A142583 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)).

1, 1, 1, 2, 5, 2, 6, 41, 31, 6, 30, 931, 940, 301, 30, 210, 44347, 51971, 21227, 3571, 210, 2310, 5339027, 6762728, 3137268, 665308, 64681, 2310, 30030, 901841261, 1212061411, 618052532, 153213712, 19579601, 1231231, 30030, 510510

Offset

1,4

Comments

Row sums are:

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

Links

Table of n, a(n) for n=1..37.

Formula

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)).

Example

{1},
{1, 1},
{2, 5, 2},
{6, 41, 31, 6},
{30, 931, 940, 301, 30},
{210, 44347, 51971, 21227, 3571, 210},
{2310, 5339027, 6762728, 3137268, 665308, 64681, 2310},
{30030, 901841261, 1212061411, 618052532, 153213712, 19579601, 1231231, 30030},
{510510, 260621176267, 365610805408, 199220508695, 54785396836, 8263116209, 688678048, 29609581, 510510},
{9699690, 94084000213783, 136937156748959, 78865165215633, 23562710455719, 4023906738627, 405611939181, 23773940267, 746876131, 9699690},
{223092870, 49770428979243299, 74603683500398660, 44869225596233918, 14278572367678410, 2670588974929140, 307118568654990, 21905488909522, 941898097240, 22309287001, 223092870}

Mathematica

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[%]

Crossrefs

Sequence in context: A267090 A067948 A142148 * A327838 A086956 A198570

Adjacent sequences: A142580 A142581 A142582 * A142584 A142585 A142586

Keyword

nonn,uned

Author

Roger L. Bagula and Gary W. Adamson, Sep 21 2008

Status

approved