A142583 - OEIS

%I #4 Jul 10 2019 21:23:44

%S 1,1,1,2,5,2,6,41,31,6,30,931,940,301,30,210,44347,51971,21227,3571,

%T 210,2310,5339027,6762728,3137268,665308,64681,2310,30030,901841261,

%U 1212061411,618052532,153213712,19579601,1231231,30030,510510

%N A triangle sequence based on a prime root product using a primorial function: f(n)=primorial(n); p(x,n)=If[n == 0, 1, f(n)*(x + 1/f(n))*Product[x + Prime[i], {i, 1, n - 1}]]; t(n,m)=coefficients(p(x,n)).

%C Row sums are:

%C {1, 2, 9, 84, 2232, 121536, 15973632, 2906039808, 889220312064, 337903091527680, 186522488129617920}.

%F f(n)=primorial(n); p(x,n)=If[n == 0, 1, f(n)*(x + 1/f(n))*Product[x + Prime[i], {i, 1, n - 1}]]; t(n,m)=coefficients(p(x,n)).

%e {1},

%e {1, 1},

%e {2, 5, 2},

%e {6, 41, 31, 6},

%e {30, 931, 940, 301, 30},

%e {210, 44347, 51971, 21227, 3571, 210},

%e {2310, 5339027, 6762728, 3137268, 665308, 64681, 2310},

%e {30030, 901841261, 1212061411, 618052532, 153213712, 19579601, 1231231, 30030},

%e {510510, 260621176267, 365610805408, 199220508695, 54785396836, 8263116209, 688678048, 29609581, 510510},

%e {9699690, 94084000213783, 136937156748959, 78865165215633, 23562710455719, 4023906738627, 405611939181, 23773940267, 746876131, 9699690},

%e {223092870, 49770428979243299, 74603683500398660, 44869225596233918, 14278572367678410, 2670588974929140, 307118568654990, 21905488909522, 941898097240, 22309287001, 223092870}

%t Clear[a, p, n] a[0] = 1; a[n_] := a[n] = Prime[n]*a[n - 1]; aa = Table[a[n], {n, 0, 20}]; p[x_, n_] = If[n == 0, 1, aa[[n]]*(x + 1/aa[[n]])*Product[x + Prime[i], {i, 1, n - 1}]]; Table[FullSimplify[ExpandAll[p[x, n]]], {n, 0, 10}]; Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 0, 10}]; Flatten[%]

%K nonn,uned

%O 1,4

%A _Roger L. Bagula_ and _Gary W. Adamson_, Sep 21 2008