A242903 G.f. A(x) satisfies: coefficient of x^n in A(x)^(2*n) equals A000172(n) = Sum_{k=0..n} C(n,k)^3, the n-th Franel number.

1, 1, 1, 1, 3, 8, 26, 89, 324, 1225, 4786, 19170, 78408, 326275, 1377772, 5891401, 25467509, 111144579, 489145720, 2168854885, 9681072845, 43473716527, 196286934526, 890640262188, 4059500301390, 18579693200838, 85360357637580, 393548515741979, 1820335724153452, 8445294476235727, 39291407672079211

Offset: 0

Author

Paul D. Hanna, May 25 2014

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + x + x^2 + x^3 + 3*x^4 + 8*x^5 + 26*x^6 + 89*x^7 + 324*x^8 +...
Form a table of coefficients in A(x)^(2*n) as follows:
[1,  0,   0,    0,    0,     0,      0,      0,       0,       0, ...];
[1,  2,   3,    4,    9,    24,     75,    252,     903,    3376, ...];
[1,  4,  10,   20,   43,   108,    316,   1020,    3537,   12908, ...];
[1,  6,  21,   56,  138,   354,   1002,   3120,   10485,   37318, ...];
[1,  8,  36,  120,  346,   960,   2756,   8448,   27723,   96440, ...];
[1, 10,  55,  220,  735,  2252,   6785,  21020,   68340,  233870, ...];
[1, 12,  78,  364, 1389,  4716,  15184,  48588,  159186,  541424, ...];
[1, 14, 105,  560, 2408,  9030,  31304, 104960,  351792, 1203244, ...];
[1, 16, 136,  816, 3908, 16096,  60184, 213152,  739162, 2570464, ...];
[1, 18, 171, 1140, 6021, 27072, 109047, 409500, 1480293, 5280932, ...]; ...
then the main diagonal forms the Franel numbers:
[1, 2, 10, 56, 346, 2252, 15184, 104960, 739162, 5280932, ...].
		

Crossrefs

Cf. A166990, A088220, A000172.

Programs

PARI

{a(n)=polcoeff(sqrt(x/serreverse(x*exp(sum(m=1, n+1, sum(k=0, m, binomial(m, k)^3)*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} A000172(n)*x^n/n ) ) ), where A000172(n) is the n-th Franel number.

[x^n] A(x)^(2*n+2) = (n+1)*A166990(n).

Convolution square-root of A088220.