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%I #30 Nov 28 2021 02:49:32
%S 1,1,5,61,1161,30201,998413,40077493,1893550865,102951388657,
%T 6331847746581,434653328279853,32944254978940825,2732662648183661545,
%U 246228744062320481309,23949858491053731087781,2501088964314980938821153,279111248034686114681365473
%N E.g.f. satisfies: A(x) = exp( x * (1 + A(x)^4)/2 ).
%H Seiichi Manyama, <a href="/A349715/b349715.txt">Table of n, a(n) for n = 0..339</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F a(n) = (1/2^n) * Sum_{k=0..n} (4*k+1)^(n-1) * binomial(n,k).
%F E.g.f.: ( -LambertW( -2*x * exp(2*x) )/(2*x) )^(1/4).
%F G.f.: 2 * Sum_{k>=0} (4*k+1)^(k-1) * x^k/(2 - (4*k+1)*x)^(k+1).
%F a(n) ~ sqrt(1 + LambertW(exp(-1))) * 2^(n-2) * n^(n-1) / (LambertW(exp(-1))^(n + 1/4) * exp(n)). - _Vaclav Kotesovec_, Nov 26 2021
%t a[n_] := (1/2^n) * Sum[(4*k + 1)^(n - 1) * Binomial[n, k], {k, 0, n}]; Array[a, 18, 0] (* _Amiram Eldar_, Nov 27 2021 *)
%o (PARI) a(n) = sum(k=0, n, (4*k+1)^(n-1)*binomial(n, k))/2^n;
%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace((-lambertw(-2*x*exp(2*x))/(2*x))^(1/4)))
%o (PARI) my(N=20, x='x+O('x^N)); Vec(2*sum(k=0, N, (4*k+1)^(k-1)*x^k/(2-(4*k+1)*x)^(k+1)))
%Y Cf. A007889, A202617, A349714, A349716, A349719, A349720, A349721.
%K nonn
%O 0,3
%A _Seiichi Manyama_, Nov 26 2021
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