A327605
Number of parts in all twice partitions of n where both partitions are strict.
Original entry on oeis.org
0, 1, 1, 5, 8, 15, 28, 49, 86, 156, 259, 412, 679, 1086, 1753, 2826, 4400, 6751, 10703, 16250, 24757, 38047, 57459, 85861, 129329, 192660, 286177, 424358, 624510, 915105, 1347787, 1961152, 2847145, 4144089, 5988205, 8638077, 12439833, 17837767, 25536016
Offset: 0
a(3) = 5 = 1+2+2 counting the parts in 3, 21, 2|1.
-
g:= proc(n, i) option remember; `if`(i*(i+1)/2 f+
[0, f[1]])(g(n-i, min(n-i, i-1)))))
end:
b:= proc(n, i) option remember; `if`(i*(i+1)/2 (f-> f+[0, f[1]*
h[2]/h[1]])(b(n-i, min(n-i, i-1))*h[1]))(g(i$2))))
end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..42);
-
b[n_, i_, k_] := b[n, i, k] = With[{}, If[n == 0, Return@{1, 0}]; If[k == 0, Return@{1, 1}]; If[i (i + 1)/2 < n, Return@{0, 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_] := b[n, n, 2][[2]];
a /@ Range[0, 42] (* Jean-François Alcover, Jun 03 2020, after Alois P. Heinz in A327622 *)
A327618
Number A(n,k) of parts in all k-times partitions of n; square array A(n,k), n>=0, k>=0, read by antidiagonals.
Original entry on oeis.org
0, 0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 5, 6, 1, 0, 1, 7, 14, 12, 1, 0, 1, 9, 25, 44, 20, 1, 0, 1, 11, 39, 109, 100, 35, 1, 0, 1, 13, 56, 219, 315, 274, 54, 1, 0, 1, 15, 76, 386, 769, 1179, 581, 86, 1, 0, 1, 17, 99, 622, 1596, 3643, 3234, 1417, 128, 1, 0, 1, 19, 125, 939, 2960, 9135, 12336, 10789, 2978, 192, 1
Offset: 0
A(2,2) = 5 = 1+2+2 counting the parts in 2, 11, 1|1.
Square array A(n,k) begins:
0, 0, 0, 0, 0, 0, 0, 0, ...
1, 1, 1, 1, 1, 1, 1, 1, ...
1, 3, 5, 7, 9, 11, 13, 15, ...
1, 6, 14, 25, 39, 56, 76, 99, ...
1, 12, 44, 109, 219, 386, 622, 939, ...
1, 20, 100, 315, 769, 1596, 2960, 5055, ...
1, 35, 274, 1179, 3643, 9135, 19844, 38823, ...
1, 54, 581, 3234, 12336, 36911, 93302, 208377, ...
-
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:
A:= (n, k)-> b(n$2, k)[2]:
seq(seq(A(n, d-n), n=0..d), d=0..14);
-
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]]]];
A[n_, k_] := b[n, n, k][[2]];
Table[A[n, d-n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, Apr 30 2020, after Alois P. Heinz *)
A327607
Number of parts in all twice partitions of n where the first partition is strict.
Original entry on oeis.org
0, 1, 3, 11, 21, 58, 128, 276, 516, 1169, 2227, 4324, 8335, 15574, 29116, 55048, 97698, 176291, 323277, 563453, 1005089, 1770789, 3076868, 5293907, 9184885, 15668638, 26751095, 45517048, 76882920, 128738414, 217219751, 360525590, 599158211, 995474365
Offset: 0
a(3) = 11 = 1+2+3+2+3 counting the parts in 3, 21, 111, 2|1, 11|1.
-
g:= proc(n) option remember; (p-> [p(n), add(p(n-j)*
numtheory[tau](j), j=1..n)])(combinat[numbpart])
end:
b:= proc(n, i) option remember; `if`(i*(i+1)/2 (f-> f+[0, f[1]*
h[2]/h[1]])(b(n-i, min(n-i, i-1))*h[1]))(g(i))))
end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..37);
-
g[n_] := g[n] = {PartitionsP[n], Sum[PartitionsP[n - j] DivisorSigma[0, j], {j, 1, n}]};
b[n_, i_] := b[n, i] = If[i(i+1)/2 < n, 0, If[n == 0, {1, 0}, Module[{h, f}, h = g[i]; f = b[n - i, Min[n - i, i - 1]] h[[1]]; b[n, i - 1] + f + {0, f[[1]] h[[2]] / h[[1]]}]]];
a[n_] := b[n, n][[2]];
a /@ Range[0, 37] (* Jean-François Alcover, Dec 05 2020, after Alois P. Heinz *)
A327608
Number of parts in all twice partitions of n where the second partition is strict.
Original entry on oeis.org
0, 1, 3, 8, 17, 34, 74, 134, 254, 470, 842, 1463, 2620, 4416, 7545, 12749, 21244, 34913, 57868, 93583, 151963, 244602, 391206, 620888, 987344, 1550754, 2435087, 3804354, 5920225, 9162852, 14179754, 21785387, 33436490, 51121430, 77935525, 118384318, 179617794
Offset: 0
a(3) = 8 = 1+2+2+3 counting the parts in 3, 21, 2|1, 1|1|1.
-
g:= proc(n, i) option remember; `if`(i*(i+1)/2 f+
[0, f[1]])(g(n-i, min(n-i, i-1)))))
end:
b:= proc(n, i) option remember; `if`(n=0, [1, 0],
`if`(i<2, 0, b(n, i-1)) +(h-> (f-> f +[0, f[1]*
h[2]/h[1]])(b(n-i, min(n-i, i))*h[1]))(g(i$2)))
end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..37);
-
g[n_, i_] := g[n, i] = If[i(i+1)/2 < n, 0, If[n == 0, {1, 0}, g[n, i - 1] + Function[f, f + {0, f[[1]]}][g[n - i, Min[n - i, i - 1]]]]];
b[n_, i_] := b[n, i] = If[n == 0, {1, 0}, If[i < 2, 0, b[n, i - 1]] + Module[{h, f}, h = g[i, i]; f = b[n - i, Min[n - i, i]] h[[1]]; f + {0, f[[1]] h[[2]]/h[[1]]}]];
a[n_] := b[n, n][[2]];
a /@ Range[0, 37] (* Jean-François Alcover, Dec 05 2020, after Alois P. Heinz *)
A327590
Number of partitions in all twice partitions of n.
Original entry on oeis.org
0, 1, 4, 10, 29, 63, 164, 339, 797, 1640, 3578, 7139, 15210, 29621, 60381, 117116, 232523, 442388, 863069, 1621560, 3105993, 5785525, 10894394, 20083143, 37434186, 68344449, 125774280, 228088127, 415668548, 747660318, 1351364816, 2413792653, 4327245170
Offset: 0
a(3) = 10 = 1+1+1+2+2+3 counting the partitions in 3, 21, 111, 2|1, 11|1, 1|1|1.
-
b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, 0, b(n, i-1)+
(p-> p+[0, p[1]])(combinat[numbpart](i)*b(n-i, min(n-i, i)))))
end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..42);
-
b[n_, i_] := b[n, i] = If[n == 0, {1, 0}, If[i<1, {0, 0}, b[n, i-1] + Function[p, p + {0, p[[1]]}][PartitionsP[i] b[n-i, Min[n-i, i]]]]];
a[n_] := b[n, n][[2]];
a /@ Range[0, 42] (* Jean-François Alcover, Dec 16 2020, after Alois P. Heinz *)
Showing 1-5 of 5 results.
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