A271619
Twice partitioned numbers where the first partition is strict.
Original entry on oeis.org
1, 1, 2, 5, 8, 18, 34, 65, 109, 223, 386, 698, 1241, 2180, 3804, 6788, 11390, 19572, 34063, 56826, 96748, 163511, 272898, 452155, 755928, 1244732, 2054710, 3382147, 5534696, 8992209, 14733292, 23763685, 38430071, 62139578, 99735806, 160183001, 256682598
Offset: 0
a(6)=34: {(6);(5)(1),(51);(4)(2),(42);(4)(11),(41)(1),(411);(33);(3)(2)(1),(31)(2),(32)(1),(321);(3)(11)(1),(31)(11),(311)(1),(3111);(22)(2),(222);(21)(2)(1),(22)(11),(211)(2),(221)(1),(2211);(21)(11)(1),(111)(2)(1),(211)(11),(1111)(2),(2111)(1),(21111);(111)(11)(1),(1111)(11),(11111)(1),(111111)}
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b:= proc(n, i) option remember; `if`(n>i*(i+1)/2, 0,
`if`(n=0, 1, b(n, i-1) +`if`(i>n, 0,
b(n-i, i-1)*combinat[numbpart](i))))
end:
a:= n-> b(n$2):
seq(a(n), n=0..50); # Alois P. Heinz, Apr 11 2016
-
With[{n = 50}, CoefficientList[Series[Product[(1 + PartitionsP[i] x^i), {i, 1, n}], {x, 0, n}], x]]
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.
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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);
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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 *)
A327594
Number of parts in all twice partitions of n.
Original entry on oeis.org
0, 1, 5, 14, 44, 100, 274, 581, 1417, 2978, 6660, 13510, 29479, 58087, 120478, 236850, 476913, 916940, 1812498, 3437043, 6657656, 12512273, 23780682, 44194499, 83117200, 152837210, 283431014, 517571202, 949844843, 1719175176, 3127751062, 5618969956, 10133425489
Offset: 0
a(2) = 5 = 1+2+2 counting the parts in 2, 11, 1|1.
a(3) = 14 = 1+2+3+2+3+3: 3, 21, 111, 2|1, 11|1, 1|1|1.
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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`(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)))
end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..37);
# second Maple program:
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-> b(n$2, 2)[2]:
seq(a(n), n=0..37);
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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_] := b[n, n, 2][[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 *)
A327552
Number of partitions in all twice partitions of n where the first partition is strict.
Original entry on oeis.org
0, 1, 2, 7, 11, 29, 63, 125, 225, 489, 930, 1704, 3260, 5859, 10868, 20026, 35062, 61660, 111789, 191119, 337432, 585847, 1003876, 1705380, 2921394, 4930357, 8311554, 14013583, 23435178, 38849655, 64847870, 106784912, 175699558, 289676875, 472418418, 772944773
Offset: 0
a(3) = 7 = 1+1+1+2+2 counting the partitions in 3, 21, 111, 2|1, 11|1.
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b:= proc(n, i) option remember; `if`(i*(i+1)/2p+[0, p[1]])(
combinat[numbpart](i)*b(n-i, min(n-i, i-1)))))
end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..36);
-
b[n_, i_] := b[n, i] = If[i(i+1)/2Jean-François Alcover, Dec 17 2020, after Alois P. Heinz *)
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