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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.

%I A242903 #6 Nov 03 2024 17:08:27
%S A242903 1,1,1,1,3,8,26,89,324,1225,4786,19170,78408,326275,1377772,5891401,
%T A242903 25467509,111144579,489145720,2168854885,9681072845,43473716527,
%U A242903 196286934526,890640262188,4059500301390,18579693200838,85360357637580,393548515741979,1820335724153452,8445294476235727,39291407672079211
%N 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.
%F A242903 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.
%F A242903 [x^n] A(x)^(2*n+2) = (n+1)*A166990(n).
%F A242903 Convolution square-root of A088220.
%e A242903 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 +...
%e A242903 Form a table of coefficients in A(x)^(2*n) as follows:
%e A242903 [1,  0,   0,    0,    0,     0,      0,      0,       0,       0, ...];
%e A242903 [1,  2,   3,    4,    9,    24,     75,    252,     903,    3376, ...];
%e A242903 [1,  4,  10,   20,   43,   108,    316,   1020,    3537,   12908, ...];
%e A242903 [1,  6,  21,   56,  138,   354,   1002,   3120,   10485,   37318, ...];
%e A242903 [1,  8,  36,  120,  346,   960,   2756,   8448,   27723,   96440, ...];
%e A242903 [1, 10,  55,  220,  735,  2252,   6785,  21020,   68340,  233870, ...];
%e A242903 [1, 12,  78,  364, 1389,  4716,  15184,  48588,  159186,  541424, ...];
%e A242903 [1, 14, 105,  560, 2408,  9030,  31304, 104960,  351792, 1203244, ...];
%e A242903 [1, 16, 136,  816, 3908, 16096,  60184, 213152,  739162, 2570464, ...];
%e A242903 [1, 18, 171, 1140, 6021, 27072, 109047, 409500, 1480293, 5280932, ...]; ...
%e A242903 then the main diagonal forms the Franel numbers:
%e A242903 [1, 2, 10, 56, 346, 2252, 15184, 104960, 739162, 5280932, ...].
%o A242903 (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)}
%o A242903 for(n=0,30,print1(a(n),", "))
%Y A242903 Cf. A166990, A088220, A000172.
%K A242903 nonn
%O A242903 0,5
%A A242903 _Paul D. Hanna_, May 25 2014