%I #21 Sep 22 2024 05:46:55
%S 1,1,2,8,50,388,3363,31132,301156,3007000,30753169,320492869,
%T 3391067666,36331532588,393353506931,4296895624750,47300050998991,
%U 524168531729460,5842914510975105,65470405191871331,737008925038212059,8331166456981245521
%N Expansion of 1/(1-x*A002294(x)).
%H Vladimir Kruchinin and D. V. Kruchinin, <a href="http://arxiv.org/abs/1103.2582">Composita and their properties</a>, arXiv:1103.2582 [math.CO], 2011-2013.
%F a(n) = Sum_{k=1..n} k/(4*n-3*k)*binomial(5*n-4*k-1,n-k), n>0; a(0) = 1.
%F From _Vaclav Kotesovec_, Sep 22 2024: (Start)
%F Recurrence: 8*(n-1)*(2*n-3)*(4*n-5)*(4*n-3)*(302869201*n^9 - 8459245881*n^8 + 104437088286*n^7 - 748013787426*n^6 + 3425181067929*n^5 - 10398368877609*n^4 + 20929298529704*n^3 - 26931712087164*n^2 + 20105202695760*n - 6634504195200)*a(n) = 5*(5*n-21)*(5*n-19)*(5*n-18)*(5*n-17)*(302869201*n^9 - 5733423072*n^8 + 47666412474*n^7 - 228372041208*n^6 + 694720947369*n^5 - 1391357951688*n^4 + 1834379283596*n^3 - 1535202635232*n^2 + 740132688960*n - 156635942400)*a(n-1) + 8*(n-1)*(2*n-3)*(4*n-5)*(4*n-3)*(302869201*n^9 - 8459245881*n^8 + 104437088286*n^7 - 748013787426*n^6 + 3425181067929*n^5 - 10398368877609*n^4 + 20929298529704*n^3 - 26931712087164*n^2 + 20105202695760*n - 6634504195200)*a(n-3) - 3*(341333589527*n^13 - 11542804387321*n^12 + 178153937603069*n^11 - 1661124543043265*n^10 + 10436085419810511*n^9 - 46636299048022863*n^8 + 152460394393134247*n^7 - 369062312013610715*n^6 + 661348648146586462*n^5 - 866258572340414716*n^4 + 805991085205781784*n^3 - 504373614185279520*n^2 + 190252933034572800*n - 32669988422400000)*a(n-4) + 5*(5*n-21)*(5*n-19)*(5*n-18)*(5*n-17)*(302869201*n^9 - 5733423072*n^8 + 47666412474*n^7 - 228372041208*n^6 + 694720947369*n^5 - 1391357951688*n^4 + 1834379283596*n^3 - 1535202635232*n^2 + 740132688960*n - 156635942400)*a(n-5).
%F a(n) ~ 5^(5*n + 7/2) / (314721 * sqrt(Pi) * n^(3/2) * 2^(8*n - 9/2)). (End)
%t Join[{1},Table[Sum[k/(4n-3k) Binomial[5n-4k-1,n-k],{k,n}],{n,30}]] (* _Harvey P. Dale_, Feb 05 2012 *)
%o (PARI) a(n)=max(1,sum(k=1,n, k/(4*n-3*k)*binomial(5*n-4*k-1,n-k)))
%Y Cf. A002294.
%K nonn
%O 0,3
%A _Vladimir Kruchinin_, Feb 14 2011