A359435
a(n) = binomial(2*n-1,n) - n^2 - 1.
Original entry on oeis.org
0, 18, 100, 425, 1666, 6370, 24228, 92277, 352594, 1351933, 5200130, 20058103, 77558534, 300539938, 1166802820, 4537567325, 17672631538, 68923264009, 269128936778, 1052049481375, 4116715363270, 16123801840973, 63205303218250, 247959266473375, 973469712823326
Offset: 3
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Table[Binomial[2*n - 1, n] - n^2 - 1, {n, 3, 30}] (* Wesley Ivan Hurt, Jan 20 2024 *)
A371003
a(n) = binomial(2*n-1,n) - binomial(n,2)*(binomial(n-1,2) + 2) - 1.
Original entry on oeis.org
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
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).
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Table[Binomial[2n-1,n]-Binomial[n,2]*(Binomial[n-1,2]+2)-1,{n,27}] (* James C. McMahon, Mar 08 2024 *)
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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
A370197
a(n) is the number of ways to place n indistinguishable balls into n distinguishable boxes with at least 4 boxes remaining empty.
Original entry on oeis.org
0, 0, 0, 0, 5, 81, 658, 3830, 18525, 80587, 330330, 1312015, 5132075, 19946915, 77383374, 300272554, 1166405717, 4536991655, 17671814690, 68922126879, 269127380699, 1052047384687, 4116712577510, 16123798186665, 63205298480275, 247959260395901, 973469705104278
Offset: 1
a(6)=81 since 6 can be written as 6+0+0+0+0+0, 0+6+0+0+0+0, etc. (6 such compositions); 5+1+0+0+0+0 (30 such compositions); 4+2+0+0+0+0 (30 such compositions); 3+3+0+0+0+0 (15 such compositions).
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
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).
Showing 1-4 of 4 results.
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