OFFSET
1,2
COMMENTS
Row sums are {1, 8, 64, 540, 4920, 48720, 524160, 6108480, 76809600, 1037836800, 15008716800, 231437606400, ...}.
The important observation here is that the modulo two pattern is the same as the Hermite product A171531 type polynomials.
REFERENCES
Eugene Jahnke and Fritz Emde, Table of Functions with Formulae and Curves, Dover Book, New York, 1945, page 32.
FORMULA
p(x,n) = p(x,n) = Sum_{k=1..n+1} [Eulerian(n + 1, k - 1)*x^(k - 1), ];
q(x,n) = p''(x,n) - x*p'(x,n) + n*p(x,n).
EXAMPLE
{1},
{4, 4},
{25, 28, 11},
{136, 234, 144, 26},
{609, 2040, 1590, 624, 57},
{2388, 15096, 19056, 9648, 2412, 120},
{8593, 95196, 208893, 148336, 54267, 8628, 247},
{29224, 532918, 1961928, 2205850, 1063000, 285786, 29272, 502},
{95689, 2739256, 16059128, 28938232, 20207530, 7250696, 1422304, 95752, 1013},
{305284, 13239252, 118078464, 329909376, 350572104, 171167736, 47500128, 6757056, 305364, 2036}
MATHEMATICA
t[n_, k_] := Sum[(-1)^j Binomial[n + 1, j](k + 1 - j)^n, {j, 0, k + 1}]
p[x_, n_] := Sum[t[n + 1, k - 1]*x^(k - 1), {k, 1, n + 1}]
b = Table[CoefficientList[D[p[x, n], {x, 2}] - x*D[p[x, n], {x, 1}] + n*p[x, n], x], {n, 1, 10}]
Flatten[%]
CROSSREFS
KEYWORD
AUTHOR
Roger L. Bagula, Dec 13 2009
STATUS
approved