A360859
Triangle read by rows. T(n, k) = binomial(n, ceil(k/2)) * binomial(n, floor(k/2)).
Original entry on oeis.org
1, 1, 1, 1, 2, 4, 1, 3, 9, 9, 1, 4, 16, 24, 36, 1, 5, 25, 50, 100, 100, 1, 6, 36, 90, 225, 300, 400, 1, 7, 49, 147, 441, 735, 1225, 1225, 1, 8, 64, 224, 784, 1568, 3136, 3920, 4900, 1, 9, 81, 324, 1296, 3024, 7056, 10584, 15876, 15876, 1, 10, 100, 450, 2025, 5400, 14400, 25200, 44100, 52920, 63504
Offset: 0
Triangle T(n, k) starts:
[0] 1;
[1] 1, 1;
[2] 1, 2, 4;
[3] 1, 3, 9, 9;
[4] 1, 4, 16, 24, 36;
[5] 1, 5, 25, 50, 100, 100;
[6] 1, 6, 36, 90, 225, 300, 400;
[7] 1, 7, 49, 147, 441, 735, 1225, 1225;
[8] 1, 8, 64, 224, 784, 1568, 3136, 3920, 4900;
[9] 1, 9, 81, 324, 1296, 3024, 7056, 10584, 15876, 15876;
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A360859 := (n, k) -> binomial(n, ceil(k/2)) * binomial(n, floor(k/2)):
seq(seq(A360859(n, k), k = 0..n), n = 0..10);
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from math import comb
def A360859_T(n,k): return comb(n,m:=k>>1)**2*(n-m)//(m+1) if k&1 else comb(n,k>>1)**2 # Chai Wah Wu, Feb 28 2023
A360857
Triangle read by rows. T(n, k) = binomial(n, ceil(k/2)) * binomial(n + 1, floor(k/2)).
Original entry on oeis.org
1, 1, 1, 1, 2, 6, 1, 3, 12, 12, 1, 4, 20, 30, 60, 1, 5, 30, 60, 150, 150, 1, 6, 42, 105, 315, 420, 700, 1, 7, 56, 168, 588, 980, 1960, 1960, 1, 8, 72, 252, 1008, 2016, 4704, 5880, 8820, 1, 9, 90, 360, 1620, 3780, 10080, 15120, 26460, 26460
Offset: 0
Table T(n, k) starts:
[0] 1;
[1] 1, 1;
[2] 1, 2, 6;
[3] 1, 3, 12, 12;
[4] 1, 4, 20, 30, 60;
[5] 1, 5, 30, 60, 150, 150;
[6] 1, 6, 42, 105, 315, 420, 700;
[7] 1, 7, 56, 168, 588, 980, 1960, 1960;
[8] 1, 8, 72, 252, 1008, 2016, 4704, 5880, 8820;
[9] 1, 9, 90, 360, 1620, 3780, 10080, 15120, 26460, 26460.
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A360857 := (n, k) -> binomial(n, ceil(k/2))*binomial(n + 1, floor(k/2)):
seq(seq(A360857(n, k), k=0..n), n=0..9);
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Table[Binomial[n,Ceiling[k/2]]Binomial[n+1,Floor[k/2]],{n,0,10},{k,0,n}]//Flatten (* Harvey P. Dale, Mar 06 2023 *)
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from math import comb
def A360857_T(n,k): return comb(n+1,m:=k>>1)**2*(n+1-m)*(n-m)//((m+1)*(n+1)) if k&1 else comb(n+1,m:=k>>1)**2*(n+1-m)//(n+1) # Chai Wah Wu, Feb 28 2023
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