A364375 - OEIS

%I #17 Aug 27 2024 09:19:30

%S 1,0,-1,2,0,-11,28,1,-206,564,38,-4711,13329,1273,-119762,344707,

%T 41884,-3251250,9445976,1381154,-92305098,269504686,45848871,

%U -2707126108,7921304973,1532928960,-81375728566,238196143730,51591751698,-2493907008116,7293147604136

%N G.f. satisfies A(x) = (1 + x*A(x)) * (1 - x*A(x)^3).

%F a(n) = Sum_{k=0..n} (-1)^k * binomial(n+2*k+1,k) * binomial(n+2*k+1,n-k) / (n+2*k+1).

%F D-finite with recurrence +2*n*(191553133*n -462036810)*(2*n+1) *(n+1)*a(n) +2*n*(6735679202*n^3 -31340869996*n^2 +39568451245*n -13340358389)*a(n-1) +6*(13937077342*n^4 -106287464449*n^3 +278022830194*n^2 -296712736455*n +108876423952)*a(n-2) +6*(42118990776*n^4 -422141236704*n^3 +1546534911485*n^2 -2448212978721*n +1409411956166)*a(n-3) +6*(72631772298*n^4 -948761263665*n^3 +4512788370945*n^2 -9254886913710*n +6888712179986)*a(n-4) +3*(10147840245*n^4 -513806508936*n^3 +5519825354705*n^2 -22028093493130*n +30003008863784)*a(n-5) +6*(-9503341830*n^4 +235269814455*n^3 -2064338754902*n^2 +7709425316943*n -10409244067330)*a(n-6) +18*(3*n-20)*(n-6) *(156488131*n-746235854) *(3*n-13)*a(n-7)=0. - _R. J. Mathar_, Jul 25 2023

%F From _Peter Bala_, Aug 24 2024: (Start)

%F G.f. A(x) satisfies (1/x) * series_reversion(x*A(x)) = 1/G(x), where G(x) is the g.f. of A364371.

%F P-recursive: fifth-order recurrence:      (2*n+1)*(2*n+2)*(3045*n^5-26680*n^4+84901*n^3-123566*n^2+86300*n-25368)*n*a(n) + 6*(18270*n^7-160080*n^6+500851*n^5-666969*n^4+307749*n^3+70849*n^2-76222*n+8288)*n*a(n-1) +  6*(54810*n^8-562455*n^7+2191158*n^6-3956204*n^5+2960986*n^4+88959*n^3-1045774*n^2+187688*n+69888)*a(n-2) + 6*(109620*n^8-1289340*n^7+5897421*n^6-13016841*n^5+13725877*n^4-5967199*n^3+2484230*n^2-3359528*n+1002624)*a(n-3) - 6*(54810*n^8-726885*n^7+3719313*n^6-9080919*n^5+10367473*n^4-4378276*n^3+1152956*n^2-2297912*n+768096)*a(n-4) + (3*n-7)*(3*n-12)*(3*n-14)*(3045*n^5-11455*n^4+8631*n^3+1507*n^2+2376*n-1368)*a(n-5) = 0 with a(0) = 1, a(1) = 0, a(2) = -1, a(3) = 2 and a(4) = 0. (End)

%p A364375 := proc(n)

%p     add( (-1)^k*binomial(n+2*k+1,k) * binomial(n+2*k+1,n-k)/(n+2*k+1),k=0..n) ;

%p end proc:

%p seq(A364375(n),n=0..80); # _R. J. Mathar_, Jul 25 2023

%o (PARI) a(n) = sum(k=0, n, (-1)^k*binomial(n+2*k+1, k)*binomial(n+2*k+1, n-k)/(n+2*k+1));

%Y Cf. A364371, A364372, A364374, A364376.

%Y Cf. A198953.

%K sign,easy

%O 0,4

%A _Seiichi Manyama_, Jul 21 2023