login
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.
2
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
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
KEYWORD
nonn,changed
AUTHOR
Paul D. Hanna, May 25 2014
STATUS
approved