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A171633
Coefficients of a Hermite-like polynomial from Eulerian polynomials: 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).
0
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
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
Sequence in context: A269152 A269097 A307552 * A221276 A117429 A132650
KEYWORD
nonn,uned,tabl
AUTHOR
Roger L. Bagula, Dec 13 2009
STATUS
approved