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 A322746 a(n) = 1/2 * (-1 + Sum_{k=0..n} binomial(2*n,2*k)*(n+1)^(n-k)*n^k). 4
 0, 1, 24, 675, 25920, 1275125, 76545000, 5425069447, 443365544448, 41047124680809, 4245890890571000, 485307363135371051, 60742714406414040000, 8262695239025750162653, 1213734518568509516047560, 191478489107270936785743375, 32288451913272713227175006208 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 LINKS Seiichi Manyama, Table of n, a(n) for n = 0..321 Wikipedia, Chebyshev polynomials. FORMULA sqrt(a(n)+1) + sqrt(a(n)) = (sqrt(n+1) + sqrt(n))^n. sqrt(a(n)+1) - sqrt(a(n)) = (sqrt(n+1) - sqrt(n))^n. a(n) = (A173174(n) - 1)/2. a(n) ~ exp(1/2) * 2^(2*n - 2) * n^n. - Vaclav Kotesovec, Dec 25 2018 EXAMPLE (sqrt(3) + sqrt(2))^2 = 5 + 2*sqrt(6) = sqrt(25) + sqrt(24). So a(2) = 24. PROG (PARI) {a(n) = 1/2*(-1+sum(k=0, n, binomial(2*n, 2*k)*(n+1)^(n-k)*n^k))} (PARI) {a(n) = (polchebyshev(n, 1, 2*n+1)-1)/2} CROSSREFS Main diagonal of A322699. Cf. A322747. Sequence in context: A268473 A291066 A160038 * A184274 A093456 A189412 Adjacent sequences:  A322743 A322744 A322745 * A322747 A322748 A322749 KEYWORD nonn AUTHOR Seiichi Manyama, Dec 25 2018 STATUS approved

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Last modified November 26 17:52 EST 2021. Contains 349343 sequences. (Running on oeis4.)