A374497
Expansion of 1/(1 - 4*x - 4*x^2)^(3/2).
Original entry on oeis.org
1, 6, 36, 200, 1080, 5712, 29792, 153792, 787680, 4009280, 20304768, 102405888, 514678528, 2579028480, 12890311680, 64283809792, 319954540032, 1589720712192, 7886437652480, 39069462835200, 193307835764736, 955361266917376, 4716674314223616, 23264437702656000
Offset: 0
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a[n_]:= Sum[(2*k+1)*Binomial[2*k,k]*Binomial[k,n-k],{k,0,n}]; Array[a,24,0] (* Stefano Spezia, May 08 2025 *)
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a(n) = binomial(n+2, 2)*sum(k=0, n\2, 2^(n-k)*binomial(n, 2*k)*binomial(2*k, k)/(k+1));
A374513
Expansion of 1/(1 - 4*x - 4*x^2)^(7/2).
Original entry on oeis.org
1, 14, 140, 1176, 8904, 62832, 421344, 2718144, 17008992, 103847744, 621292672, 3654187264, 21182563584, 121263109632, 686660004864, 3851149940736, 21416533501440, 118199459288064, 647926485764096, 3529938203545600, 19124354344775680
Offset: 0
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a[n_]:=2^(n-4) Pochhammer[n+1, 6]*Hypergeometric2F1[(1-n)/2, -n/2, 4, 2]/45; Array[a,21,0] (* Stefano Spezia, Jul 10 2024 *)
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a(n) = binomial(n+6, 3)/20*sum(k=0, n\2, 2^(n-k)*binomial(n+3, n-2*k)*binomial(2*k+3, k));
A377189
Expansion of 1/(1 - 4*x^2 - 4*x^3)^(5/2).
Original entry on oeis.org
1, 0, 10, 10, 70, 140, 490, 1260, 3570, 9660, 25872, 69300, 182490, 480480, 1252680, 3255252, 8412690, 21655920, 55535480, 141921780, 361577216, 918529040, 2327337740, 5882631040, 14836032770, 37339221192, 93794645700, 235186913780, 588736957920, 1471462327160
Offset: 0
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a(n) = sum(k=0, n\2, (-4)^k*binomial(-5/2, k)*binomial(k, n-2*k));
A387402
a(n) = Sum_{k=0..n} (1-i)^k * (1+i)^(n-k) * binomial(n+2,k) * binomial(n+2,n-k), where i is the imaginary unit.
Original entry on oeis.org
1, 6, 32, 160, 780, 3752, 17920, 85248, 404640, 1918400, 9090048, 43064320, 204032192, 966887040, 4583424000, 21735350272, 103114538496, 489392157696, 2323701678080, 11037970513920, 52454251902976, 249373626208256, 1186024281341952, 5642924625100800, 26858183388774400, 127880625111662592
Offset: 0
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[&+[2^(n-k) * Binomial(n+2,n-2*k) * Binomial(2*k+2,k): k in [0..Floor (n/2)]]: n in [0..35]]; // Vincenzo Librandi, Sep 04 2025
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Table[Sum[2^(n-k)*Binomial[n+2,n-2*k]*Binomial[2*k+2,k],{k,0,Floor[n/2]}],{n,0,30}] (* Vincenzo Librandi, Sep 04 2025 *)
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a(n) = sum(k=0, n\2, 2^(n-k)*binomial(n+2, n-2*k)*binomial(2*k+2, k));
Showing 1-4 of 4 results.