250323

PARI


Fuss–Catalan number

f_k(x) = Sum_{j>=0} binomial(k*j+1, n)/(k*j+1) * x^j

がf_k(x) = 1 + x*f_k(x)^k を満たすことを確認してみた。

N=10;
K=10;

a(n, k) = binomial(k*n+1, n)/(k*n+1);
b(n, k) = my(x='x+O('x^(n+1))); sum(j=0, n, a(j, k)*x^j);
for(k=0, K, f = b(N, k); if(f==1+x*f^k, print([k, Vec(f)])));

出力結果

[0, [1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0]]

[1, [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]]

[2, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796]]

[3, [1, 1, 3, 12, 55, 273, 1428, 7752, 43263, 246675, 1430715]]

[4, [1, 1, 4, 22, 140, 969, 7084, 53820, 420732, 3362260, 27343888]]

[5, [1, 1, 5, 35, 285, 2530, 23751, 231880, 2330445, 23950355, 250543370]]

[6, [1, 1, 6, 51, 506, 5481, 62832, 749398, 9203634, 115607310, 1478314266]]

[7, [1, 1, 7, 70, 819, 10472, 141778, 1997688, 28989675, 430321633, 6503352856]]

[8, [1, 1, 8, 92, 1240, 18278, 285384, 4638348, 77652024, 1329890705, 23190029720]]

[9, [1, 1, 9, 117, 1785, 29799, 527085, 9706503, 184138713, 3573805950, 70625252863]]

[10, [1, 1, 10, 145, 2470, 46060, 910252, 18730855, 397089550, 8612835715, 190223180840]]