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 A206849 a(n) = Sum_{k=0..n} binomial(n^2, k^2). 11
 1, 2, 6, 137, 13278, 4098627, 8002879629, 66818063663192, 1520456935214867934, 167021181249536494996841, 102867734705055054467692090431, 179314863425920182637610314008444247, 1094998941099523423274757578750950802034789 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS Seiichi Manyama, Table of n, a(n) for n = 0..57 Vaclav Kotesovec, Limits, graph for 500 terms FORMULA Ignoring the initial term a(0), equals the logarithmic derivative of A206848. Equals the row sums of triangle A226234. From Vaclav Kotesovec, Mar 03 2014: (Start) 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) EXAMPLE 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; ... MATHEMATICA Table[Sum[Binomial[n^2, k^2], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 03 2014 *) PROG (PARI) {a(n)=sum(k=0, n, binomial(n^2, k^2))} for(n=0, 20, print1(a(n), ", ")) CROSSREFS Cf. A206848 (exp), A226234, A206847, A206850, A206851. Sequence in context: A274695 A156515 A254223 * A368395 A090907 A159478 Adjacent sequences: A206846 A206847 A206848 * A206850 A206851 A206852 KEYWORD nonn AUTHOR Paul D. Hanna, Feb 15 2012 STATUS approved

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