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).

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

Author

Roger L. Bagula, Dec 13 2009

Keywords

• nonn
• uned
• tabl

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.

Examples

			{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}
		

References

• Eugene Jahnke and Fritz Emde, Table of Functions with Formulae and Curves, Dover Book, New York, 1945, page 32.

Programs

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[%]

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).