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));
A374511
Expansion of 1/(1 - 4*x - 4*x^2)^(5/2).
Original entry on oeis.org
1, 10, 80, 560, 3640, 22512, 134400, 781440, 4451040, 24939200, 137865728, 753625600, 4080643840, 21916106240, 116877312000, 619457482752, 3265293719040, 17128725519360, 89462514606080, 465434423336960, 2412895587536896, 12468681310412800, 64242981906022400
Offset: 0
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a[n_]:=2^(n-3) Pochhammer[n+1, 4]*Hypergeometric2F1[(1-n)/2, -n/2, 3, 2]/3; Array[a,23,0] (* Stefano Spezia, Jul 10 2024 *)
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a(n) = binomial(n+4, 2)/6*sum(k=0, n\2, 2^(n-k)*binomial(n+2, n-2*k)*binomial(2*k+2, k));
A377190
Expansion of 1/(1 - 4*x^2 - 4*x^3)^(7/2).
Original entry on oeis.org
1, 0, 14, 14, 126, 252, 1050, 2772, 8778, 24948, 72072, 204204, 570570, 1585584, 4351776, 11879868, 32162130, 86582496, 231703472, 616900284, 1634721088, 4312944064, 11333823228, 29673291648, 77423101938, 201367680696, 522180220044, 1350350044316, 3482928560880
Offset: 0
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a(n) = sum(k=0, n\2, (-4)^k*binomial(-7/2, k)*binomial(k, n-2*k));
A387403
a(n) = Sum_{k=0..n} (1-i)^k * (1+i)^(n-k) * binomial(n+3,k) * binomial(n+3,n-k), where i is the imaginary unit.
Original entry on oeis.org
1, 8, 50, 280, 1484, 7616, 38304, 190080, 934560, 4564736, 22189024, 107476096, 519180480, 2502850560, 12046666752, 57912029184, 278136798720, 1334832967680, 6402435630080, 30695114813440, 147110418036736, 704860523102208, 3376580007936000, 16172904859238400
Offset: 0
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[&+[2^(n-k) * Binomial(n+3,n-2*k) * Binomial(2*k+3,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+3,n-2*k]*Binomial[2*k+3,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+3, n-2*k)*binomial(2*k+3, k));
Showing 1-4 of 4 results.