A024788 -id:A024788 - cp's OEIS frontend

A302347 a(n) = Sum of (Y(2,p)^2) over the partitions p of n, Y(2,p) = number of part sizes with multiplicity 2 or greater in p.

0, 0, 1, 1, 3, 4, 10, 13, 25, 34, 59, 80, 127, 172, 260, 349, 505, 673, 946, 1248, 1711, 2238, 3010, 3902, 5162, 6637, 8663, 11051, 14253, 18051, 23047, 28988, 36677, 45840, 57538, 71485, 89082, 110062, 136269, 167487, 206138, 252132, 308640, 375777, 457698

Offset: 0

Emily Anible, Apr 05 2018

nonn

This sequence is part of the contribution to the b^2 term of C_{1-b,2}(q) for(1-b,2)-colored partitions - partitions in which we can label parts any of an indeterminate 1-b colors, but are restricted to using only 2 of the colors per part size. This formula is known to match the Han/Nekrasov-Okounkov hooklength formula truncated at hooks of size two up to the linear term in b.

It is of interest to enumerate and determine specific characteristics of partitions of n, considering each partition individually.

			For a(6), we sum over partitions of six. For each partition, we count 1 for each part which appears more than once, then square the total in each partition.
6............0^2 = 0
5,1..........0^2 = 0
4,2..........0^2 = 0
4,1,1........1^2 = 1
3,3..........1^2 = 1
3,2,1........0^2 = 0
3,1,1,1......1^2 = 1
2,2,2........1^2 = 1
2,2,1,1......2^2 = 4
2,1,1,1,1....1^2 = 1
1,1,1,1,1,1..1^2 = 1
--------------------
Total.............10

Alois P. Heinz, Table of n, a(n) for n = 0..2000
Guo-Niu Han, The Nekrasov-Okounkov hook length formula: refinement, elementary proof, extension and applications, arXiv:0805.1398 [math.CO], 2008.
Guo-Niu Han, The Nekrasov-Okounkov hook length formula: refinement, elementary proof, extension and applications, Annales de l'institut Fourier, Tome 60 (2010) no. 1, pp. 1-29.
W. J. Keith, Restricted k-color partitions, Ramanujan Journal (2016) 40: 71.

Cf. A024786, A024788, A302300.

b:= proc(n, i, p) option remember; `if`(n=0 or i=1, (
      `if`(n>1, 1, 0)+p)^2, add(b(n-i*j, i-1,
      `if`(j>1, 1, 0)+p), j=0..n/i))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..60);  # Alois P. Heinz, Apr 05 2018

Array[Total[Count[Split@ #, ?(Length@ # > 1 &)]^2 & /@ IntegerPartitions[#]] &, 44] (* _Michael De Vlieger, Apr 07 2018 *)
b[n_, i_, p_] := b[n, i, p] = If[n == 0 || i == 1, (
     If[n > 1, 1, 0] + p)^2, Sum[b[n - i*j, i - 1,
     If[j > 1, 1, 0] + p], {j, 0, n/i}]];
a[n_] := b[n, n, 0];
a /@ Range[0, 60] (* Jean-François Alcover, Jun 06 2021, after Alois P. Heinz *)

def sum_square_freqs_greater_one(freq_list):
    tot = 0
    for f in freq_list:
        count = 0
        for i in f:
            if i > 1:
                count += 1
        tot += count*count
    return tot
def frequencies(partition, n):
    tot = 0
    freq_list = []
    i = 0
    for p in partition:
        freq = [0 for i in range(n+1)]
        for i in p:
            freq[i] += 1
        for f in freq:
            if f == 0:
                tot += 1
        freq_list.append(freq)
    return freq_list

a(n) = Sum_{p in P(n)} (H(2,p)^2 + 2*A024786 - 2*A024788), where P(n) is the set of partitions of n, and H(2,p) is the hooks of length 2 in partition p.
G.f: (q^2*(1+q^4))/((1-q^2)*(1-q^4))*Product_{j>=1} 1/(1-q^j).
a(n) ~ sqrt(3) * exp(Pi*sqrt(2*n/3)) / (8*Pi^2). - Vaclav Kotesovec, May 22 2018

A179385 The n-th term is the sum of all the 1's generated from all the combinations of prime numbers and ones possible, that add to n, when each prime is only allowed once and any number of ones are allowed.

Original entry on oeis.org

1, 2, 4, 7, 10, 15, 20, 27, 35, 44, 55, 67, 81, 97, 115, 135, 158, 183, 212, 244, 280, 320, 364, 413, 467, 526, 591, 661, 737, 820, 909, 1007, 1112, 1226, 1349, 1481, 1624, 1778, 1943, 2121, 2311, 2515, 2734, 2968, 3219, 3486, 3771, 4075, 4399, 4744, 5112, 5502

Offset: 1

Joseph Foley, Jul 12 2010

nonn

			n=7 gives 11111 11, 2111 11, 311 11, 5 11, 5 2, 32 11. (Grouped in 5's) no. of 1's: 7, 5, 4, 2, 0, 2. Sum is 20, therefore a(7) = 20.
n=12 gives 11111 11111 11, 11111 11111 2, 11111 311 11, 11111 32 11, 11111 5 11, 5 2111 11, 5 311 11, 5 32 11, 7111 11, 721 11, 73 11, 73 2, 75, eleven 1, no. of 1's: 12, 10, 9, 7, 7, 5, 4, 2, 5, 3, 2, 0, 0, 1. Sum is 67, therefore a(12) = 67.
1: 1 => 1 2: 11, 2 => 2 3: 111, 21 => 4 4: 1111, 211, 22, 31 => 7 5: 11111, 2111, 311, 23 => 10 6: 11111 1, 2111 1, 311 1, 23 1, 5 1 => 15 and so on.

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 175 terms from Robert G. Wilson v)

Cf. A000586.
Cf. A000070, A024786, A024787, A024788, A024789, A024790, A024791, A024792, A024793, A024794.
Partial sums of A280271.

b:= proc(n,i) option remember; if n<=0 then 0 elif i=0 then n else b(n, i-1) +b(n-ithprime(i), i-1) fi end: # R. J. Mathar, Jul 14 2010
a:= n-> b(n, numtheory[pi](n)): seq(a(n), n=1..80); # Alois P. Heinz

fQ[lst_List] := Sort@ Flatten@ Most@ Split@ lst == Rest@ Union@ lst; f[n_] := Sum[ Count[ Select[ IntegerPartitions[n, {k}, Join[{1}, Prime@ Range@ PrimePi@n]], fQ@# &], 1, 2], {k, n}]; Array[f, 50] (* improved by Robert G. Wilson v, Jul 20 2010 *)
(* second program: *)
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1] + If[Prime[i] > n, 0, b[n - Prime[i], i - 1]]]];
a[n_] := Sum[k*b[n - k, PrimePi[n - k]], {k, 1, n}];
Table[a[n], {n, 1, 80}] (* Jean-François Alcover, Aug 29 2016, after Alois P. Heinz *)

a(n) = my(r); r = x/(1-x)^2 + O(x^(n+1)); forprime(p=2,n,r*=1+x^p); polcoeff(r,n) \\ Max Alekseyev, Jul 14 2010

a(n) = Sum_{k=1..n} k * A000586(n-k). - Max Alekseyev, Jul 14 2010

Corrected and extended by R. J. Mathar, Jul 14 2010

A141450 Upper right triangle of the number of m's in all partitions of n.

Original entry on oeis.org

1, 1, 2, 1, 1, 4, 1, 1, 3, 7, 1, 1, 2, 4, 12, 1, 1, 2, 4, 8, 19, 1, 1, 2, 3, 6, 11, 30, 1, 1, 2, 3, 6, 9, 19, 45, 1, 1, 2, 3, 5, 8, 15, 26, 67, 1, 1, 2, 3, 5, 8, 13, 21, 41, 97, 1, 1, 2, 3, 5, 7, 12, 18, 31, 56, 139, 1, 1, 2, 3, 5, 7, 12, 17, 28, 45, 83, 195, 1, 1, 2, 3, 5, 7, 11, 16, 25, 38, 63

Offset: 1

Robert G. Wilson v, Aug 07 2008

The "last" column read from the bottom is A000041.

Mirror of triangle A066633. - Omar E. Pol, May 01 2012

			A000070: 1, 2, 4, 7, 12, 19, 30, 45, 67, 97, 139, 195, 272, 373, 508, ...,
A024786: 0, 1, 1, 3, 4, 8, 11, 19, 26, 41, 56, 83, 112, 160, 213, ...,
A024787: 0, 0, 1, 1, 2, 4, 6, 9, 15, 21, 31, 45, 63, 87, 122, ...,
A024788: 0, 0, 0, 1, 1, 2, 3, 6, 8, 13, 18, 28, 38, 55, 74, ...,
A024789: 0, 0, 0, 0, 1, 1, 2, 3, 5, 8, 12, 17, 25, 35, 50, ...,
A024790: 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 12, 16, 24, 33, ...,
A024791: 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 11, 16, 23, ...,
A024792: 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 11, 15, ...,
A024793: 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 11, ...,
A024794: 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, ...

Cf. A000041, A000070, A024786, A024787, A024788, A024789, A024790, A024791, A024792, A024793, A024794.

(* First do ) Needs["Combinatorica`"] (* then *) f[n_, m_] := Count[Flatten@ Partitions@ n, m]; Table[ f[n, m], {n, 13}, {m, n, 1, -1}]

A304825 Sum of binomial(Y(2,p), 2) over the partitions p of n, where Y(2,p) is the number of part sizes with multiplicity 2 or greater in p.

Original entry on oeis.org

1, 1, 3, 4, 9, 12, 22, 30, 50, 68, 105, 142, 210, 281, 400, 531, 736, 967, 1311, 1707, 2274, 2935, 3851, 4930, 6389, 8116, 10402, 13121, 16658, 20872, 26275, 32719, 40880, 50613, 62807, 77343, 95389, 116874, 143331, 174789, 213251, 258903, 314367, 380079, 459462

Offset: 6

Emily Anible, May 19 2018

nonn

			For a(8), we sum over the partitions of eight. For each partition p, we take binomial(Y(2,p),2): that is, the number of parts with multiplicity at least two choose 2.
8................B(0,2) = 0
7,1..............B(0,2) = 0
6,2..............B(0,2) = 0
6,1,1............B(1,2) = 0
5,3..............B(0,2) = 0
5,2,1............B(0,2) = 0
5,1,1,1..........B(1,2) = 0
4,4..............B(1,2) = 0
4,3,1............B(0,2) = 0
4,2,2............B(1,2) = 0
4,2,1,1..........B(1,2) = 0
4,1,1,1,1........B(1,2) = 0
3,3,2............B(1,2) = 0
3,3,1,1..........B(2,2) = 1
3,2,2,1..........B(1,2) = 0
3,2,1,1,1........B(1,2) = 0
3,1,1,1,1,1......B(1,2) = 0
2,2,2,2..........B(1,2) = 0
2,2,2,1,1........B(2,2) = 1
2,2,1,1,1,1......B(2,2) = 1
2,1,1,1,1,1,1....B(1,2) = 0
1,1,1,1,1,1,1,1..B(1,2) = 0
---------------------------
Total.....................3

Cf. A024786, A302347.

b:= proc(n, i, p) option remember; `if`(n=0 or i=1,
      binomial(`if`(n>1, 1, 0)+p, 2), add(
      b(n-i*j, i-1, `if`(j>1, 1, 0)+p), j=0..n/i))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=6..60);  # Alois P. Heinz, May 19 2018

Array[Total[Binomial[Count[Split@#, _?(Length@# >= 2 &)], 2] & /@IntegerPartitions[#]] &, 50]
(* Second program: *)
b[n_, i_, p_] := b[n, i, p] = If[n == 0 || i == 1,
     Binomial[If[n > 1, 1, 0] + p, 2], Sum[
     b[n-i*j, i-1, If[j>1, 1, 0]+p], {j, 0, n/i}]];
a[n_] := b[n, n, 0];
a /@ Range[6, 60] (* Jean-François Alcover, May 30 2021, after Alois P. Heinz *)

a(n) = (A301313(n) - A024788(n))/4.

G.f.: q^6 /((1-q^2)*(1-q^4))*Product_{j>=1} 1/(1-q^j).

cp's OEIS Frontend

A302347 a(n) = Sum of (Y(2,p)^2) over the partitions p of n, Y(2,p) = number of part sizes with multiplicity 2 or greater in p.

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A179385 The n-th term is the sum of all the 1's generated from all the combinations of prime numbers and ones possible, that add to n, when each prime is only allowed once and any number of ones are allowed.

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A141450 Upper right triangle of the number of m's in all partitions of n.

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Mathematica

A304825 Sum of binomial(Y(2,p), 2) over the partitions p of n, where Y(2,p) is the number of part sizes with multiplicity 2 or greater in p.

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Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

Formula