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A243951
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Self-convolution square-root of A243950, which is the sums of the squares of the q-binomial coefficients for q=2 in rows of triangle A022166.
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2
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1, 1, 5, 45, 781, 23981, 1371885, 145101805, 29560055405, 11546945197165, 8881721878376045, 13338290506465706605, 39879639563413780322925, 234862804790553590007179885, 2768979430068663216466330446445, 64586918396493458414460474344516205, 3024204274887062319005574660727125346925
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OFFSET
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0,3
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LINKS
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FORMULA
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a(n) ~ c * 2^(n^2/2-1), where c = 18.0796893855819714431... if n is even and c = 18.02126069886312898683... if n is odd (constants same as for A243950). - Vaclav Kotesovec, Jun 23 2014
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EXAMPLE
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G.f.: A(x) = 1 + x + 5*x^2 + 45*x^3 + 781*x^4 + 23981*x^5 + 1371885*x^6 +...
where
A(x)^2 = 1 + 2*x + 11*x^2 + 100*x^3 + 1677*x^4 + 49974*x^5 + 2801567*x^6 + 293257480*x^7 + 59426801521*x^8 +...+ A243950(n)*x^n +...
The terms in this sequence appear to be divisible by 5 everywhere except
a(n) == 1 (mod 5) when n = {0,1,4,5,20,21,24,25,100,101,104,105,120,121,124, 125,500,501,...}.
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MATHEMATICA
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a[n_] := SeriesCoefficient[Sqrt[Sum[x^m Sum[QBinomial[m, k, 2]^2, {k, 0, m}], {m, 0, n}]], {x, 0, n}]; Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Apr 09 2016 *)
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PROG
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(PARI) {A022166(n, k)=polcoeff(x^k/prod(j=0, k, 1-2^j*x+x*O(x^n)), n)}
{a(n)=polcoeff(sqrt(sum(m=0, n, x^m*sum(k=0, m, A022166(m, k)^2) +x*O(x^n))), n)}
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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