A026816 -id:A026816 - cp's OEIS frontend

A326598 Sum of the largest parts of the partitions of n into 10 parts.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 5, 9, 17, 27, 46, 69, 108, 158, 234, 329, 471, 645, 891, 1198, 1614, 2125, 2808, 3637, 4718, 6029, 7699, 9709, 12243, 15265, 19013, 23473, 28933, 35381, 43211, 52396, 63436, 76343, 91710, 109580, 130720, 155171, 183884

Offset: 0

Wesley Ivan Hurt, Jul 13 2019

nonn

Index entries for sequences related to partitions

Cf. A026816, A326588, A326589, A326590, A326591, A326592, A326593, A326594, A326595, A326596, A326597.

Table[Total[IntegerPartitions[n,{10}][[;;,1]]],{n,0,50}] (* Harvey P. Dale, May 02 2025 *)

a(n) = Sum_{r=1..floor(n/10)} Sum_{q=r..floor((n-r)/9)} Sum_{p=q..floor((n-q-r)/8)} Sum_{o=p..floor((n-p-q-r)/7)} Sum_{m=o..floor((n-o-p-q-r)/6)} Sum_{l=m..floor((n-m-o-p-q-r)/5)} Sum_{k=l..floor((n-l-m-o-p-q-r)/4)} Sum_{j=k..floor((n-k-l-m-o-p-q-r)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p-q-r)/2)} (n-i-j-k-l-m-o-p-q-r).

a(n) = A326588(n) - A326589(n) - A326590(n) - A326591(n) - A326592(n) - A326593(n) - A326594(n) - A326595(n) - A326596(n) - A326597(n).

A060029 Expansion of (1-x-x^N)/((1-x)(1-x^2)(1-x^3)...(1-x^N)) for N = 10.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 4, 4, 7, 8, 11, 12, 18, 19, 26, 29, 37, 40, 51, 53, 65, 68, 79, 80, 92, 87, 94, 84, 82, 58, 45, -1, -36, -109, -180, -297, -413, -594, -780, -1042, -1325, -1704, -2112, -2647, -3228, -3961, -4772, -5769, -6867, -8206, -9682, -11446, -13402, -15710

Offset: 0

N. J. A. Sloane, Mar 17 2001

Difference of the number of partitions of n+9 into 9 parts and the number of partitions of n+9 into 10 parts. - Wesley Ivan Hurt, Apr 16 2019

Ray Chandler, Table of n, a(n) for n = 0..1000
P. A. MacMahon, Perpetual reciprocants, Proc. London Math. Soc., 17 (1886), 139-151; Coll. Papers II, pp. 584-596.
Index entries for sequences related to partitions
Index entries for linear recurrences with constant coefficients, signature (1, 1, 0, 0, -1, 0, -1, 0, 0, 0, -1, 1, 1, 1, 2, 0, 0, -1, -1, -1, -1, -3, 0, 0, 1, 1, 2, 2, 1, 1, 0, 0, -3, -1, -1, -1, -1, 0, 0, 2, 1, 1, 1, -1, 0, 0, 0, -1, 0, -1, 0, 0, 1, 1, -1).

Cf. A026815, A026816.

Cf. For other values of N: A060022 (N=3), A060023 (N=4), A060024 (N=5), A060025 (N=6), A060026 (N=7), A060027 (N=8), A060028 (N=9), this sequence (N=10).

CoefficientList[Series[(1-x-x^10)/Times@@(1-x^Range[10]),{x,0,60}],x] (* Harvey P. Dale, May 15 2016 *)

a(n) = A026815(n+9) - A026816(n+9). - Wesley Ivan Hurt, Apr 16 2019

A008639 Number of partitions of n into at most 10 parts.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 55, 75, 97, 128, 164, 212, 267, 340, 423, 530, 653, 807, 984, 1204, 1455, 1761, 2112, 2534, 3015, 3590, 4242, 5013, 5888, 6912, 8070, 9418, 10936, 12690, 14663, 16928, 19466, 22367, 25608, 29292, 33401, 38047

Offset: 0

N. J. A. Sloane

For n > 9: also number of partitions of n into parts <= 10: a(n) = A026820(n, 10). - 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 359
Index entries for related partition-counting sequences
Index entries for linear recurrences with constant coefficients, signature (1, 1, 0, 0, -1, 0, -1, 0, 0, 0, -1, 1, 1, 1, 2, 0, 0, -1, -1, -1, -1, -3, 0, 0, 1, 1, 2, 2, 1, 1, 0, 0, -3, -1, -1, -1, -1, 0, 0, 2, 1, 1, 1, -1, 0, 0, 0, -1, 0, -1, 0, 0, 1, 1, -1).

Essentially same as A026816.
a(n) = A008284(n + 10, 10), n >= 0.
Cf. A266778 (first differences), A288345 (partial sums).

CoefficientList[ Series[ 1/ Product[ 1 - x^n, {n, 1, 10} ], {x, 0, 60} ], x ]

Vec(1/prod(k=1,10,1-x^k)+O(x^99)) \\ Charles R Greathouse IV, May 06 2015

G.f.: 1/Product_{k=1..10} (1 - x^k). - David Neil McGrath, Apr 29 2015

a(n) = a(n-10) + A008638(n). - Vladimír Modrák, Sep 29 2020

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

Original entry on oeis.org

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

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

A326598 Sum of the largest parts of the partitions of n into 10 parts.

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A060029 Expansion of (1-x-x^N)/((1-x)(1-x^2)(1-x^3)...(1-x^N)) for N = 10.

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A008639 Number of partitions of n into at most 10 parts.

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

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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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