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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 (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 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 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 March 24 04:28 EDT 2023. Contains 361454 sequences. (Running on oeis4.)