A243951 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.

1, 1, 5, 45, 781, 23981, 1371885, 145101805, 29560055405, 11546945197165, 8881721878376045, 13338290506465706605, 39879639563413780322925, 234862804790553590007179885, 2768979430068663216466330446445, 64586918396493458414460474344516205, 3024204274887062319005574660727125346925

Offset

0,3

Links

Paul D. Hanna, Table of n, a(n) for n = 0..80

Formula

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

Example

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,...}.

Mathematica

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 *)

Prog

(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), ", "))

Crossrefs

Cf. A243950, A022166.

Sequence in context: A295227 A167812 A155104 * A290941 A211051 A322661

Adjacent sequences: A243948 A243949 A243950 * A243952 A243953 A243954

Keyword

nonn

Author

Paul D. Hanna, Jun 21 2014

Status

approved