A359435 -id:A359435 - cp's OEIS frontend

A371003 a(n) = binomial(2n-1,n) - binomial(n,2)(binomial(n-1,2) + 2) - 1.

0, 0, 0, 4, 45, 281, 1358, 5790, 23229, 90667, 350130, 1348315, 5194995, 20051019, 77548994, 300527354, 1166786517, 4537546535, 17672605394, 68923231539, 269128896899, 1052049432887, 4116715304850, 16123801771169, 63205303135475, 247959266375901, 973469712709278

Offset: 1

Enrique Navarrete, Mar 07 2024

nonn

a(n) is the number of ways to place n indistinguishable balls into n distinguishable boxes with at least 3 boxes remaining empty.

a(n) is also the number of weak compositions of n into n parts in which at least three parts are zero.

			a(5)=45 since 5 can be written as 5+0+0+0+0, 0+5+0+0+0, etc. (5 such compositions); 4+1+0+0+0 (20 such compositions); 3+2+0+0+0 (20 such compositions).

Cf. A010763, A352027, A359435.

Table[Binomial[2n-1,n]-Binomial[n,2]*(Binomial[n-1,2]+2)-1,{n,27}] (* James C. McMahon, Mar 08 2024 *)

from math import comb
def A371003(n): return comb((n<<1)-1,n)-n-((m:=(n-1)**2)*(m+3)>>2) # Chai Wah Wu, Mar 29 2024

A371036 a(n) is the number of ways to place n indistinguishable balls into n distinguishable boxes with at least one box remaining empty and not all balls placed in a single box.

Original entry on oeis.org

0, 0, 6, 30, 120, 455, 1708, 6426, 24300, 92367, 352704, 1352065, 5200286, 20058285, 77558744, 300540178, 1166803092, 4537567631, 17672631880, 68923264389, 269128937198, 1052049481837, 4116715363776, 16123801841525, 63205303218850, 247959266474025, 973469712824028

Offset: 1

Enrique Navarrete, Mar 08 2024

nonn

a(n) is also the number of weak compositions of n into n parts in which at least one part is zero and the composition does not contain a single nonzero part.

			a(4)=30 since 4 can be written as 3+1+0+0, 0+3+1+0, etc. (12 such compositions); 2+2+0+0 (6 such compositions); 2+1+1+0 (12 such compositions).

Cf. A001700, A010763, A048775, A352027, A359435.

a(n) = binomial(2n-1,n)-n-1, n > 1; a(1)=0.
a(n) = A048775(n-1)-1, n > 1.
a(n) = A001700(n-1)-(n+1), n > 1.

cp's OEIS Frontend

A371003 a(n) = binomial(2n-1,n) - binomial(n,2)(binomial(n-1,2) + 2) - 1.

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A371036 a(n) is the number of ways to place n indistinguishable balls into n distinguishable boxes with at least one box remaining empty and not all balls placed in a single box.

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cp's OEIS Frontend

A371003 a(n) = binomial(2*n-1,n) - binomial(n,2)*(binomial(n-1,2) + 2) - 1.

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Mathematica

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A371036 a(n) is the number of ways to place n indistinguishable balls into n distinguishable boxes with at least one box remaining empty and not all balls placed in a single box.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A371003 a(n) = binomial(2n-1,n) - binomial(n,2)(binomial(n-1,2) + 2) - 1.