A024786 -id:A024786 - cp's OEIS frontend

A291553 Column 3 of A060244.

0, 0, 1, 2, 4, 8, 13, 22, 35, 54, 81, 121, 174, 250, 352, 491, 675, 924, 1246, 1674, 2226, 2944, 3862, 5046, 6541, 8449, 10846, 13869, 17641, 22365, 28214, 35485, 44443, 55494, 69036, 85650, 105894, 130594, 160561, 196923, 240847, 293907, 357722, 434477, 526448

Offset: 1

Vaclav Kotesovec, Aug 26 2017

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Cf. A060244, A024786.

nmax = 50; col = 3; Flatten[{0, 0, CoefficientList[Coefficient[Normal[Series[Product[Product[1/(1 - x^(i - j)*y^j), {j, 0, i}], {i, 2, nmax + col}], {x, 0, col}, {y, 0, nmax}]], x^col], y]}]
nmax = 50; Rest[CoefficientList[Series[(x^3 * (1 + x - x^4))/((1-x)^2 * (1+x) * (1 + x + x^2)) / QPochhammer[x], {x, 0, nmax}], x]]
Table[Sum[(Floor[k/2] - Floor[(k-1)/3]) * PartitionsP[n-k], {k, 3, n}], {n, 1, 50}]

G.f.: x^3 * (1 + x - x^4) / ((1 - x)^2 * (1 + x) * (1 + x + x^2)) * Product_{k>=1} 1/(1 - x^k).
a(n) = Sum_{k=3..n} (floor(k/2) - floor((k-1)/3)) * A000041(n-k).
a(n) ~ exp(Pi*sqrt(2*n/3)) / (4*sqrt(3)*Pi^2).
a(n) ~ n * A000041(n) / Pi^2.

A301313 a(n) = Sum_{p in P} binomial(H(2,p),2), where P is the set of partitions of n, and H(2,p) = number of hooks of size 2 in p.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 6, 7, 18, 24, 49, 66, 116, 158, 255, 346, 525, 707, 1030, 1374, 1936, 2560, 3519, 4608, 6207, 8056, 10673, 13735, 17942, 22906, 29569, 37469, 47864, 60235, 76249, 95335, 119705, 148770, 185447, 229182, 283810, 348903, 429498, 525411, 643244

Offset: 0

Emily Anible, Apr 03 2018

nonn

This sequence is part of the contribution to the quadratic b^2 term of a 2-truncation of the Han/Nekrasov-Okounkov hooklength formula (2-truncation here being the limiting of hook sizes counted by the formula to only those of size 1 or 2). Exploring this sequence may lead to more general formulas regarding the hooklength formula for larger hooks, or the entire contribution to the quadratic term of the formula.

			For n=6, we sum over the partitions of 6. For each partition, we calculate binomial(number of hooks of size 2 in partition, 2):
6............binomial(1,2) = 0
5,1..........binomial(1,2) = 0
4,2..........binomial(2,2) = 1
4,1,1........binomial(2,2) = 1
3,3..........binomial(2,2) = 1
3,2,1........binomial(0,2) = 0
3,1,1,1......binomial(2,2) = 1
2,2,2........binomial(2,2) = 1
2,2,1,1......binomial(2,2) = 1
2,1,1,1,1....binomial(1,2) = 0
1,1,1,1,1,1..binomial(1,2) = 0
------------------------------
Total........................6

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1900 from Alois P. Heinz)
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.

Cf. A024786, A138785.

b:= proc(n, i, p, l) option remember; `if`(n=0, p*(p-1)/2,
      `if`(i>n, 0, b(n, i+1, p, 1)+add(b(n-i*j, i+1, p+
      `if`(j>1, 1, 0)+l, 0), j=1..n/i)))
    end:
a:= n-> b(n, 1, 0$2):
seq(a(n), n=0..50);  # Alois P. Heinz, Apr 05 2018

b[n_, i_, p_, l_] := b[n, i, p, l] = If[n == 0, p*(p-1)/2, If[i > n, 0, b[n, i+1, p, 1] + Sum[b[n-i*j, i+1, p+If[j>1, 1, 0]+l, 0], {j, 1, n/i}]] ];
a[n_] := b[n, 1, 0, 0];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Apr 28 2018, after Alois P. Heinz *)
Table[Sum[(2*k - 5 - (-1)^(k/2))*(1 + (-1)^k)/4 * PartitionsP[n-k], {k, 1, n}], {n, 0, 60}] (* Vaclav Kotesovec, Oct 06 2018 *)

G.f.: (q^4+3*q^6)/((1-q^2)*(1-q^4))*Product_{j>=1} 1/(1-q^j). - Emily Anible, May 18 2018

a(n) ~ sqrt(3) * exp(Pi*sqrt((2*n)/3)) / (4*Pi^2). - Vaclav Kotesovec, Oct 06 2018

a(10)-a(44) from Alois P. Heinz, Apr 03 2018

A302300 a(n) = Sum_{p in P} (Sum_{k_j = 1} 1)^2, where P is the set of partitions of n, and the k_j are the frequencies in p.

Original entry on oeis.org

0, 1, 1, 5, 6, 12, 21, 33, 50, 79, 116, 169, 246, 346, 487, 675, 927, 1254, 1702, 2263, 3014, 3966, 5210, 6766, 8795, 11303, 14531, 18521, 23583, 29803, 37654, 47231, 59206, 73792, 91867, 113778, 140788, 173377, 213289, 261318, 319764, 389846, 474745, 576164

Offset: 0

Emily Anible, Apr 04 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 once, then square the total in each partition.
   6............1^2 = 1
   5,1..........2^2 = 4
   4,2..........2^2 = 4
   4,1,1........1^2 = 1
   3,3..........0^2 = 0
   3,2,1........3^2 = 9
   3,1,1,1......1^2 = 1
   2,2,2........0^2 = 0
   2,2,1,1......0^2 = 0
   2,1,1,1,1....1^2 = 1
   1,1,1,1,1,1..0^2 = 0
   --------------------
   Total.............21

Alois P. Heinz, Table of n, a(n) for n = 0..4000
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, A197126.

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@ Map[Count[Split@ #, ?(Length@ # == 1 &)]^2 &, IntegerPartitions[#]] &, 43] (* _Michael De Vlieger, Apr 05 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 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
def sum_square_freqs_of_one(freq_part):
    tot = 0
    for f in freq_part:
        count = 0
        for i in f:
            if i == 1:
                count += 1
        tot += count*count
    return tot
import sympy.combinatorics
def A302300(n): # rewritten by R. J. _Mathar, 2023-03-24
    a =0
    if n ==0 :
        return 0
    part = sympy.combinatorics.IntegerPartition([n])
    partlist = []
    while True:
        part = part.next_lex()
        partlist.append(part.partition)
        if len(part.partition) <=1 :
            break
    freq_part = frequencies(partlist, n)
    return sum_square_freqs_of_one(freq_part)
for n in range(20): print(A302300(n))

a(n) = Sum_{p in P} (Sum_{k_j = 1} 1)^2, where P is the set of partitions of n, and k_j are the frequencies in p.

A330040 Number of non-isomorphic cover graphs of lattice quotients of essential lattice congruences of the weak order on the symmetric group S_n.

Original entry on oeis.org

1, 1, 3, 19, 748, 2027309

Offset: 1

Torsten Muetze, Nov 28 2019

			For n=3, the weak order on S_3 has the cover relations 123<132, 123<213, 132<312, 213<231, 312<321, 231<321, and there are four essential lattice congruences, namely {}, {132=312}, {213=231}, {132=312,213=231}. The cover graph of the first one is a 6-cycle, the cover graph of the middle two is a 5-cycle, and the cover graph of the last one is a 4-cycle. These are 3 non-isomorphic graphs, showing that a(3)=3.

Hung Phuc Hoang, Torsten Mütze, Combinatorial generation via permutation languages. II. Lattice congruences, arXiv:1911.12078 [math.CO], 2019.
V. Pilaud and F. Santos, Quotientopes, arXiv:1711.05353 [math.CO], 2017-2019; Bull. Lond. Math. Soc., 51 (2019), no. 3, 406-420.

Cf. A091687, A001246, A052528, A024786, A123663, A330039, A330042.

A330042 Number of non-isomorphic regular cover graphs of lattice quotients of essential lattice congruences of the weak order on the symmetric group S_n.

Original entry on oeis.org

1, 1, 3, 10, 51, 335, 2909

Offset: 1

Torsten Muetze, Nov 28 2019

			For n=3, the weak order on S_3 has the cover relations 123<132, 123<213, 132<312, 213<231, 312<321, 231<321, and there are four essential lattice congruences, namely {}, {132=312}, {213=231}, {132=312,213=231}. The cover graph of the first one is a 6-cycle, the cover graph of the middle two is a 5-cycle, and the cover graph of the last one is a 4-cycle. These are 3 non-isomorphic regular graphs, showing that a(3)=3.

Hung Phuc Hoang, Torsten Mütze, Combinatorial generation via permutation languages. II. Lattice congruences, arXiv:1911.12078 [math.CO], 2019.
V. Pilaud and F. Santos, Quotientopes, arXiv:1711.05353 [math.CO], 2017-2019; Bull. Lond. Math. Soc., 51 (2019), no. 3, 406-420.

Cf. A091687, A001246, A052528, A024786, A123663, A330039, A330040.

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

A276426 Triangle read by rows: T(n,k) is the number of integer partitions of n having k distinct odd parts (n>=0).

Original entry on oeis.org

1, 0, 1, 1, 1, 0, 3, 2, 2, 1, 0, 6, 1, 3, 5, 3, 0, 11, 4, 5, 8, 9, 0, 20, 9, 1, 7, 15, 19, 1, 0, 32, 21, 3, 11, 24, 38, 4, 0, 51, 41, 9, 15, 39, 69, 12, 0, 80, 73, 23, 22, 58, 123, 27, 1, 0, 119, 128, 49, 1, 30, 90, 202, 60, 3, 0, 175, 213, 98, 4, 42, 130, 328, 118, 9

Offset: 0

Emeric Deutsch, Sep 19 2016

Sum of entries in row n = A000041(n).
T(2n,0) = A000041(n); T(2n+1,0) = 0.
Sum(k*T(n,k), k>=0) = A024786(n+1).

			T(4,0) = 2 because we have [4], [2,2];
T(4,1) = 2 because we have [1,1,2], [1,1,1,1];
T(4,2) = 1 because we have [1,3];
Triangle starts:
1;
0,1;
1,1;
0,3;
2,2,1.

Alois P. Heinz, Rows n = 0..1000, flattened

Cf. A000041, A024786, A264052.

G := product((1-x^(2*j-1)+t*x^(2*j-1))/(1-x^j), j = 1 .. 100): Gser := simplify(series(G, x = 0, 32)); for n from 0 to 27 do P[n] := sort(coeff(Gser, x, n)) end do: for n from 0 to 27 do seq(coeff(P[n], t, i), i = 0 .. degree(P[n])) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(n, i) option remember; expand(
      `if`(n=0, 1, `if`(i<1, 0, add(b(n-i*j, i-1)*
      `if`(j>0 and i::odd, x, 1), j=0..n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
seq(T(n), n=0..25);  # Alois P. Heinz, Sep 20 2016

b[n_, i_] := b[n, i] = Expand[If[n==0, 1, If[i<1, 0, Sum[b[n-i*j, i-1]*If[j > 0 && OddQ[i], x, 1], {j, 0, n/i}]]]]; T[n_] := Function[p, Table[ Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, n]]; Table[T[n], {n, 0, 25}] // Flatten (* Jean-François Alcover, Feb 07 2017, after Alois P. Heinz *)

G.f.: G(t,x) = Product_{j>=1} ((1-(1-t)*x^{2*j-1})/(1-x^j)).

A302348 a(n) = Sum_{p in P} (H(2,p)^2)/2, where P is the set of partitions of n, and H(2,p) is the number of hooks of length 2 in p.

Original entry on oeis.org

0, 0, 1, 1, 4, 5, 14, 18, 37, 50, 90, 122, 199, 270, 415, 559, 820, 1096, 1556, 2060, 2847, 3736, 5057, 6576, 8747, 11279, 14788, 18916, 24493, 31097, 39838, 50225, 63737, 79833, 100471, 125076, 156237, 193394, 239956, 295443, 364334, 446349, 547360, 667440

Offset: 0

Emily Anible, Apr 05 2018

nonn

This sequence is part of the contribution to the b^2 term of the Han/Nekrasov-Okounkov hooklength formula truncated at hooks of size two.

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 hook of length 2, then square the total in each partition. We divide the final result in half to get a(6).
6............1^2 = 1
5,1..........1^2 = 1
4,2..........2^2 = 4
4,1,1........2^2 = 4
3,3..........2^2 = 4
3,2,1........0^2 = 0
3,1,1,1......2^2 = 4
2,2,2........2^2 = 4
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.............28/2=14

Alois P. Heinz, Table of n, a(n) for n = 0..1000
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, A302347.

b:= proc(n, i, p, l) option remember; `if`(n=0, p^2,
      `if`(i>n, 0, b(n, i+1, p, 1)+add(b(n-i*j, i+1, p+
      `if`(j>1, 1, 0)+l, 0), j=1..n/i)))
    end:
a:= n-> b(n, 1, 0$2)/2:
seq(a(n), n=0..50);  # Alois P. Heinz, Apr 06 2018

b[n_, i_, p_, l_] := b[n, i, p, l] = If[n == 0, p^2, If[i > n, 0, b[n, i + 1, p, 1] + Sum[b[n - i*j, i+1, p + If[j > 1, 1, 0]+l, 0], {j, 1, n/i}]]];
a[n_] := b[n, 1, 0, 0]/2;
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, May 18 2018, after Alois P. Heinz *)

G.f: (q^2*(1+q^2+2*q^4))/((1-q^2)*(1-q^4)*Product_{i>0} (1-q^i)).

A316861 a(n) = Sum_{p in P} y(1)*y(2), where P is the set of partitions of n, and y(k) is the number of parts with multiplicity at least k in p.

Original entry on oeis.org

0, 0, 1, 1, 4, 7, 13, 22, 38, 58, 93, 139, 208, 302, 438, 616, 869, 1200, 1650, 2239, 3026, 4038, 5374, 7081, 9292, 12103, 15704, 20236, 25992, 33191, 42237, 53490, 67524, 84860, 106341, 132736, 165212, 204928, 253518, 312629, 384585, 471734, 577276, 704584, 858078

Offset: 0

Emily Anible, Jul 15 2018

nonn

Also (1/2)*Sum_{p in P} H(1)*H(2), where P is the set of partitions of n, and H(k) is the number of k-hooks in p.

			For n=6, we sum over the partitions of 6. For each partition, we count the parts with multiplicity at least one, and those of at least two.
6............y(1)*y(2) = 1*0 = 0
5,1..........y(1)*y(2) = 2*0 = 0
4,2..........y(1)*y(2) = 2*0 = 0
4,1,1........y(1)*y(2) = 2*1 = 2
3,3..........y(1)*y(2) = 1*1 = 1
3,2,1........y(1)*y(2) = 3*0 = 0
3,1,1,1......y(1)*y(2) = 2*1 = 2
2,2,2........y(1)*y(2) = 1*1 = 1
2,2,1,1......y(1)*y(2) = 2*2 = 4
2,1,1,1,1....y(1)*y(2) = 2*1 = 2
1,1,1,1,1,1..y(1)*y(2) = 1*1 = 1
--------------------------------
Total.........................13

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000097, A116646, A000070, A024786.

b:= proc(n, i, x, y) option remember;
     `if`(n=0, x*y, `if`(i<1, 0, add(b(n-i*j, i-1,
     `if`(j>0, 1, 0)+x, `if`(j>1, 1, 0)+y), j=0..n/i)))
    end:
a:= n-> b(n$2, 0$2):
seq(a(n), n=0..55);  # Alois P. Heinz, Jul 30 2018

Array[Total[
   Count[Split@#, (_?(Length@# >= 1 &))] Count[
       Split@#, (_?(Length@# >= 2 &))] & /@
    IntegerPartitions[#]] &, 50]
(* Second program: *)
b[n_, i_, x_, y_] := b[n, i, x, y] = If[n == 0, x*y, If[i < 1, 0, Sum[b[n - i*j, i - 1, If[j > 0, 1, 0] + x, If[j > 1, 1, 0] + y], {j, 0, n/i}]]];
a[n_] := b[n, n, 0, 0];
a /@ Range[0, 55] (* Jean-François Alcover, Sep 16 2019, after Alois P. Heinz *)

seq(n)={Vec(x*(1 + x^2 + x^3)/((1 - x)^2*(1 + x)*(1 + x + x^2)*prod(i=1, n-1, 1 - x^i + O(x^n))) + O(x^n), -n)} \\ Andrew Howroyd, Jul 15 2018

G.f.: (q^3/((1-q)(1-q^2)) + q^2/(1-q^2) - q^3/(1-q^3))*Product_{j>=1} 1/(1-q^j).
a(n) = A000097(n+3) + A116646(n).
In general, Sum_{n>=0} q^n Sum_{p in P} y(s)*y(t) for s < t is given by (q^(s+t)/((1-q^s)(1-q^t)) + q^t/(1-q^t) - q^(s+t)/(1-q^(s+t))) * Product_{j>=1} 1/(1-q^j).

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}]

cp's OEIS Frontend

A291553 Column 3 of A060244.

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A301313 a(n) = Sum_{p in P} binomial(H(2,p),2), where P is the set of partitions of n, and H(2,p) = number of hooks of size 2 in p.

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A302300 a(n) = Sum_{p in P} (Sum_{k_j = 1} 1)^2, where P is the set of partitions of n, and the k_j are the frequencies in p.

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A330040 Number of non-isomorphic cover graphs of lattice quotients of essential lattice congruences of the weak order on the symmetric group S_n.

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A330042 Number of non-isomorphic regular cover graphs of lattice quotients of essential lattice congruences of the weak order on the symmetric group S_n.

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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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A276426 Triangle read by rows: T(n,k) is the number of integer partitions of n having k distinct odd parts (n>=0).

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A302348 a(n) = Sum_{p in P} (H(2,p)^2)/2, where P is the set of partitions of n, and H(2,p) is the number of hooks of length 2 in p.

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