A243425 G.f. A(x) satisfies: coefficient of x^n in A(x)^(2*n) equals A005260(n) = Sum_{k=0..n} C(n,k)^4.

1, 1, 3, 9, 60, 417, 3430, 29927, 278316, 2693437, 26976407, 277394148, 2916106328, 31220964707, 339508802940, 3741551907530, 41714692453164, 469827584596185, 5339334757945439, 61165396353689573, 705720529604453193, 8195208178337460065, 95724512701573485819, 1124070800784913396731

Offset: 0

Author

Paul D. Hanna, Jun 04 2014

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 9*x^3 + 60*x^4 + 417*x^5 + 3430*x^6 +...
Form a table of coefficients in A(x)^(2*n) for n>=0, which begins:
[1,  0,   0,    0,     0,      0,      0,       0,        0, ...];
[1,  2,   7,   24,   147,   1008,   8135,   70296,   648172, ...];
[1,  4,  18,   76,   439,   2940,  22936,  194300,  1761411, ...];
[1,  6,  33,  164,   960,   6378,  48526,  403440,  3598050, ...];
[1,  8,  52,  296,  1810,  12128,  90972,  744656,  6542519, ...];
[1, 10,  75,  480,  3105,  21252, 158845, 1286240, 11157705, ...];
[1, 12, 102,  724,  4977,  35100, 263844, 2125020, 18253680, ...];
[1, 14, 133, 1036,  7574,  55342, 421484, 3395016, 28975933, ...];
[1, 16, 168, 1424, 11060,  84000, 651848, 5277696, 44916498, ...]; ...
then the main diagonal forms A005260(n) = Sum_{k=0..n} C(n,k)^4.
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..600

Crossrefs

Cf. A242903.

Programs

PARI

{a(n)=polcoeff(sqrt(x/serreverse(x*exp(sum(m=1, n+1, sum(k=0, m, binomial(m, k)^4)*x^m/m +x^2*O(x^n))))), n)}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f.: sqrt( x / Series_Reversion( x*exp( Sum_{n>=1} A005260(n)*x^n/n ) ) ), where A005260(n) = Sum_{k=0..n} C(n,k)^4.

a(n) ~ c * d^n / n^(5/2), where d= 13.142352254618115022093263384837224..., c = 0.051491668112404252102416729094836... . - Vaclav Kotesovec, Jun 05 2014