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 A306609 a(n) = Sum_{k=0..n} k*binomial(4*n+2,2*k) 1
 0, 15, 465, 11102, 236997, 4751010, 91474890, 1712391420, 31398038701, 566621243642, 10097483539038, 178113001428004, 3115342162844450, 54103694774702292, 933929099838928692, 16037182307150776056, 274132978890654857853, 4667160114821964359530, 79177297937966956038102, 1338972240005810710258452 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS Seiichi Manyama, Table of n, a(n) for n = 0..828 MathOverflow, Specific partial sum of even/odd binomial coefficients FORMULA a(n) = (n+1/2)*(16^n - binomial(4*n,2*n)) = (2*n+1)*A000346(2*n-1). -512*(4*n + 1)*(86*n + 213)*(3 + 4*n)*(n + 1)*a(n) + 32*(2336*n^4 + 8800*n^3 + 10524*n^2 + 11540*n + 9703)*a(n + 1) - 2*(n + 2)*(544*n^3 - 1072*n^2 + 1138*n + 8055)*a(n + 2) - (2*n + 5)*(26*n - 31)*(n + 3)*(n + 2)*a(n + 3) = 0. a(n) ~ 16^n * (n - sqrt(n/(2*Pi)) + 1/2). MAPLE f:= n -> (n+1/2)*(16^n-binomial(4*n, 2*n)): map(f, [\$0..30]); PROG (GAP) List([0..30], n->Sum([0..n], k->k*Binomial(4*n+2, 2*k))); # Muniru A Asiru, Mar 01 2019 CROSSREFS Cf. A000346. Sequence in context: A209488 A320097 A177080 * A272905 A005815 A120600 Adjacent sequences:  A306606 A306607 A306608 * A306610 A306611 A306612 KEYWORD nonn AUTHOR Robert Israel, Feb 28 2019 STATUS approved

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Last modified August 24 03:20 EDT 2019. Contains 326260 sequences. (Running on oeis4.)