A087897 -id:A087897 - cp's OEIS frontend

A238780 Number of palindromic partitions of n whose greatest part has multiplicity 4.

0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 2, 2, 3, 2, 5, 4, 7, 5, 10, 8, 14, 11, 20, 16, 26, 21, 37, 31, 48, 40, 65, 55, 85, 72, 113, 97, 145, 125, 190, 165, 242, 211, 313, 274, 396, 348, 505, 446, 633, 561, 801, 713, 998, 890, 1249, 1118, 1548, 1389, 1922, 1730

Offset: 0

Clark Kimberling, Mar 05 2014

Palindromic partitions are defined at A025065.

			a(8) counts these partitions (written as palindromes):  3333, 11222211.

Cf. A025065, A087897, A238779.

z = 40; p[n_, k_] := Select[IntegerPartitions[n], (Count[OddQ[Transpose[Tally[#]][[2]]], True] <= 1) && (Count[#, Max[#]] == k) &]
Table[p[n, 1], {n, 1, 12}]
t1 = Table[Length[p[n, 1]], {n, 1, z}] (* A000009(n-1), n>=1 *)
Table[p[n, 2], {n, 1, 12}]
t2 = Table[Length[p[n, 2]], {n, 1, z}] (* A238779 *)
Table[p[n, 3], {n, 1, 12}]
t3 = Table[Length[p[n, 3]], {n, 1, z}] (* A087897(n-3), n>=3 *)
Table[p[n, 4], {n, 1, 12}]
t4 = Table[Length[p[n, 4]], {n, 1, z}] (* A238780 *)
(* Peter J. C. Moses, Mar 03 2014 *)

A266138 Expansion of Product_{k>=1} 1/(1 - k(x^(2k+1))).

Original entry on oeis.org

1, 0, 0, 1, 0, 2, 1, 3, 2, 5, 7, 7, 11, 13, 24, 26, 35, 44, 69, 78, 112, 150, 188, 245, 318, 429, 537, 729, 924, 1177, 1534, 1965, 2518, 3287, 4108, 5394, 6857, 8604, 11022, 14073, 17899, 22549, 28900, 36182, 45954, 58395, 72912, 92118, 116201, 146279

Offset: 0

Vaclav Kotesovec, Dec 21 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A006906, A077335, A087897, A265951, A266137.

nmax=80; CoefficientList[Series[Product[1/(1-k*(x^(2*k+1))), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ c * 3^(n/7), where
c = 617630.638335... if mod(n,7) = 0
c = 617630.321433... if mod(n,7) = 1
c = 617630.360795... if mod(n,7) = 2
c = 617630.429073... if mod(n,7) = 3
c = 617630.357078... if mod(n,7) = 4
c = 617630.421636... if mod(n,7) = 5
c = 617630.341606... if mod(n,7) = 6.

A277210 Expansion of Product_{k>=1} 1/(1 - x^(3*k+1)).

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 4, 3, 3, 4, 5, 4, 6, 6, 7, 7, 9, 8, 11, 11, 12, 13, 16, 15, 18, 20, 22, 22, 27, 27, 31, 33, 37, 38, 45, 46, 51, 55, 62, 63, 72, 76, 84, 89, 99, 103, 116, 122, 133, 142, 158, 164, 181, 193, 210, 222, 245, 257, 281, 299, 324, 343, 376, 396, 429, 457, 495, 522, 568, 601, 649, 689

Offset: 0

Ilya Gutkovskiy, Oct 05 2016

nonn

Number of partitions of n into parts larger than 1 and congruent to 1 mod 3.

More generally, the ordinary generating function for the number of partitions of n into parts larger than 1 and congruent to 1 mod m (for m>0) is Product_{k>=1} 1/(1 - x^(m*k+1)).

			a(14) = 2, because we have [10, 4] and [7, 7].

Index entries for related partition-counting sequences

Cf. A016777, A035382, A087897, A117957.

CoefficientList[Series[(1 - x)/QPochhammer[x, x^3], {x, 0, 85}], x]

G.f.: Product_{k>=1} 1/(1 - x^(3*k+1)).

a(n) ~ Pi^(1/3) * Gamma(1/3) * exp(sqrt(2*n)*Pi/3) / (2^(13/6)*3^(3/2)*n^(7/6)). - Vaclav Kotesovec, Oct 06 2016

A323089 Number of strict integer partitions of n using 1 and numbers that are not perfect powers.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 3, 3, 4, 4, 5, 6, 7, 9, 10, 12, 14, 16, 20, 22, 26, 31, 34, 40, 46, 51, 59, 66, 75, 86, 96, 110, 123, 139, 157, 176, 199, 221, 248, 278, 309, 346, 385, 427, 476, 528, 586, 650, 719, 795, 880, 973, 1074, 1186, 1307, 1439, 1584, 1744, 1915, 2104

Offset: 0

Gus Wiseman, Jan 04 2019

nonn

			A list of all strict integer partitions using 1 and numbers that are not perfect powers begins:
  1: (1)         8: (5,2,1)      12: (12)         14: (14)
  2: (2)         9: (7,2)        12: (11,1)       14: (13,1)
  3: (3)         9: (6,3)        12: (10,2)       14: (12,2)
  3: (2,1)       9: (6,2,1)      12: (7,5)        14: (11,3)
  4: (3,1)       9: (5,3,1)      12: (7,3,2)      14: (11,2,1)
  5: (5)        10: (10)         12: (6,5,1)      14: (10,3,1)
  5: (3,2)      10: (7,3)        12: (6,3,2,1)    14: (7,6,1)
  6: (6)        10: (7,2,1)      13: (13)         14: (7,5,2)
  6: (5,1)      10: (6,3,1)      13: (12,1)       14: (6,5,3)
  6: (3,2,1)    10: (5,3,2)      13: (11,2)       14: (6,5,2,1)
  7: (7)        11: (11)         13: (10,3)       15: (15)
  7: (6,1)      11: (10,1)       13: (10,2,1)     15: (14,1)
  7: (5,2)      11: (7,3,1)      13: (7,6)        15: (13,2)
  8: (7,1)      11: (6,5)        13: (7,5,1)      15: (12,3)
  8: (6,2)      11: (6,3,2)      13: (7,3,2,1)    15: (12,2,1)
  8: (5,3)      11: (5,3,2,1)    13: (6,5,2)      15: (11,3,1)

Cf. A001597, A007916, A052410, A087897, A276431, A303707, A305630, A323088, A323090.

perpowQ[n_]:=GCD@@FactorInteger[n][[All,2]]>1;
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&And@@Not/@perpowQ/@#&]],{n,65}]

O.g.f.: (1 + x) * Product_{n in A007916} (1 + x^n).

A334305 a(n) is the number of partitions of n of the form [k,k,b(1),b(2),...], where k>=b(1)>b(2)>...>=2.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 2, 0, 2, 1, 2, 2, 3, 2, 4, 4, 4, 6, 6, 7, 8, 10, 10, 13, 14, 16, 18, 22, 22, 28, 30, 34, 39, 44, 48, 56, 62, 69, 78, 88, 96, 110, 122, 134, 152, 168, 186, 208, 231, 254, 284, 314, 346, 384, 425, 466, 518, 570, 626, 692, 762, 834, 922, 1010

Offset: 0

Victor Mishnyakov, Elena Lanina, Apr 22 2020

a(n)>0 if n>=2k>=4.

			a(4)=1 because we have [2,2]; a(6)=2 because we have [2,2,2] and [3,3].
G.f.= x^4+2x^6+2x^8+x^9+2x^10+2x^11+3x^12+2x^13+...

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A000009, A000041, A025147, A087897.

nmax = 100; CoefficientList[Series[Sum[x^(2 k) Product[(1 + x^i), {i, 2, k}], {k, 2, nmax/2}], {x, 0, nmax}], x]
Flatten[{{0, 0, 0}, Table[PartitionsQ[n + 3] - 2*(-1)^n + 2*Sum[(-1)^k * PartitionsQ[n - k + 3], {k, 1, n - 2}], {n, 3, 70}]}] (* Vaclav Kotesovec, Apr 24 2020 *)

G.f.: Sum_{k>=2} x^(2k) Product_{i=2..k} (1+x^i).
From Vaclav Kotesovec, Apr 24 2020: (Start)
For n>=3, a(n) + a(n+1) = A087897(n+4).
a(n) ~ exp(Pi*sqrt(n/3)) * Pi / (16 * 3^(3/4) * n^(5/4)). (End)

A239302 Triangular array: T(n,k) = number of partitions x(1) > x(2) > ... > x(k) of n+2 such that x(1) = x(2) + k, for n >= 1.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 2, 1, 1, 1, 1, 0, 1, 2, 2, 1, 1, 1, 1, 0, 1, 2, 2, 2, 1, 1, 1, 1, 0, 1, 3, 2, 2, 2, 1, 1, 1, 1, 0, 1, 3, 3, 2, 2, 2, 1, 1, 1, 1, 0, 1, 4, 3, 3, 2, 2, 2, 1, 1, 1, 1, 0, 1, 5, 4, 3, 3, 2, 2, 2, 1

Offset: 1

Clark Kimberling, Mar 14 2014

The columns are identical, and the limit of the reversal of rows is A087897(n+3).

Sum of row n equals A111133(n+3).

			First 17 rows:
1
0 1
1 0 1
1 1 0 1
1 1 1 0 1
1 1 1 1 0 1
2 1 1 1 1 0 1
2 2 1 1 1 1 0 1
2 2 2 1 1 1 1 0 1
3 2 2 2 1 1 1 1 0 1
3 3 2 2 2 1 1 1 1 0 1
4 3 3 2 2 2 1 1 1 1 0 1
5 4 3 3 2 2 2 1 1 1 1 0 1
5 5 4 3 3 2 2 2 1 1 1 1 0 1
6 5 5 4 3 3 2 2 2 1 1 1 1 0 1
8 6 5 5 4 3 3 2 2 2 1 1 1 1 0 1
8 8 6 5 5 4 3 3 2 2 2 1 1 1 1 0 1
To account for row 7, start with the strict partitions (A000009) of 9 that have more than one part:  81, 72, 63, 621, 54, 531, 432.  Next, form (part 1) - (part 2) for each of those partitions, getting 7, 5, 3, 4, 1, 2, 1; finally, note that the numbers of occurrences of 1,2,3,4,5,6,7, respectively, are 2,1,1,1,1,0,1.

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A000009, A111133, A087897.

z = 25; d[n_] := d[n] = Rest[Select[IntegerPartitions[n], DeleteDuplicates[#] == # &]]; t[n_] := t[n] = Table[d[n][[k, 1]] - d[n][[k, 2]], {k, 1, -1 + PartitionsQ[n]}]; u = Table[Count[t[n], j], {n, 3, z}, {j, 1, n - 2}]; TableForm[u] (* A239302 as an array *)
v = Flatten[u]  (* A239302 as a sequence *)

A277264 Expansion of Product_{k>=1} 1/(1 - x^(5*k+1)).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 2, 1, 1, 0, 1, 2, 2, 1, 1, 1, 3, 3, 2, 1, 2, 3, 4, 3, 2, 2, 5, 5, 5, 3, 3, 5, 8, 6, 5, 4, 7, 9, 10, 7, 6, 8, 12, 12, 11, 8, 11, 15, 17, 14, 13, 13, 19, 21, 20, 16, 19, 23, 28, 26, 23, 23, 31, 34, 35, 30, 31, 37, 46, 44, 41, 39, 48, 55, 59, 52, 52, 59, 71, 73, 71, 65, 75, 87, 94

Offset: 0

Ilya Gutkovskiy, Oct 07 2016

nonn

Number of partitions of n into parts larger than 1 and congruent to 1 mod 5.

			a(22) = 2, because we have [16, 6] and [11, 11].

Index entries for related partition-counting sequences

Cf. A016861, A087897, A109697 (partial sums), A117957, A277210.

CoefficientList[Series[(1 - x)/QPochhammer[x, x^5], {x, 0, 100}], x]

G.f.: Product_{k>=1} 1/(1 - x^(5*k+1)).

a(n) ~ Pi^(1/5) * Gamma(1/5) * exp(Pi*sqrt(2*n/15)) / (2^(21/10) * 3^(3/5) * 5^(9/10) * n^(11/10)). - Vaclav Kotesovec, Oct 09 2016

A277349 Expansion of Product_{k>=1} 1/(1 - x^(6*k+1)).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 2, 1, 1, 0, 0, 1, 2, 2, 1, 1, 0, 1, 3, 3, 2, 1, 1, 1, 3, 4, 3, 2, 1, 2, 4, 5, 5, 3, 2, 2, 5, 7, 6, 5, 3, 3, 6, 9, 9, 7, 5, 4, 7, 11, 12, 10, 7, 6, 9, 14, 16, 14, 11, 8, 11, 17, 20, 19, 15, 12, 14, 21, 26, 25, 21, 17, 18, 26, 32, 33, 28, 23, 24, 32, 41

Offset: 0

Ilya Gutkovskiy, Oct 10 2016

nonn

Number of partitions of n into parts larger than 1 and congruent to 1 mod 6.

			a(26) = 2, because we have [19, 7] and [13, 13].

Robert Israel, Table of n, a(n) for n = 0..3000
Index entries for related partition-counting sequences

Cf. A016921, A087897, A109701 (partial sums), A117957, A277210, A277264.

N:= 100:
G:= 1/mul(1-x^m,m=7..N,6):
S:= series(G,x,N+1):
seq(coeff(S,x,j),j=0..N); # Robert Israel, Jan 23 2019

CoefficientList[Series[(1 - x)/QPochhammer[x, x^6], {x, 0, 100}], x]

G.f.: Product_{k>=1} 1/(1 - x^(6*k+1)).

a(n) ~ Pi^(1/6) * Gamma(1/6) * exp(sqrt(n)*Pi/3) / (24*sqrt(3)*n^(13/12)). - Vaclav Kotesovec, Oct 10 2016

A303903 Expansion of (1 - x^2)*Product_{k>=3} (1 + x^Fibonacci(k)).

Original entry on oeis.org

1, 0, 0, 1, -1, 1, 0, -1, 2, -1, 0, 1, -2, 2, 0, -1, 2, -2, 1, 0, -2, 3, -1, 0, 2, -3, 2, 0, -2, 3, -2, 0, 1, -3, 3, 0, -1, 3, -3, 2, 0, -3, 4, -2, 0, 2, -4, 3, 0, -2, 3, -3, 1, 0, -3, 4, -1, 0, 3, -4, 3, 0, -3, 5, -3, 0, 2, -5, 4, 0, -2, 4, -4, 2, 0, -4, 5, -2, 0, 3, -5, 3, 0, -3, 4, -3

Offset: 0

Ilya Gutkovskiy, May 02 2018

sign

First differences of A000119.

Ilya Gutkovskiy, Scatter plot of a(n) up to n=50000
Index entries for sequences related to partitions

Cf. A000045, A000119, A003107, A087897, A238999, A239002.

nmax = 85; CoefficientList[Series[(1 - x^2) Product[1 + x^Fibonacci[k], {k, 3, 21}], {x, 0, nmax}], x]

A341449 Heinz numbers of integer partitions into odd parts > 1.

Original entry on oeis.org

1, 5, 11, 17, 23, 25, 31, 41, 47, 55, 59, 67, 73, 83, 85, 97, 103, 109, 115, 121, 125, 127, 137, 149, 155, 157, 167, 179, 187, 191, 197, 205, 211, 227, 233, 235, 241, 253, 257, 269, 275, 277, 283, 289, 295, 307, 313, 331, 335, 341, 347, 353, 365, 367, 379, 389

Offset: 1

Gus Wiseman, Feb 15 2021

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The sequence of partitions together with their Heinz numbers begins:
      1: ()        97: (25)       197: (45)       307: (63)
      5: (3)      103: (27)       205: (13,3)     313: (65)
     11: (5)      109: (29)       211: (47)       331: (67)
     17: (7)      115: (9,3)      227: (49)       335: (19,3)
     23: (9)      121: (5,5)      233: (51)       341: (11,5)
     25: (3,3)    125: (3,3,3)    235: (15,3)     347: (69)
     31: (11)     127: (31)       241: (53)       353: (71)
     41: (13)     137: (33)       253: (9,5)      365: (21,3)
     47: (15)     149: (35)       257: (55)       367: (73)
     55: (5,3)    155: (11,3)     269: (57)       379: (75)
     59: (17)     157: (37)       275: (5,3,3)    389: (77)
     67: (19)     167: (39)       277: (59)       391: (9,7)
     73: (21)     179: (41)       283: (61)       401: (79)
     83: (23)     187: (7,5)      289: (7,7)      415: (23,3)
     85: (7,3)    191: (43)       295: (17,3)     419: (81)

Note: A-numbers of ranking sequences are in parentheses below.
Partitions with no ones are A002865 (A005408).
The case of even parts is A035363 (A066207).
These partitions are counted by A087897.
The version for factorizations is A340101.
A000009 counts partitions into odd parts (A066208).
A001222 counts prime factors.
A027193 counts partitions of odd length/maximum (A026424/A244991).
A056239 adds up prime indices.
A078408 counts partitions with odd parts, length, and sum (A300272).
A112798 lists the prime indices of each positive integer.
A257991/A257992 count odd/even prime indices.
Cf. A026804 (A340932), A160786 (A340931), A340386, A340604, A340607, A340785, A340854/A340855, A341446.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],OddQ[#]&&OddQ[Times@@primeMS[#]]&]

cp's OEIS Frontend

A238780 Number of palindromic partitions of n whose greatest part has multiplicity 4.

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A266138 Expansion of Product_{k>=1} 1/(1 - k*(x^(2*k+1))).

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A277210 Expansion of Product_{k>=1} 1/(1 - x^(3*k+1)).

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A323089 Number of strict integer partitions of n using 1 and numbers that are not perfect powers.

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A334305 a(n) is the number of partitions of n of the form [k,k,b(1),b(2),...], where k>=b(1)>b(2)>...>=2.

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A239302 Triangular array: T(n,k) = number of partitions x(1) > x(2) > ... > x(k) of n+2 such that x(1) = x(2) + k, for n >= 1.

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A277264 Expansion of Product_{k>=1} 1/(1 - x^(5*k+1)).

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A277349 Expansion of Product_{k>=1} 1/(1 - x^(6*k+1)).

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A266138 Expansion of Product_{k>=1} 1/(1 - k(x^(2k+1))).