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A244432 E.g.f.: exp( Sum_{n>=1} Pell(n)*x^n/n ), where Pell(n) = A000129(n).

%I A244432 #32 Nov 28 2023 00:22:25
%S A244432 1,1,3,17,137,1437,18547,284221,5042289,101635289,2294115299,
%T A244432 57323597289,1570795420537,46836600355573,1509632295204243,
%U A244432 52303597825637333,1938434314587648353,76521195859545355569,3205495988651927796931,142018837513142207290561
%N A244432 E.g.f.: exp( Sum_{n>=1} Pell(n)*x^n/n ), where Pell(n) = A000129(n).
%C A244432 a(n) == 0 (mod 7^k) for n >= 7*k, for k>=1 (conjecture).
%H A244432 Vaclav Kotesovec, <a href="/A244432/b244432.txt">Table of n, a(n) for n = 0..390</a>
%F A244432 E.g.f.: ( (1 + x/S) / (1 - S*x) )^(sqrt(2)/4) where S = sqrt(2) + 1.
%F A244432 E.g.f.: exp( Integral 1/(1-2*x-x^2) dx ).
%F A244432 a(n) ~ n! * 2^(3*sqrt(2)/8) * n^(sqrt(2)/4-1) * (1+sqrt(2))^(n-1/(2*sqrt(2))) / GAMMA(1/(2*sqrt(2))). - _Vaclav Kotesovec_, Jun 28 2014
%F A244432 a(0) = a(1) = 1; a(n) = (2*n-1) * a(n-1) + (n-1) * (n-2) * a(n-2). - _Ilya Gutkovskiy_, Aug 13 2021
%F A244432 E.g.f.: exp((1/sqrt(2)) * arctanh(x*sqrt(2)/(1-x))). - _Fabian Pereyra_, Oct 11 2023
%F A244432 a(n) = n!*Sum_{k=0..n} binomial(n-1,k-1)*binomial(1/sqrt(8),k)*(1+sqrt(2))^(n-k)*(sqrt(8))^k. - _Fabian Pereyra_, Oct 19 2023
%e A244432 E.g.f.: A(x) = 1 + x + 3*x^2 + 17*x^3/3! + 137*x^4/4! + 1437*x^5/5! + ...
%e A244432 where
%e A244432 log(A(x)) = x + 2*x^2/2 + 5*x^3/3 + 12*x^4/4 + 29*x^5/5 + 70*x^6/6 + 169*x^7/7 + 408*x^8/8 + 985*x^9/9 + ... + A000129(n)*x^n/n + ...
%o A244432 (PARI) {a(n)=n!*polcoeff(exp(intformal(1/(1-2*x-x^2 +x*O(x^n)))), n)}
%o A244432 for(n=0, 30, print1(a(n), ", "))
%Y A244432 Cf. A000129, A244430.
%K A244432 nonn
%O A244432 0,3
%A A244432 _Paul D. Hanna_, Jun 27 2014