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 A206850 G.f.: exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n^2, k^2) * x^k ). 9

%I #10 Sep 07 2013 20:37:07

%S 1,1,2,4,8,56,522,5972,424954,16560881,1528544877,483389731955,

%T 70609119680761,53933819677734187,58734216507052608587,

%U 38789122414735365076327,202547156817505166242299130,712808848212730366850407506134,2914935606380176735260119042755221

%N G.f.: exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n^2, k^2) * x^k ).

%C Equals antidiagonal sums of triangle A228902.

%e G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 8*x^4 + 56*x^5 + 522*x^6 + 5972*x^7 +...

%e such that, by definition, the logarithm equals the series:

%e log(A(x)) = x*(1+x) + x^2*(1 + 4*x + x^2)/2

%e + x^3*(1 + 9*x + 126*x^2 + x^3)/3

%e + x^4*(1 + 16*x + 1820*x^2 + 11440*x^3 + x^4)/4

%e + x^5*(1 + 25*x + 12650*x^2 + 2042975*x^3 + 2042975*x^4 + x^5)/5

%e + x^6*(1 + 36*x + 58905*x^2 + 94143280*x^3 + 7307872110*x^4 + 600805296*x^5 + x^6)/6

%e + x^7*(1 + 49*x + 211876*x^2 + 2054455634*x^3 + 3348108992991*x^4 + 63205303218876*x^5 + 262596783764*x^6 + x^7)/7 +...

%e + x^n*(Sum_{k=0..n} binomial(n^2, k^2)*x^k)/n +...

%o (PARI) {a(n)=polcoeff(exp(sum(m=1, n, sum(k=0, m, binomial(m^2, k^2)*x^k)*x^m/m)+x*O(x^n)), n)}

%o for(n=0,25,print1(a(n),", "))

%Y Cf. A206851 (log), A228902, A206830, A167006.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Feb 13 2012

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Last modified June 13 17:32 EDT 2024. Contains 373391 sequences. (Running on oeis4.)