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

0,5

Links

Table of n, a(n) for n=0..30.

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.

Example

 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, ...].

Prog

 (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), ", "))

Crossrefs

Cf. A166990, A088220, A000172.

Sequence in context: A148816 A148817 A148818 * A081497 A028270 A124383

Adjacent sequences: A242900 A242901 A242902 * A242904 A242905 A242906

Keyword

nonn

Author

Paul D. Hanna, May 25 2014

Status

approved