A273528
Triangle T(n,m) (n >= 1, 0 <= m < n) giving coefficients of (n-1)! P_n, where P_n is the polynomial formula for row n of A213086.
Original entry on oeis.org
1, 0, 1, 0, 1, 1, 0, 1, 3, 2, 0, 2, 9, 10, 3, 0, 2, 25, 50, 35, 8, 0, -12, 86, 270, 260, 102, 14, 0, -120, 140, 1344, 2030, 1260, 350, 36, 0, -1248, -1016, 7336, 15862, 13048, 5236, 1024, 78, 0, -9216, -22464, 28528, 124488, 139776, 76104, 22152, 3312, 200, 0, -90720, -322344, 1860, 1036990, 1514205, 1018563, 379890, 80760, 9165, 431
Offset: 1
Row T(5) = {0, 2, 9, 10, 3}, so P_5(k) = (1/4!)(2k + 9k^2 + 10k^3 + 3k^4), which gives 1, 7, 25, 65, 140, 266, ..., that is A001296 (row 5 of A213086), for k >=1.
Triangle begins:
{1},
{0, 1},
{0, 1, 1},
{0, 1, 3, 2},
{0, 2, 9, 10, 3},
{0, 2, 25, 50, 35, 8},
{0, -12, 86, 270, 260, 102, 14},
...
A002219
a(n) is the number of partitions of 2n that can be obtained by adding together two (not necessarily distinct) partitions of n.
Original entry on oeis.org
1, 3, 6, 14, 25, 53, 89, 167, 278, 480, 760, 1273, 1948, 3089, 4682, 7177, 10565, 15869, 22911, 33601, 47942, 68756, 96570, 136883, 189674, 264297, 362995, 499617, 678245, 924522, 1243098, 1676339, 2237625, 2988351, 3957525, 5247500, 6895946, 9070144, 11850304
Offset: 1
Here are the seven partitions of 5: 1^5, 1^3 2, 1 2^2, 1^2 3, 2 3, 1 4, 5. Adding these together in pairs we get a(5) = 25 partitions of 10: 1^10, 1^8 2, 1^6 2^2, etc. (we get all partitions of 10 into parts of size <= 5 - there are 30 such partitions - except for five of them: we do not get 2 4^2, 3^2 4, 2^3 4, 1 3^3, 2^5). - _N. J. A. Sloane_, Jun 03 2012
From _Gus Wiseman_, Oct 27 2022: (Start)
The a(1) = 1 through a(4) = 14 partitions:
(11) (22) (33) (44)
(211) (321) (422)
(1111) (2211) (431)
(3111) (2222)
(21111) (3221)
(111111) (3311)
(4211)
(22211)
(32111)
(41111)
(221111)
(311111)
(2111111)
(11111111)
(End)
- N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
A122768 counts distinct submultisets of partitions.
-
g:= proc(n, i) option remember;
`if`(n=0, 1, `if`(i>1, g(n, i-1), 0)+`if`(i>n, 0, g(n-i, i)))
end:
b:= proc(n, i, s) option remember;
`if`(i=1 and s<>{} or n in s, g(n, i), `if`(i<1 or s={}, 0,
b(n, i-1, s)+ `if`(i>n, 0, b(n-i, i, map(x-> {`if`(x>n-i, NULL,
max(x, n-i-x)), `if`(xn, NULL, max(x-i, n-x))}[], s)))))
end:
a:= n-> b(2*n, n, {n}):
seq(a(n), n=1..25); # Alois P. Heinz, Jul 10 2012
-
b[n_, i_, s_] := b[n, i, s] = If[MemberQ[s, 0 | n], 0, If[n == 0, 1, If[i < 1, 0, b[n, i-1, s] + If[i <= n, b[n-i, i, Select[Flatten[Transpose[{s, s-i}]], 0 <= # <= n-i &]], 0]]]]; A006827[n_] := b[2*n, 2*n, {n}]; a[n_] := PartitionsP[2*n] - A006827[n]; Table[Print[an = a[n]]; an, {n, 1, 25}] (* Jean-François Alcover, Nov 12 2013, after Alois P. Heinz *)
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
subptns[s_]:=primeMS/@Divisors[Times@@Prime/@s];
Table[Length[Select[IntegerPartitions[2n],MemberQ[Total/@subptns[#],n]&]],{n,10}] (* Gus Wiseman, Oct 27 2022 *)
-
from itertools import combinations_with_replacement
from sympy.utilities.iterables import partitions
def A002219(n): return len({tuple(sorted((p+q).items())) for p, q in combinations_with_replacement(tuple(Counter(p) for p in partitions(n)),2)}) # Chai Wah Wu, Sep 20 2023
A357976
Numbers with a divisor having the same sum of prime indices as their quotient.
Original entry on oeis.org
1, 4, 9, 12, 16, 25, 30, 36, 40, 48, 49, 63, 64, 70, 81, 84, 90, 100, 108, 112, 120, 121, 144, 154, 160, 165, 169, 192, 196, 198, 210, 220, 225, 252, 256, 264, 270, 273, 280, 286, 289, 300, 324, 325, 336, 351, 352, 360, 361, 364, 390, 400, 432, 441, 442, 448
Offset: 1
The terms together with their prime indices begin:
1: {}
4: {1,1}
9: {2,2}
12: {1,1,2}
16: {1,1,1,1}
25: {3,3}
30: {1,2,3}
36: {1,1,2,2}
40: {1,1,1,3}
48: {1,1,1,1,2}
49: {4,4}
For example, 40 has factorization 8*5, and both factors have the same sum of prime indices 3, so 40 is in the sequence.
The partitions with these Heinz numbers are counted by
A002219.
Positions of nonzero terms in
A357879.
Cf.
A033879,
A033880,
A064914,
A181819,
A213086,
A235130,
A237194,
A276107,
A300273,
A321144,
A357975.
-
filter:= proc(n) local F,s,t,i,R;
F:= ifactors(n)[2];
F:= map(t -> [numtheory:-pi(t[1]),t[2]], F);
s:= add(t[1]*t[2],t=F)/2;
if not s::integer then return false fi;
try
R:= Optimization:-Maximize(0, [add(F[i][1]*x[i],i=1..nops(F)) = s, seq(x[i]<= F[i][2],i=1..nops(F))], assume=nonnegint, depthlimit=20);
catch "no feasible integer point found; use feasibilitytolerance option to adjust tolerance": return false;
end try;
true
end proc:
filter(1):= true:
select(filter, [$1..1000]); # Robert Israel, Oct 26 2023
-
sumprix[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>k*PrimePi[p]]];
Select[Range[100],MemberQ[sumprix/@Divisors[#],sumprix[#]/2]&]
A213074
Irregular triangle read by rows: coefficients c(n,k) (n>=2, 0<=k<= floor((n-2)/2)) in formula for number of restricted partitions.
Original entry on oeis.org
1, 1, 1, 2, 1, 3, 1, 7, 8, 1, 10, 14, 1, 17, 50, 36, 1, 24, 89, 78, 1, 36, 207, 368, 200, 1, 49, 340, 701, 431, 1, 70, 685, 2190, 2756, 1188, 1, 93, 1075, 3935, 5564, 2658
Offset: 2
Triangle c(n,k) begins:
n\k
- 0 1 2 3 4 5 ...
---------------------------------
2 1
3 1
4 1 2
5 1 3
6 1 7 8
7 1 10 14
8 1 17 50 36
9 1 24 89 78
10 1 36 207 368 200
11 1 49 340 701 431
12 1 70 685 2190 2756 1188
13 1 93 1075 3935 5564 2658
...
-
with(combinat):
h:= proc(n, m) option remember;
`if`(m>1, map(x-> map(y-> sort([x[], y[]]), h(n, 1))[],
h(n, m-1)), `if`(m=1, map(x->map(y-> `if`(y>1, y-1, NULL), x),
{partition(n)[]}), {[]}))
end:
T:= proc(n) local i, g, t;
g:= floor((n+1)/2);
subs(solve({seq(nops(h(n, t))=add(c||i *binomial(t+g, g+i),
i=0..n-g-1), t=1..n-g)}, {seq(c||i, i=0..n-g-1)}),
[seq(c||i, i=0..n-g-1)])[]
end:
seq(T(n), n=2..10); # Alois P. Heinz, Jul 11 2012
-
nmax = 13; mmax = 5;
T[n_, m_] := T[n, m] = Module[{ip, lg, i}, ip = IntegerPartitions[n]; lg = Length[ip]; i[0] = 1; Table[ Join[ Sequence @@ Table[ip[[i[k]]], {k, 1, m}]] // Sort, Evaluate[Sequence @@ Table[{i[k], i[k - 1], lg}, {k, 1, m}]]] // Flatten[#, m - 1] & // Union // Length]; T[_, 0] = 1;
U[n_, m_] := With[{g = Floor[(n + 1)/2]}, If[n == 1, 1, Sum[Binomial[m + g, g + k] c[n, k], {k, 0, n - g - 1}]]];
Do[TT = Table[T[n , m] - U[n , m], {n, 1, nmax}, {m, 0, mm}] // Flatten; c[_, 0] = 1; sol = Solve[Thread[TT == 0]][[1]]; cc = Table[c[n, k], {n, 2, nmax}, {k, 0, Floor[(n - 2)/2]}] /. sol // Flatten; Print[cc], {mm, 2, mmax}];
cc (* Jean-François Alcover, May 25 2016 *)
A002220
a(n) is the number of partitions of 3n that can be obtained by adding together three (not necessarily distinct) partitions of n.
Original entry on oeis.org
1, 4, 10, 30, 65, 173, 343, 778, 1518, 3088, 5609, 10959, 18990, 34441, 58903, 102044, 167499, 282519, 451529, 737208, 1160102, 1836910, 2828466, 4410990, 6670202, 10161240, 15186315, 22758131, 33480869
Offset: 1
From _Gus Wiseman_, Apr 20 2024: (Start)
The a(1) = 1 through a(3) = 10 triquanimous partitions:
(111) (222) (333)
(2211) (3321)
(21111) (32211)
(111111) (33111)
(222111)
(321111)
(2211111)
(3111111)
(21111111)
(111111111)
(End)
- N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
The Heinz numbers of these partitions are given by
A371955.
A371783 counts k-quanimous partitions.
A002221
a(n) is the number of partitions of 4n that can be obtained by adding together four (not necessarily distinct) partitions of n.
Original entry on oeis.org
1, 5, 15, 55, 140, 448, 1022, 2710, 6048, 14114, 28831, 64091, 123649, 251295, 476835, 916972, 1654044, 3080159, 5377431, 9624588, 16490017, 28433473, 47423409, 80279375
Offset: 1
- N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
A002222
a(n) is the number of partitions of 5n that can be obtained by adding together five (not necessarily distinct) partitions of n.
Original entry on oeis.org
1, 6, 21, 91, 266, 994, 2562, 7764, 19482, 51212, 116028, 288541, 612463, 1375609, 2862437, 6036606, 11846488, 24080685, 45506290
Offset: 1
- N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
A284645
Number of partitions of n^2 that are the sum of n not necessarily distinct partitions of n.
Original entry on oeis.org
1, 1, 3, 10, 55, 266, 1974, 11418, 88671, 613756, 4884308
Offset: 0
a(0) = 1: the empty partition.
a(1) = 1: 1.
a(2) = 3: 22, 211, 1111.
a(3) = 10: 333, 3321, 32211, 33111, 222111, 321111, 2211111, 3111111, 21111111, 111111111. (Two of the A206226(3) = 12 partitions are not counted here: 3222, 22221.)
Showing 1-8 of 8 results.
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