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A206849
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a(n) = Sum_{k=0..n} binomial(n^2, k^2).
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11
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1, 2, 6, 137, 13278, 4098627, 8002879629, 66818063663192, 1520456935214867934, 167021181249536494996841, 102867734705055054467692090431, 179314863425920182637610314008444247, 1094998941099523423274757578750950802034789
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OFFSET
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0,2
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LINKS
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FORMULA
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Ignoring the initial term a(0), equals the logarithmic derivative of A206848.
Equals the row sums of triangle A226234.
Limit n->infinity a(n)^(1/n^2) = 2
Lim sup n->infinity a(n)/(2^(n^2)/n) = sqrt(2/Pi) * JacobiTheta3(0,exp(-4)) = Sqrt[2/Pi] * EllipticTheta[3, 0, 1/E^4] = 0.827112271364145742...
Lim inf n->infinity a(n)/(2^(n^2)/n) = sqrt(2/Pi) * JacobiTheta2(0,exp(-4)) = Sqrt[2/Pi] * EllipticTheta[2, 0, 1/E^4] = 0.587247586271786487...
(End)
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EXAMPLE
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L.g.f.: L(x) = 2*x + 6*x^2/2 + 137*x^3/3 + 13278*x^4/4 + 4098627*x^5/5 +...
where exponentiation yields the g.f. of A206848:
exp(L(x)) = 1 + 2*x + 5*x^2 + 53*x^3 + 3422*x^4 + 826606*x^5 + 1335470713*x^6 +...
Illustration of terms: by definition,
a(1) = C(1,0) + C(1,1);
a(2) = C(4,0) + C(4,1) + C(4,4);
a(3) = C(9,0) + C(9,1) + C(9,4) + C(9,9);
a(4) = C(16,0) + C(16,1) + C(16,4) + C(16,9) + C(16,16);
a(5) = C(25,0) + C(25,1) + C(25,4) + C(25,9) + C(25,16) + C(25,25);
a(6) = C(36,0) + C(36,1) + C(36,4) + C(36,9) + C(36,16) + C(36,25) + C(36,36); ...
Numerically, the above evaluates to be:
a(1) = 1 + 1 = 2;
a(2) = 1 + 4 + 1 = 6;
a(3) = 1 + 9 + 126 + 1 = 137;
a(4) = 1 + 16 + 1820 + 11440 + 1 = 13278;
a(5) = 1 + 25 + 12650 + 2042975 + 2042975 + 1 = 4098627;
a(6) = 1 + 36 + 58905 + 94143280 + 7307872110 + 600805296 + 1 = 8002879629; ...
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MATHEMATICA
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Table[Sum[Binomial[n^2, k^2], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 03 2014 *)
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PROG
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(PARI) {a(n)=sum(k=0, n, binomial(n^2, k^2))}
for(n=0, 20, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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