230908

PARI


A357782, A357783 and A357784

出力してみた。

a(n) = if(n==0, 1, 2 * sum(k=0, n-1, binomial(n-1,k)*c(k)));
b(n) = if(n==0, 0,     sum(k=0, n-1, binomial(n-1,k)*a(k)));
c(n) = if(n==0, 0,     sum(k=0, n-1, binomial(n-1,k)*b(k)));

for(n=0, 20, print1(a(n),", "))
for(n=0, 20, print1(b(n),", "))
for(n=0, 20, print1(c(n),", "))

出力結果

1, 0, 0, 2, 12, 50, 184, 686, 2996, 16642, 110328, 784190, 5645876, 40685762, 296458344, 2226254766, 17564381332, 147289101090, 1312394060536, 12305546886398, 119906479624084, 

0, 1, 1, 1, 3, 21, 131, 705, 3515, 17389, 91739, 547889, 3746227, 28241373, 224124083, 1821051233, 15023818091, 126366334125, 1094358852075, 9858890038513, 92983173940419, 

0, 0, 1, 3, 7, 17, 61, 343, 2231, 14301, 88561, 542011, 3397483, 22638993, 164336085, 1299899087, 10991061663, 97070035205, 881323166809, 8173386231395, 77489746906355,