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A240722 Expansion of log'((-1-sqrt(1-8*x)+sqrt(2+2*sqrt(1-8*x)+8*x))/(4*x)).

%I A240722 #32 Nov 09 2024 00:44:21
%S A240722 1,7,46,323,2326,17062,126764,950819,7184422,54604082,416990564,
%T A240722 3197008718,24592858876,189719297068,1467180979096,11370769197347,
%U A240722 88291427756294,686715981892666,5349188181210548,41723881423351898,325845079538136596
%N A240722 Expansion of log'((-1-sqrt(1-8*x)+sqrt(2+2*sqrt(1-8*x)+8*x))/(4*x)).
%H A240722 Vincenzo Librandi, <a href="/A240722/b240722.txt">Table of n, a(n) for n = 0..200</a>
%F A240722 a(n) = (n+1)*Sum_{k=0..n} (binomial(k+1, n-k) * binomial(n+2*k+2, n+k+1) / (n+k+2)).
%F A240722 A(x) = B'(x)/B(x) where B(x) = (-1-sqrt(1-8*x) + sqrt(2+2*sqrt(1-8*x)+8*x)) / (4*x) is the g.f. of A186997.
%F A240722 a(n) ~ 2^(3*n+2)/sqrt(3*Pi*n). - _Vaclav Kotesovec_, Apr 12 2014
%F A240722 Conjecture: -(3*n-1)*(2*n+1)*(n+1)*a(n) +2*(21*n^3+14*n^2-2*n-1)*a(n-1) +8*n*(3*n+2)*(2*n-1)*a(n-2)=0. - _R. J. Mathar_, Jun 14 2016
%F A240722 From _Peter Bala_, Jul 27 2024: (Start)
%F A240722 The following remarks assume an offset of 1.
%F A240722 a(n) = [x^n] c(x + x^2)^n, where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108.
%F A240722 It follows that the Gauss congruences hold: a(n*p^r) == a(n*p^(r-1)) (mod p^r) for all primes p and positive integers n and r.
%F A240722 Calculation suggests that the stronger supercongruences a(n*p^r) == a(n*p^(r-1)) (mod p^(2*r)) hold for all primes p >= 3. (End)
%t A240722 CoefficientList[Series[(1/(16*x*(1 + x)*(-1 + 8*x)))*(8 - Sqrt[2]*Sqrt[1 + Sqrt[1 - 8*x] + 4*x] - 3*Sqrt[2 - 16*x]*Sqrt[1 + Sqrt[1 - 8*x] + 4*x] + 8*x*(-7 - 8*x + Sqrt[2]*Sqrt[1 + Sqrt[1 - 8*x] + 4*x])), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Apr 12 2014 *)
%o A240722 (Maxima)
%o A240722 a(n):=(n+1)*sum((binomial(k+1,n-k)*binomial(n+2*k+2,n+k+1))/(n+k+2),k,0,n);
%o A240722 (PARI) for(n=0, 30, print1((n+1)*sum(k=0, n, (binomial(k+1, n-k)*binomial(n +2*k +2, n+k+1))/(n+k+2)), ", ")) \\ _G. C. Greubel_, Sep 30 2018
%o A240722 (Magma) [(n+1)*(&+[Binomial(k+1,n-k)*Binomial(n+2*k+2, n+k+1)/(n+k+2): k in [0..n]]): n in [0..30]]; // _G. C. Greubel_, Sep 30 2018
%Y A240722 Cf. A186997.
%K A240722 nonn,easy
%O A240722 0,2
%A A240722 _Vladimir Kruchinin_, Apr 11 2014