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A171633 Coefficients of a Hermite like polynomial from Eulerian polynomials:p(x,n)=Sum[Eulerian(n + 1, k - 1)*x^(k - 1), {k, 1, n + 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 (list; table; graph; refs; listen; history; text; internal format)
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

LINKS

Table of n, a(n) for n=1..39.

FORMULA

p(x,n)=p(x,n)=Sum[Eulerian(n + 1, k - 1)*x^(k - 1), {k, 1, n + 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: A092897 A269152 A269097 * A221276 A117429 A132650

Adjacent sequences:  A171630 A171631 A171632 * A171634 A171635 A171636

KEYWORD

nonn,uned,tabl

AUTHOR

Roger L. Bagula, Dec 13 2009

STATUS

approved

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Last modified December 13 09:05 EST 2017. Contains 295957 sequences.