A360153
a(n) = Sum_{k=0..floor(n/3)} binomial(2*n-6*k,n-3*k).
Original entry on oeis.org
1, 2, 6, 21, 72, 258, 945, 3504, 13128, 49565, 188260, 718560, 2753721, 10588860, 40835160, 157871241, 611669250, 2374441380, 9233006541, 35956933050, 140220970200, 547490880981, 2140055896770, 8373651697800, 32795094564081, 128550662334522
Offset: 0
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A360153 := proc(n)
add(binomial(2*n-6*k,n-3*k),k=0..n/3) ;
end proc:
seq(A360153(n),n=0..70) ; # R. J. Mathar, Mar 12 2023
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a[n_] := Sum[Binomial[2*n - 6*k, n - 3*k], {k, 0, Floor[n/3]}]; Array[a, 26, 0] (* Amiram Eldar, Jan 28 2023 *)
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a(n) = sum(k=0, n\3, binomial(2*n-6*k, n-3*k));
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my(N=30, x='x+O('x^N)); Vec(1/(sqrt(1-4*x)*(1-x^3)))
A106187
Sequence array for central binomial numbers A000984.
Original entry on oeis.org
1, 2, 1, 6, 2, 1, 20, 6, 2, 1, 70, 20, 6, 2, 1, 252, 70, 20, 6, 2, 1, 924, 252, 70, 20, 6, 2, 1, 3432, 924, 252, 70, 20, 6, 2, 1, 12870, 3432, 924, 252, 70, 20, 6, 2, 1, 48620, 12870, 3432, 924, 252, 70, 20, 6, 2, 1, 184756, 48620, 12870, 3432, 924, 252, 70, 20, 6, 2, 1
Offset: 0
Triangle begins:
1;
2, 1;
6, 2, 1;
20, 6, 2, 1;
70, 20, 6, 2, 1;
252, 70, 20, 6, 2, 1;
...
The matrix inverse starts:
1;
-2,1;
-2,-2,1;
-4,-2,-2,1;
-10,-4,-2,-2,1;
-28,-10,-4,-2,-2,1;
-84,-28,-10,-4,-2,-2,1;
-264,-84,-28,-10,-4,-2,-2,1;
apparently related to A002420. - _R. J. Mathar_, Apr 08 2013
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A106187 := proc(n,k)
binomial(2*(n-k),n-k) ;
end proc: # R. J. Mathar, Apr 08 2013
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T[n_, k_] := (((2*n - 2*k)!)/((n - k)!)^2); Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Detlef Meya, Aug 11 2024 *)
A360149
a(n) = Sum_{k=0..floor(n/2)} binomial(2*n+k,n-2*k).
Original entry on oeis.org
1, 2, 7, 27, 107, 429, 1731, 7012, 28478, 115864, 471991, 1924483, 7852083, 32053208, 130893949, 534673600, 2184482707, 8926392419, 36479840422, 149095843951, 609400587426, 2490900041118, 10181669553847, 41618414303969, 170118507902985, 695366323719302
Offset: 0
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A360149 := proc(n)
add(binomial(2*n+k,n-2*k),k=0..floor(n/2)) ;
end proc:
seq(A360149(n),n=0..40) ; # R. J. Mathar, Mar 02 2023
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a[n_] := Sum[Binomial[2*n + k, n - 2*k], {k, 0, Floor[n/2]}]; Array[a, 26, 0] (* Amiram Eldar, Jan 28 2023 *)
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a(n) = sum(k=0, n\2, binomial(2*n+k, n-2*k));
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my(N=30, x='x+O('x^N)); Vec(1/(sqrt(1-4*x)*(1-x^2*(2/(1+sqrt(1-4*x)))^5)))
Showing 1-3 of 3 results.