_Paul D. Hanna (pauldhanna(AT)juno.com), _, Feb 01 2011

proposed

approved

a(n) equals the coefficient of x^(2n-1) in the (2n-1)-th iteration of x/(1-x^2) for n>=1.

Compare a(n) to (2n-1)^(2n-2), which is the coefficient of x^(2n-1) in the (2n-1)-th iteration of x/(1-x).

The coefficients of x^(2k-1), k>=1, in the n-th iterations of x/(1-x^2) begin:

(PARI) {a(n)=local(A=x, G=x/(1-x^2)); for(i=1, 2*n-1, A=subst(G, x, A+x*O(x^(2*n)))); polcoeff(A, 2*n-1)}

Cf. A185751.

allocated for Paul D. Hanna

a(n) equals the coefficient of x^(2n-1) in the (2n-1)-th iteration of x/(1-x^2) for n>=1.

1, 3, 35, 763, 24639, 1057991, 56776733, 3658333743, 275216215907, 23680746142603, 2293972666771246, 247075461463313473, 29290441677755271415, 3790060387175131847103, 531544501037995251654986

1,2

The coefficients of x^(2k-1), k>=1, in the n-th iterations of x/(1-x^2) begin:

n=1: [(1), 1, 1, 1, 1, 1, 1, 1, ...];

n=2: [1, 2, 5, 13, 34, 89, 233, 610, 1597, ...];

n=3: [1,(3), 12, 51, 221, 965, 4227, 18540, ...];

n=4: [1, 4, 22, 130, 789, 4848, 29975, 185953, ...];

n=5: [1, 5,(35), 265, 2070, 16420, 131353, 1055966, ...];

n=6: [1, 6, 51, 471, 4501, 43771, 429939, 4249026, ...];

n=7: [1, 7, 70,(763), 8624, 99344, 1157226, 13575289, ...];

n=8: [1, 8, 92, 1156, 15086, 200880, 2707230, 36768138, ...];

n=9: [1, 9, 117, 1665,(24639), 372363, 5699493, 87963975, ...];

n=10:[1, 10, 145, 2305, 38140, 644965, 11052481, ...];

n=11:[1, 11, 176, 3091, 56551,(1057991), 20067377, ...];

n=12:[1, 12, 210, 4038, 80939, 1659824, 34522269, ...];

n=13:[1, 13, 247, 5161, 112476, 2508870,(56776733), ...];

n=14:[1, 14, 287, 6475, 152439, 3674503, 89886811, ...];

n=15:[1, 15, 330, 7995, 202210, 5238010, 137730384,(3658333743), ...]; ...

coefficients in parenthesis form the initial terms of this sequence.

(PARI) {a(n)=local(A=x, G=x/(1-x^2)); for(i=1, 2*n-1, A=subst(G, x, A+x*O(x^(2*n)))); polcoeff(A, 2*n-1)}

Cf. A185751.

allocated

nonn

Paul D. Hanna (pauldhanna(AT)juno.com), Feb 01 2011

approved

proposed

allocated for Paul D. Hanna

allocated

approved