A026811 -id:A026811 - cp's OEIS frontend

A008641 Number of partitions of n into at most 12 parts.

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 100, 133, 172, 224, 285, 366, 460, 582, 725, 905, 1116, 1380, 1686, 2063, 2503, 3036, 3655, 4401, 5262, 6290, 7476, 8877, 10489, 12384, 14552, 17084, 19978, 23334, 27156, 31570, 36578, 42333, 48849, 56297

Offset: 0

N. J. A. Sloane

With a different offset, number of partitions of n in which the greatest part is 12.

Also number of partitions of n into parts <= 12: a(n)=A026820(n,12). [Reinhard Zumkeller, Jan 21 2010]

A. Cayley, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 10, p. 415.
H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958, p. 2.

T. D. Noe, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 361
Index entries for linear recurrences with constant coefficients, signature (1, 1, 0, 0, -1, 0, -1, 0, 0, 0, 0, 1, -1, 0, 2, 1, 1, 0, 0, -1, -1, -2, -1, -1, 0, -2, 0, 1, 2, 2, 2, 2, 1, 1, 0, -1, -2, -1, -4, -1, -2, -1, 0, 1, 1, 2, 2, 2, 2, 1, 0, -2, 0, -1, -1, -2, -1, -1, 0, 0, 1, 1, 2, 0, -1, 1, 0, 0, 0, 0, -1, 0, -1, 0, 0, 1, 1, -1).

a(n) = A008284(n+12, 12), n >= 0.

Cf. A026810, A026811, A026812, A026813, A026814, A026815, A026816.

1/(1-x)/(1-x^2)/(1-x^3)/(1-x^4)/(1-x^5)/(1-x^6)/(1-x^7)/(1-x^8)/(1-x^9)/(1-x^10)/(1-x^11)/(1-x^12)
with(combstruct):ZL13:=[S,{S=Set(Cycle(Z,card<13))}, unlabeled]:seq(count(ZL13,size=n),n=0..46); # Zerinvary Lajos, Sep 24 2007
B:=[S,{S = Set(Sequence(Z,1 <= card),card <=12)},unlabelled]: seq(combstruct[count](B, size=n), n=0..46); # Zerinvary Lajos, Mar 21 2009

CoefficientList[ Series[ 1/ Product[ 1 - x^n, {n, 1, 12} ], {x, 0, 60} ], x ]
Table[ Length[ Select[ Partitions[n], First[ # ] == 12 & ]], {n, 1, 60} ]

G.f.: 1/Product_{k=1..12}(1-x^k).

a(n) = a(n-1) + a(n-2) - a(n-5) - a(n-7) + a(n-12) - a(n-13) + 2*a(n-15) + a(n-16) + a(n-17) - a(n-20) - a(n-21) - 2*a(n-22) - a(n-23) - a(n-24) - 2*a(n-26) + a(n-28) + 2*a(n-29) + 2*a(n-30) + 2*a(n-31) + 2*a(n-32) + a(n-33) + a(n-34) - a(n-36) - 2*a(n-37) - a(n-38) - 4*a(n-39) - a(n-40) - 2*a(n-41) - a(n-42) + a(n-44) + a(n-45) + 2*a(n-46) + 2a(n-47) + 2*a(n-48) + 2*a(n-49) + a(n-50) - 2*a(n-52) - a(n-54) - a(n-55) - 2*a(n-56) - a(n-57) - a(n-58) + a(n-61) + a(n-62) + 2*a(n-63) - a(n-65) + a(n-66) - a(n-71) - a(n-73) + a(n-76) + a(n-77) - a(n-78). - David Neil McGrath, Jul 28 2015

More terms from Robert G. Wilson v, Dec 11 2000

A008766 Expansion of (1+x^5)/((1-x)(1-x^2)(1-x^3)*(1-x^4)).

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 10, 13, 18, 23, 29, 36, 45, 54, 65, 77, 91, 106, 123, 141, 162, 184, 208, 234, 263, 293, 326, 361, 399, 439, 482, 527, 576, 627, 681, 738, 799, 862, 929, 999, 1073, 1150, 1231, 1315, 1404, 1496, 1592, 1692, 1797, 1905, 2018, 2135, 2257, 2383, 2514, 2649, 2790, 2935

Offset: 0

N. J. A. Sloane

From Washington Bomfim, Jan 14 2021: (Start)
Let \n,m\ be the number of partitions of n into m non-distinct parts.
For n >= 1, \n,5\ = round((2*n^3-15*n^2+60*n-110*[n mod 2 = 0]-65*[n mod 2])/144).
For n >= 10, \n,5\ = A026811(n) - A026811(n-10).
(End)

Washington Bomfim, Table of n, a(n) for n = 0..9999 (first 1000 terms from G. C. Greubel)
Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-1,-1,1,-1,2,-1).

Cf. A026811, A001401.

a:=[1,1,2,3,5,7,10,13,18];; for n in [10..60] do a[n]:=2*a[n-1]-a[n-2]+a[n-3]-a[n-4]-a[n-5]+a[n-6]-a[n-7]+2*a[n-8]-a[n-9]; od; a; # G. C. Greubel, Sep 10 2019

R:=PowerSeriesRing(Integers(), 60); Coefficients(R!( (1+x^5)/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)) )); // G. C. Greubel, Sep 10 2019

seq(coeff(series((1+x^5)/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)), x, n+1), x, n), n = 0 .. 60); # G. C. Greubel, Sep 10 2019

CoefficientList[Series[(1+x^5)/(1-x)/(1-x^2)/(1-x^3)/(1-x^4),{x,0,60}],x] (* or *) LinearRecurrence[{2,-1,1,-1,-1,1,-1,2,-1}, {1,1,2,3,5,7,10, 13,18}, 60] (* Harvey P. Dale, Jul 24 2016 *)

Vec((1+x^5)/(1-x)/(1-x^2)/(1-x^3)/(1-x^4) +O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

seq(x) = { a = vector(x+1); my(N = 5);
for(n=0,x, a[n+1]=round((2*N^3-15*N^2+60*N-110*!(N%2)-65*(N%2))/144); N++);a};
seq(60) \\ Washington Bomfim, Jan 14 2021

def A008766_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+x^5)/((1-x)*(1-x^2)*(1-x^3)*(1-x^4))).list()
A008766_list(60) # G. C. Greubel, Sep 10 2019

a(n) = round((2*N^3 - 15*N^2 + 60*N - 110*[N mod 2=0] - 65*[N mod 2])/144), where N = n+5. - Washington Bomfim, Jan 14 2021

Terms a(45) onward added by G. C. Greubel, Sep 10 2019

A309427 Number of prime parts in the partitions of n into 5 parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 3, 5, 11, 17, 24, 32, 46, 57, 77, 92, 118, 141, 175, 204, 249, 287, 342, 390, 459, 517, 600, 672, 771, 859, 975, 1078, 1214, 1336, 1495, 1636, 1818, 1982, 2190, 2378, 2615, 2830, 3097, 3340, 3641, 3915, 4250, 4557, 4930, 5273, 5687, 6068

Offset: 0

Wesley Ivan Hurt, Aug 01 2019

nonn

			Figure 1: The partitions of n into 5 parts for n = 10, 11, ..
                                                       1+1+1+1+10
                                                        1+1+1+2+9
                                                        1+1+1+3+8
                                                        1+1+1+4+7
                                                        1+1+1+5+6
                                            1+1+1+1+9   1+1+2+2+8
                                            1+1+1+2+8   1+1+2+3+7
                                            1+1+1+3+7   1+1+2+4+6
                                            1+1+1+4+6   1+1+2+5+5
                                            1+1+1+5+5   1+1+3+3+6
                                1+1+1+1+8   1+1+2+2+7   1+1+3+4+5
                                1+1+1+2+7   1+1+2+3+6   1+1+4+4+4
                                1+1+1+3+6   1+1+2+4+5   1+2+2+2+7
                    1+1+1+1+7   1+1+1+4+5   1+1+3+3+5   1+2+2+3+6
                    1+1+1+2+6   1+1+2+2+6   1+1+3+4+4   1+2+2+4+5
                    1+1+1+3+5   1+1+2+3+5   1+2+2+2+6   1+2+3+3+5
        1+1+1+1+6   1+1+1+4+4   1+1+2+4+4   1+2+2+3+5   1+2+3+4+4
        1+1+1+2+5   1+1+2+2+5   1+1+3+3+4   1+2+2+4+4   1+3+3+3+4
        1+1+1+3+4   1+1+2+3+4   1+2+2+2+5   1+2+3+3+4   2+2+2+2+6
        1+1+2+2+4   1+1+3+3+3   1+2+2+3+4   1+3+3+3+3   2+2+2+3+5
        1+1+2+3+3   1+2+2+2+4   1+2+3+3+3   2+2+2+2+5   2+2+2+4+4
        1+2+2+2+3   1+2+2+3+3   2+2+2+2+4   2+2+2+3+4   2+2+3+3+4
        2+2+2+2+2   2+2+2+2+3   2+2+2+3+3   2+2+3+3+3   2+3+3+3+3
--------------------------------------------------------------------------
  n  |     10          11          12          13          14        ...
--------------------------------------------------------------------------
a(n) |     17          24          32          46          57        ...
--------------------------------------------------------------------------
- _Wesley Ivan Hurt_, Sep 08 2019

Index entries for sequences related to partitions

Cf. A010051, A026811.

Table[Sum[Sum[Sum[Sum[(PrimePi[i] - PrimePi[i - 1]) + (PrimePi[j] - PrimePi[j - 1]) + (PrimePi[k] - PrimePi[k - 1]) + (PrimePi[l] - PrimePi[l - 1]) + (PrimePi[n - i - j - k - l] - PrimePi[n - i - j - k - l - 1]), {i, j, Floor[(n - j - k - l)/2]}], {j, k, Floor[(n - k - l)/3]}], {k, l, Floor[(n - l)/4]}], {l, Floor[n/5]}], {n, 0, 50}]

a(n) = Sum_{l=1..floor(n/5)} Sum_{k=l..floor((n-1)/4)} Sum_{j=k..floor((n-k-l)/3)} Sum_{i=j..floor((n-j-k-l)/2)} (A010051(i) + A010051(j) + A010051(k) + A010051(l) + A010051(n-i-j-k-l)).

A026925 Number of partitions of n into an odd number of parts, the greatest being 5; also, a(n+9) = number of partitions of n+4 into an even number of parts, each <=5.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 1, 3, 3, 6, 6, 10, 10, 16, 17, 25, 27, 37, 40, 53, 57, 73, 79, 99, 107, 131, 142, 171, 184, 218, 235, 275, 296, 343, 368, 422, 452, 514, 550, 621, 663, 743, 792, 883, 939, 1041, 1106, 1220, 1294, 1421, 1505, 1646, 1740

Offset: 1

Clark Kimberling

nonn

5th column of A262920.

a(n) + A026929(n) =A026811(n). - R. J. Mathar, Aug 22 2019

A382864 Triangle read by rows: T(n,k) = T(n-k,k-1) + T(n-k,k) with T(0,0) = 1 for 0 <= k <= A003056(n).

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 2, 0, 1, 2, 1, 0, 1, 3, 1, 0, 1, 3, 2, 0, 1, 4, 3, 0, 1, 4, 4, 1, 0, 1, 5, 5, 1, 0, 1, 5, 7, 2, 0, 1, 6, 8, 3, 0, 1, 6, 10, 5, 0, 1, 7, 12, 6, 1, 0, 1, 7, 14, 9, 1, 0, 1, 8, 16, 11, 2, 0, 1, 8, 19, 15, 3, 0, 1, 9, 21, 18, 5, 0, 1, 9, 24, 23, 7

Offset: 0

Seiichi Manyama, Apr 07 2025

			First few rows are:
  1;
  0, 1;
  0, 1;
  0, 1, 1;
  0, 1, 1;
  0, 1, 2;
  0, 1, 2,  1;
  0, 1, 3,  1;
  0, 1, 3,  2;
  0, 1, 4,  3;
  0, 1, 4,  4, 1;
  0, 1, 5,  5, 1;
  0, 1, 5,  7, 2;
  0, 1, 6,  8, 3;
  0, 1, 6, 10, 5;
  0, 1, 7, 12, 6, 1;
  ...

Seiichi Manyama, Rows n = 0..481, flattened

Row sums give A000009.
Columns 0..10 give A000007, A000012, A004526(n-1), A069905(n-3), A026810(n-6), A026811(n-10), A026812(n-15), A026813(n-21), A026814(n-28), A026815(n-36), A026816(n-45).
Cf. A003056, A008284, A291954, A291960, A291968, A292047, A292049.

G.f. of column k: x^(k*(k+1)/2) / Product_{j=1..k} (1-x^j).

T(n,k) = |A292047(n,k)| = |A292049(n,k)|.

cp's OEIS Frontend

A008641 Number of partitions of n into at most 12 parts.

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A008766 Expansion of (1+x^5)/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)).

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A309427 Number of prime parts in the partitions of n into 5 parts.

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A026925 Number of partitions of n into an odd number of parts, the greatest being 5; also, a(n+9) = number of partitions of n+4 into an even number of parts, each <=5.

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A382864 Triangle read by rows: T(n,k) = T(n-k,k-1) + T(n-k,k) with T(0,0) = 1 for 0 <= k <= A003056(n).

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A008766 Expansion of (1+x^5)/((1-x)(1-x^2)(1-x^3)*(1-x^4)).