A110145
a(n) = Sum_{k=0..n} C(n,k)^2*mod(k,2).
Original entry on oeis.org
0, 1, 4, 10, 32, 126, 472, 1716, 6400, 24310, 92504, 352716, 1351616, 5200300, 20060016, 77558760, 300533760, 1166803110, 4537591960, 17672631900, 68923172032, 269128937220, 1052049834576, 4116715363800, 16123800489472, 63205303218876, 247959271674352
Offset: 0
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Table[Sum[Binomial[n,k]^2 Mod[k,2],{k,0,n}],{n,0,30}] (* Harvey P. Dale, Feb 21 2013 *)
Table[(Binomial[2 n, n] - Binomial[n, n/2] Cos[Pi n/2])/2, {n, 0, 30}] (* Vladimir Reshetnikov, Oct 04 2016 *)
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a(n) = sum(k=0, n, binomial(n, k)^2*(k % 2)); \\ Michel Marcus, Oct 05 2016
A037976
a(n) = (1/4)*(binomial(4*n, 2*n) - (-1)^n*binomial(2*n, n) + (1-(-1)^n)*binomial(2*n, n)^2).
Original entry on oeis.org
0, 4, 16, 436, 3200, 78004, 675808, 15919320, 150266880, 3450748180, 34461586016, 774842070600, 8061900244736, 178065876017176, 1912172640160960, 41596867935469936, 458156035085377536
Offset: 0
- The right-hand side of a binomial coefficient identity in H. W. Gould, Combinatorial Identities, Morgantown, 1972. (See (3.75) on page 31.)
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[(1/4)*((2*n+1)*Catalan(2*n) -(-1)^n*(n+1)*Catalan(n) +(1-(-1)^n)*(n+1)^2*Catalan(n)^2): n in [0..30]]; // G. C. Greubel, Jun 22 2022
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A037976 := proc(n)
binomial(4*n,2*n)/4-(-1)^n*binomial(2*n,n)/4+(1-(-1)^n)*binomial(2*n,n)^2/4 ;
end proc:
seq(A037976(n),n=0..30) ; # R. J. Mathar, Jul 26 2015
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With[{B=Binomial}, Table[(1/4)*(B[4*n,2*n] +B[2*n,n]^2 -2*(-1)^n*B[B[2*n,n] +1, 2]), {n,0,30}]] (* G. C. Greubel, Jun 22 2022 *)
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b=binomial; [(1/4)*(b(4*n, 2*n) -(-1)^n*b(2*n, n) +(1-(-1)^n)*b(2*n, n)^2) for n in (0..30)] # G. C. Greubel, Jun 22 2022
A098772
a(n) = Sum_{k=0..n} binomial(2*n,2*k)^2.
Original entry on oeis.org
1, 2, 38, 452, 6470, 92252, 1352540, 20056584, 300546630, 4537543340, 68923356788, 1052049129144, 16123803193628, 247959261273752, 3824345320438520, 59132290704871952, 916312070771835462, 14226520736453485260, 221256270142955957252, 3446310324328958045400, 53753604366737011495220
Offset: 0
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Table[Sum[Binomial[2n,2k]^2,{k,0,n}],{n,0,20}] (* Harvey P. Dale, Jan 21 2016 *)
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makelist((binomial(4*n,2*n)+(-1)^n*binomial(2*n,n))/2,n,0,12); /* Emanuele Munarini, Feb 01 2017 */
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a(n) = sum(k=0, n, binomial(2*n, 2*k)^2); \\ Michel Marcus, Feb 01 2017
Showing 1-3 of 3 results.
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