A363045
Number of partitions of n whose greatest part is a multiple of 3.
Original entry on oeis.org
1, 0, 0, 1, 1, 2, 4, 5, 7, 11, 14, 19, 27, 34, 45, 60, 77, 99, 130, 163, 208, 265, 333, 417, 526, 651, 810, 1004, 1237, 1519, 1869, 2278, 2780, 3382, 4101, 4958, 5995, 7210, 8669, 10398, 12444, 14859, 17730, 21086, 25057, 29718, 35186, 41584, 49100, 57842, 68075, 79991
Offset: 0
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b:= proc(n, i) option remember; `if`(n=0, 1,
`if`(i<1, 0, b(n, i-1)+b(n-i, min(n-i, i))))
end:
a:= n-> add(b(n-3*i, min(n-3*i, 3*i)), i=0..n/3):
seq(a(n), n=0..60); # Alois P. Heinz, May 14 2023
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b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1] + b[n - i, Min[n - i, i]]]];
a[n_] := Sum[b[n - 3*i, Min[n - 3*i, 3*i]], {i, 0, n/3}];
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Oct 23 2023, after Alois P. Heinz *)
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my(N=60, x='x+O('x^N)); Vec(sum(k=0, N, x^(3*k)/prod(j=1, 3*k, 1-x^j)))
A363046
Number of partitions of n whose greatest part is a multiple of 4.
Original entry on oeis.org
1, 0, 0, 0, 1, 1, 2, 3, 6, 7, 11, 14, 21, 26, 36, 45, 62, 76, 100, 124, 162, 199, 255, 314, 399, 488, 612, 748, 932, 1134, 1400, 1699, 2086, 2520, 3072, 3700, 4488, 5384, 6494, 7766, 9326, 11112, 13283, 15778, 18788, 22245, 26386, 31150, 36825, 43345, 51070, 59953
Offset: 0
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b:= proc(n, i) option remember; `if`(n=0, 1,
`if`(i<1, 0, b(n, i-1)+b(n-i, min(n-i, i))))
end:
a:= n-> add(b(n-4*i, min(n-4*i, 4*i)), i=0..n/4):
seq(a(n), n=0..60); # Alois P. Heinz, May 14 2023
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my(N=60, x='x+O('x^N)); Vec(sum(k=0, N, x^(4*k)/prod(j=1, 4*k, 1-x^j)))
A363047
Number of partitions of n whose greatest part is a multiple of 5.
Original entry on oeis.org
1, 0, 0, 0, 0, 1, 1, 2, 3, 5, 8, 11, 15, 21, 28, 38, 49, 64, 82, 105, 134, 168, 211, 263, 327, 406, 501, 616, 757, 926, 1133, 1378, 1676, 2031, 2460, 2970, 3581, 4306, 5173, 6197, 7419, 8855, 10561, 12565, 14934, 17712, 20982, 24805, 29294, 34529, 40658, 47785, 56104
Offset: 0
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b:= proc(n, i) option remember; `if`(n=0, 1,
`if`(i<1, 0, b(n, i-1)+b(n-i, min(n-i, i))))
end:
a:= n-> add(b(n-5*i, min(n-5*i, 5*i)), i=0..n/5):
seq(a(n), n=0..52); # Alois P. Heinz, May 14 2023
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my(N=60, x='x+O('x^N)); Vec(sum(k=0, N, x^(5*k)/prod(j=1, 5*k, 1-x^j)))
A373029
Triangle T(n,k), n >= 0, 0 <= k <= n, read by rows, where T(n,k) is the number of distinct partitions p of n such that max(p) is a multiple of k.
Original entry on oeis.org
1, 0, 1, 0, 1, 1, 0, 2, 1, 1, 0, 2, 1, 1, 1, 0, 3, 1, 1, 1, 1, 0, 4, 2, 2, 1, 1, 1, 0, 5, 3, 1, 2, 1, 1, 1, 0, 6, 3, 1, 2, 2, 1, 1, 1, 0, 8, 4, 3, 2, 2, 2, 1, 1, 1, 0, 10, 5, 3, 2, 3, 2, 2, 1, 1, 1, 0, 12, 6, 4, 2, 3, 3, 2, 2, 1, 1, 1, 0, 15, 7, 6, 3, 3, 4, 3, 2, 2, 1, 1, 1, 0, 18, 9, 6, 4, 3, 4, 4, 3, 2, 2, 1, 1, 1
Offset: 0
Triangle begins:
1;
0, 1;
0, 1, 1;
0, 2, 1, 1;
0, 2, 1, 1, 1;
0, 3, 1, 1, 1, 1;
0, 4, 2, 2, 1, 1, 1;
0, 5, 3, 1, 2, 1, 1, 1;
0, 6, 3, 1, 2, 2, 1, 1, 1;
0, 8, 4, 3, 2, 2, 2, 1, 1, 1;
0, 10, 5, 3, 2, 3, 2, 2, 1, 1, 1;
0, 12, 6, 4, 2, 3, 3, 2, 2, 1, 1, 1;
0, 15, 7, 6, 3, 3, 4, 3, 2, 2, 1, 1, 1;
0, 18, 9, 6, 4, 3, 4, 4, 3, 2, 2, 1, 1, 1;
Showing 1-4 of 4 results.