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A157163 Product_{n>=1} (1 + 2*a(n)*x^n) = Sum_{k>=0} binomial(2*k, k)*x^k = 1/sqrt(1 - 4*x), with the central binomial numbers A000984(n).

%I A157163 #31 Apr 30 2025 23:16:41
%S A157163 1,3,4,27,48,156,576,2955,7168,27792,95232,352188,1290240,5105856,
%T A157163 17743872,77010795,252641280,1000224768,3616800768,14484040464,
%U A157163 52102692864,208963943424,764877471744,3025006038012,11258183024640,44968060784640,166308918329344
%N A157163 Product_{n>=1} (1 + 2*a(n)*x^n) = Sum_{k>=0} binomial(2*k, k)*x^k = 1/sqrt(1 - 4*x), with the central binomial numbers A000984(n).
%C A157163 In the original problem 2*a(n) = [2, 6, 8, 54, 96, 312, 1152, 5910, 14336, 55584, 190464, 704376, ...] appears.
%H A157163 Robert Israel, <a href="/A157163/b157163.txt">Table of n, a(n) for n = 1..1665</a>
%H A157163 Giedrius Alkauskas, <a href="http://arxiv.org/abs/0801.0805">One curious proof of Fermat's little theorem</a>, arXiv:0801.0805 [math.NT], 2008.
%H A157163 Giedrius Alkauskas, <a href="https://www.jstor.org/stable/40391097">A curious proof of Fermat's little theorem</a>, Amer. Math. Monthly 116(4) (2009), 362-364.
%H A157163 Giedrius Alkauskas, <a href="https://arxiv.org/abs/1609.09842">Algebraic functions with Fermat property, eigenvalues of transfer operator and Riemann zeros, and other open problems</a>, arXiv:1609.09842 [math.NT], 2016.
%H A157163 Art of Problem Solving, <a href="https://artofproblemsolving.com/community/c7h261410">Product formula for a generating function</a>
%H A157163 H. Gingold, H. W. Gould, and Michael E. Mays, <a href="https://www.researchgate.net/publication/268023169_Power_product_expansions">Power Product Expansions</a>, Utilitas Mathematica 34 (1988), 143-161.
%H A157163 H. Gingold and A. Knopfmacher, <a href="http://dx.doi.org/10.4153/CJM-1995-062-9">Analytic properties of power product expansions</a>, Canad. J. Math. 47 (1995), 1219-1239.
%H A157163 W. Lang, <a href="/A157162/a157162.txt">Recurrences for the general problem</a>.
%F A157163 Recurrence I: With FP(n,m) the set of partitions of n with m distinct parts (which could be called fermionic partitions fp):
%F A157163 a(n) = binomial(2*n, n)/2 - Sum_{m=2..maxm(n)} 2^(m-1)*(Sum_{fp from FP(n,m)} (Product_{j=1..m} a(k[j]))), with maxm(n) = A003056(n) and the distinct parts k[j], j = 1..m, of the partition fp of n, n >= 3. Inputs a(1) = 1, a(2) = 3. See the array A008289(n,m) for the cardinality of the set FP(n,m).
%F A157163 Recurrence II: With P(n,m) the set of all partitions of n with m parts, and the multinomial numbers M0 (given for every partition under A048996):
%F A157163 a(n) = (1/2)*Sum_{d|n, 1<d<n} (d/n)*(-2*a(d))^(n/d) + (1/2)*Sum_{m=1..n-1} (-1)^(m-1)*(1/m)*(Sum{p=(1^e(1),...,n^e(n)) from P(n,m)} M0(p)*cbi(1)^e(1)*...*cbi(n)^e(n)), n >= 2; a(1) = 1, with cbi(n) = binomial(2*n, n) = A000984(n). The exponents e(j) >= 0 satisfy Sum_{j=1..n} j*e(j) =n and Sum_{j=1..n} e(j) = m. The M0 numbers are m!/(Product_{j=1..n} (e(j))!).
%F A157163 Recurrence II (rewritten, due to email from V. Jovovic, Mar 10 2009):
%F A157163 a(n) = ((-2*a(1))^n + Sum_{d|n, 1<d<n} (d*(-2*a(d))^(n/d) + B(n))/(2*n), with the o.g.f. x*(log(1/sqrt(1 - 4*x)))' = 2*x/(1 - 4*x) for the sequence {B(n)}; hence B(n) = (1/2)*4^n.
%e A157163 Recurrence I: a(4) = binomial(8, 4)/2 - 2*a(1)*a(3) = 35 - 8 = 27.
%e A157163 Recurrence II: a(4) = (1/2)*(1/2)*(-2*a(2))^2 + (1/2)*(1*cbi(4) - (1/2)*(2*cbi(1)*cbi(3) + 1*cbi(2)^2) + (1/3)*3*cbi(1)^2*cbi(2)) = 27.
%e A157163 Recurrence II (rewritten): a(4)= (1/8)*((-2)^4 + 2*(-2*a(2))^2 + (1/2)*4^4) = 27.
%p A157163 N:= 100: # for a(1)..a(N)
%p A157163 S:= convert(series(-ln(1-4*x)/2,x,N+1),polynom):
%p A157163 for n from 1 to N do
%p A157163   a[n]:= coeff(S,x,n)/2;
%p A157163   S:= S - add((-1)^(k-1)*(2*a[n])^k*x^(k*n)/k, k=1..N/n)
%p A157163 od:
%p A157163 seq(a[n],n=1..N); # _Robert Israel_, Jan 03 2019
%o A157163 (PARI) a(n) = if (n==1, 1, (1/(2*n))*((-2*a(1))^n + sumdiv(n, d, if ((d!=1) && (d!=n), d*(-2*a(d))^(n/d), 0)) + 4^n/2)); \\ after 2nd Recurrence II; _Michel Marcus_, Jul 06 2015
%Y A157163 Cf. A147542 (Fibonacci), A157161 (Catalan).
%K A157163 nonn,easy
%O A157163 1,2
%A A157163 _Wolfdieter Lang_, Aug 10 2009