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
A360858
Triangle read by rows. T(n, k) = binomial(n + 1, ceil(k/2)) * binomial(n, floor(k/2)).
Original entry on oeis.org
1, 1, 2, 1, 3, 6, 1, 4, 12, 18, 1, 5, 20, 40, 60, 1, 6, 30, 75, 150, 200, 1, 7, 42, 126, 315, 525, 700, 1, 8, 56, 196, 588, 1176, 1960, 2450, 1, 9, 72, 288, 1008, 2352, 4704, 7056, 8820, 1, 10, 90, 405, 1620, 4320, 10080, 17640, 26460, 31752
Offset: 0
Triangle T(n, k) starts:
[0] 1;
[1] 1, 2;
[2] 1, 3, 6;
[3] 1, 4, 12, 18;
[4] 1, 5, 20, 40, 60;
[5] 1, 6, 30, 75, 150, 200;
[6] 1, 7, 42, 126, 315, 525, 700;
[7] 1, 8, 56, 196, 588, 1176, 1960, 2450;
[8] 1, 9, 72, 288, 1008, 2352, 4704, 7056, 8820;
[9] 1, 10, 90, 405, 1620, 4320, 10080, 17640, 26460, 31752.
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A360858 := (n, k) -> binomial(n + 1, ceil(k/2))*binomial(n, floor(k/2)):
seq(seq(A360858(n, k), k = 0..n), n = 0..9);
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from math import comb
def A360858_T(n,k): return comb(n,m:=k>>1)**2*(n+1)//(m+1 if k&1 else n+1-m) # Chai Wah Wu, Feb 28 2023
A360861
a(n) = Sum_{k=0..n} binomial(n, ceiling(k/2)) * binomial(n, floor(k/2)).
Original entry on oeis.org
1, 2, 7, 22, 81, 281, 1058, 3830, 14605, 54127, 208110, 782761, 3027038, 11501478, 44668692, 170974710, 666220005, 2564271875, 10018268150, 38728479647, 151631858378, 588229029258, 2307174835212, 8975958379817, 35258881445606, 137501193282026, 540821096592028
Offset: 0
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A360861[n_]:=(Binomial[2n+1,n]+Binomial[n,Floor[n/2]]^2)/2;
Array[A360861,30,0] (* Paolo Xausa, Dec 11 2023 *)
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a(n):=(1/2)*(binomial(2*n+1,n)+(binomial(n,floor(n/2)))^2); /* Tani Akinari, Jul 12 2023 */
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from math import comb
def A360861(n): return sum(comb(n,m:=k>>1)**2*(n-m)//(m+1) for k in range(1,n+1,2)) + sum(comb(n,k>>1)**2 for k in range(0,n+1,2)) # Chai Wah Wu, Feb 28 2023
Showing 1-3 of 3 results.