A327631
Number T(n,k) of parts in all proper k-times partitions of n; triangle T(n,k), n >= 1, 0 <= k <= n-1, read by rows.
Original entry on oeis.org
1, 1, 2, 1, 5, 3, 1, 11, 21, 12, 1, 19, 61, 74, 30, 1, 34, 205, 461, 432, 144, 1, 53, 474, 1652, 2671, 2030, 588, 1, 85, 1246, 6795, 17487, 23133, 15262, 3984, 1, 127, 2723, 20966, 76264, 148134, 158452, 88194, 19980, 1, 191, 6277, 69812, 360114, 1002835, 1606434, 1483181, 734272, 151080
Offset: 1
T(4,0) = 1:
4 (1 part).
T(4,1) = 11 = 2 + 2 + 3 + 4:
4-> 31 (2 parts)
4-> 22 (2 parts)
4-> 211 (3 parts)
4-> 1111 (4 parts)
T(4,2) = 21 = 3 + 4 + 3 + 3 + 4 + 4:
4-> 31 -> 211 (3 parts)
4-> 31 -> 1111 (4 parts)
4-> 22 -> 112 (3 parts)
4-> 22 -> 211 (3 parts)
4-> 22 -> 1111 (4 parts)
4-> 211-> 1111 (4 parts)
T(4,3) = 12 = 4 + 4 + 4:
4-> 31 -> 211 -> 1111 (4 parts)
4-> 22 -> 112 -> 1111 (4 parts)
4-> 22 -> 211 -> 1111 (4 parts)
Triangle T(n,k) begins:
1;
1, 2;
1, 5, 3;
1, 11, 21, 12;
1, 19, 61, 74, 30;
1, 34, 205, 461, 432, 144;
1, 53, 474, 1652, 2671, 2030, 588;
1, 85, 1246, 6795, 17487, 23133, 15262, 3984;
1, 127, 2723, 20966, 76264, 148134, 158452, 88194, 19980;
...
-
b:= proc(n, i, k) option remember; `if`(n=0, [1, 0],
`if`(k=0, [1, 1], `if`(i<2, 0, b(n, i-1, k))+
(h-> (f-> f +[0, f[1]*h[2]/h[1]])(h[1]*
b(n-i, min(n-i, i), k)))(b(i$2, k-1))))
end:
T:= (n, k)-> add(b(n$2, i)[2]*(-1)^(k-i)*binomial(k, i), i=0..k):
seq(seq(T(n, k), k=0..n-1), n=1..12);
-
b[n_, i_, k_] := b[n, i, k] = If[n == 0, {1, 0}, If[k == 0, {1, 1}, If[i < 2, 0, b[n, i - 1, k]] + Function[h, Function[f, f + {0, f[[1]]*h[[2]]/ h[[1]]}][h[[1]]*b[n - i, Min[n - i, i], k]]][b[i, i, k - 1]]]];
T[n_, k_] := Sum[b[n, n, i][[2]]*(-1)^(k - i)*Binomial[k, i], {i, 0, k}];
Table[T[n, k], {n, 1, 12}, {k, 0, n - 1}] // Flatten (* Jean-François Alcover, Jan 07 2020, after Alois P. Heinz *)
A327644
Number of proper many times partitions of n.
Original entry on oeis.org
1, 1, 2, 4, 14, 44, 244, 1196, 9366, 62296, 584016, 5120548, 60244028, 627389924, 8378159376, 106097674780, 1652301306958, 23655318730276, 409987534384504, 6742903763089068, 130675390985884516, 2396246933608687036, 50636625943991790784, 1032841246318579471748
Offset: 0
a(3) = 4: 3, 3->21, 3->111, 3->21->111.
a(4) = 14: 4, 4->31, 4->22, 4->211, 4->1111, 4->31->211, 4->31->1111, 4->22->112, 4->22->211, 4->22->1111, 4->211->1111, 4->31->211->1111, 4->22->112->1111, 4->22->211->1111.
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b:= proc(n, i, k) option remember; `if`(n=0 or k=0, 1, `if`(i>1,
b(n, i-1, k), 0) +b(i$2, k-1)*b(n-i, min(n-i, i), k))
end:
a:= n-> add(add(b(n$2, i)*(-1)^(k-i)*
binomial(k, i), i=0..k), k=0..max(0, n-1)):
seq(a(n), n=0..23);
-
b[n_, i_, k_] := b[n, i, k] = If[n == 0 || k == 0, 1, If[i > 1, b[n, i - 1, k], 0] + b[i, i, k - 1] b[n - i, Min[n - i, i], k]];
a[n_] := Sum[b[n, n, i] (-1)^(k - i) Binomial[k, i], {k, 0, Max[0, n - 1]}, {i, 0, k}];
a /@ Range[0, 23] (* Jean-François Alcover, Dec 09 2020, after Alois P. Heinz *)
A327647
Number of parts in all proper many times partitions of n into distinct parts.
Original entry on oeis.org
0, 1, 1, 3, 6, 15, 38, 133, 446, 1913, 7492, 36293, 175904, 953729, 5053294, 31353825, 188697696, 1268175779, 8356974190, 61775786301, 448436391810, 3579695446911, 27848806031468, 239229189529685, 2019531300063238, 18477179022470655, 165744369451885256
Offset: 0
a(4) = 6 = 1 + 2 + 3, counting the (final) parts in 4, 4->31, 4->31->211.
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b:= proc(n, i, k) option remember; `if`(n=0, [1, 0],
`if`(k=0, [1, 1], `if`(i*(i+1)/2 (f-> f +[0, f[1]*h[2]/h[1]])(h[1]*
b(n-i, min(n-i, i-1), k)))(b(i$2, k-1)))))
end:
a:= n-> add(add(b(n$2, i)[2]*(-1)^(k-i)*
binomial(k, i), i=0..k), k=0..max(0, n-2)):
seq(a(n), n=0..27);
-
b[n_, i_, k_] := b[n, i, k] = If[n == 0, {1, 0}, If[k == 0, {1, 1}, If[i(i + 1)/2 < n, 0, b[n, i - 1, k] + Function[h, Function[f, f + {0, f[[1]]* h[[2]]/h[[1]]}][h[[1]] b[n-i, Min[n - i, i - 1], k]]][b[i, i, k - 1]]]]];
a[n_] := Sum[Sum[b[n, n, i][[2]] (-1)^(k-i) Binomial[k, i], {i, 0, k}], {k, 0, Max[0, n-2]}];
a /@ Range[0, 27] (* Jean-François Alcover, May 03 2020, after Maple *)
Showing 1-3 of 3 results.
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