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 A318109 a(n) = Sum_{k=0..n} (3*n-2*k)!/((n-k)!^3*k!)*(-2)^k. 5
 1, 4, 46, 652, 10186, 168304, 2884456, 50723824, 909192538, 16538659384, 304391739796, 5655971294824, 105929883322576, 1997228410630912, 37871584674309376, 721672204654077952, 13811327854028171098, 265324110145941691912, 5114208160758838538044, 98874597697991698311832, 1916741738060370782929036 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Diagonal of rational function 1/(1 - (x + y + z - 2*x*y*z)). LINKS Seiichi Manyama, Table of n, a(n) for n = 0..766 (terms 0..100 from Gheorghe Coserea) FORMULA G.f. y=A(x) satisfies: 0 = x*(x - 1)*(4*x - 1)*(8*x^2 + 20*x - 1)*y'' + (96*x^4 + 64*x^3 - 120*x^2 + 42*x - 1)*y' + 4*(2*x + 1)*(4*x^2 - 2*x + 1)*y. From Peter Bala, Mar 16 2023: (Start) n^2*(3*n - 4)*a(n) = (3*n - 2)*(21*n^2 - 35*n + 10)*a(n-1) - 4*(9*n^3 - 30*n^2 + 29*n - 6)*a(n-2) - 8*(3*n - 1)*(n - 2)^2*a(n-3) with a(0) = 1, a(1) = 4 and a(2) = 46. Conjecture: the supercongruence a(n*p^r) == a(n*p^(r-1)) (mod p^(2*r)) holds for positive integers n and r and all primes p >= 5. (End) a(n) ~ (1 + sqrt(3))^(3*n + 1) / (2*Pi*sqrt(3)*n). - Vaclav Kotesovec, Mar 17 2023 EXAMPLE A(x) = 1 + 4*x + 46*x^2 + 652*x^3 + 10186*x^4 + 168304*x^5 + 2884456*x^6 + ... PROG (PARI) a(n) = sum(k=0, n, (3*n-2*k)!/((n-k)!^3*k!)*(-2)^k); vector(21, n, a(n-1)) CROSSREFS Cf. A000172, A124435, A318107, A318108. Sequence in context: A236956 A113264 A264717 * A234527 A126739 A191870 Adjacent sequences: A318106 A318107 A318108 * A318110 A318111 A318112 KEYWORD nonn,easy AUTHOR Gheorghe Coserea, Sep 20 2018 STATUS approved

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Last modified September 26 15:19 EDT 2023. Contains 365660 sequences. (Running on oeis4.)