, Total[s]!, If[i<1, 0, Expand[ Sum[ If[j>0 && MemberQ[s, j], 0, If[j == 0, 1, x]*b[n-i*j, i-1, If[j == 0, s, s ~Union~ {j}]]/j!], {j, 0, n/i}]]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, n, {}]]; Table[T[n], {n, 0, 16}] // Flatten (* Jean-François Alcover, Feb 11 2015, after Alois P. Heinz *)
A182473
Number of partitions of n into exactly 2 different parts with distinct multiplicities.
Original entry on oeis.org
0, 0, 0, 0, 1, 3, 3, 8, 9, 12, 16, 22, 20, 31, 35, 34, 44, 51, 53, 62, 65, 68, 87, 86, 89, 95, 118, 108, 126, 127, 138, 142, 162, 154, 188, 160, 193, 189, 227, 204, 228, 221, 258, 238, 282, 247, 311, 272, 320, 284, 344, 318, 373, 327, 398, 334, 407, 380, 450
Offset: 0
a(4) = 1: [2,1,1], part 2 occurs once and part 1 occurs twice.
a(5) = 3: [2,1,1,1], [2,2,1], [3,1,1].
a(6) = 3: [2,1,1,1,1], [3,1,1,1], [4,1,1].
a(7) = 8: [2,1,1,1,1,1], [2,2,1,1,1], [2,2,2,1], [3,1,1,1,1], [3,2,2], [3,3,1], [4,1,1,1], [5,1,1].
Cf.
A242900 (the same for compositions).
A131661
Number of compositions of n such that the cardinality of the set of parts is 2.
Original entry on oeis.org
0, 0, 2, 5, 14, 22, 44, 68, 107, 172, 261, 396, 606, 950, 1414, 2238, 3418, 5411, 8368, 13297, 20840, 33268, 52549, 84120, 133775, 214611, 343025, 551064, 883600, 1421767, 2284870, 3680296, 5924725, 9551161, 15393855, 24834827, 40061700
Offset: 1
Cf.
A242900 (with distinct multiplicities).
-
with(numtheory):
a:= n-> add(add(add(binomial(j+(n-i*j)/d, j), d=select(x->xAlois P. Heinz, Feb 01 2014
-
Rest@ CoefficientList[ Series[ Sum[ x^(i + j)*(x^i + x^j - 2)/((x^i - 1)*(x^j - 1)*(x^i + x^j - 1)), {i, 2, 37}, {j, i - 1}], {x, 0, 37}], x] (* Robert G. Wilson v, Sep 16 2007 *)
A242911
Half the number of compositions of n into exactly two different parts with equal multiplicities.
Original entry on oeis.org
1, 1, 2, 5, 3, 6, 14, 10, 5, 56, 6, 15, 153, 51, 8, 502, 9, 217, 1756, 25, 11, 7023, 264, 30, 24363, 1852, 14, 93629, 15, 6576, 352782, 40, 3827, 1377543, 18, 45, 5200379, 105812, 20, 20063228, 21, 352942, 77607976, 55, 23, 301906830, 5172, 185320, 1166803215
Offset: 3
a(6) = 5 because there are 10 compositions of 6 into exactly two different parts with equal multiplicities: [1,5], [5,1], [2,4], [4,2], [1,1,2,2], [1,2,1,2], [1,2,2,1], [2,1,1,2], [2,1,2,1], [2,2,1,1].
-
a:= n-> add(iquo(d-1, 2)*binomial(2*n/d, n/d),
d=numtheory[divisors](n))/2:
seq(a(n), n=3..60);
-
a[n_] := DivisorSum[n, Quotient[#-1, 2]*Binomial[2n/#, n/#]&]/2; Table[ a[n], {n, 3, 60}] (* Jean-François Alcover, Feb 28 2017, translated from Maple *)
Showing 1-4 of 4 results.