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

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

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.

Vaclav Kotesovec, <a href="/A243425/b243425.txt">Table of n, a(n) for n = 0..600</a>

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.

G.f.: A(x) = 1 + x + 3*x^2 + 9*x^3 + 60*x^4 + 417*x^5 + 3430*x^6 +...

(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)}

Cf. A242903.

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allocated for Paul D. Hanna

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

0,3

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.

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.

(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), ", "))

Cf. A242903.

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Paul D. Hanna, Jun 04 2014

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