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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).

%I A171633 #6 Jul 20 2019 08:07:58
%S A171633 1,4,4,25,28,11,136,234,144,26,609,2040,1590,624,57,2388,15096,19056,
%T A171633 9648,2412,120,8593,95196,208893,148336,54267,8628,247,29224,532918,
%U A171633 1961928,2205850,1063000,285786,29272,502,95689,2739256,16059128
%N 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).
%C A171633 Row sums are {1, 8, 64, 540, 4920, 48720, 524160, 6108480, 76809600, 1037836800, 15008716800, 231437606400, ...}.
%C A171633 The important observation here is that the modulo two pattern is the same as the Hermite product A171531 type polynomials.
%D A171633 Eugene Jahnke and Fritz Emde, Table of Functions with Formulae and Curves, Dover Book, New York, 1945, page 32.
%F A171633 p(x,n) = p(x,n) = Sum_{k=1..n+1} [Eulerian(n + 1, k - 1)*x^(k - 1), ];
%F A171633 q(x,n) = p''(x,n) - x*p'(x,n) + n*p(x,n).
%e A171633 {1},
%e A171633 {4, 4},
%e A171633 {25, 28, 11},
%e A171633 {136, 234, 144, 26},
%e A171633 {609, 2040, 1590, 624, 57},
%e A171633 {2388, 15096, 19056, 9648, 2412, 120},
%e A171633 {8593, 95196, 208893, 148336, 54267, 8628, 247},
%e A171633 {29224, 532918, 1961928, 2205850, 1063000, 285786, 29272, 502},
%e A171633 {95689, 2739256, 16059128, 28938232, 20207530, 7250696, 1422304, 95752, 1013},
%e A171633 {305284, 13239252, 118078464, 329909376, 350572104, 171167736, 47500128, 6757056, 305364, 2036}
%t A171633 t[n_, k_] := Sum[(-1)^j Binomial[n + 1, j](k + 1 - j)^n, {j, 0, k + 1}]
%t A171633 p[x_, n_] := Sum[t[n + 1, k - 1]*x^(k - 1), {k, 1, n + 1}]
%t A171633 b = Table[CoefficientList[D[p[x, n], {x, 2}] - x*D[p[x, n], {x, 1}] + n*p[x, n], x], {n, 1, 10}]
%t A171633 Flatten[%]
%K A171633 nonn,uned,tabl
%O A171633 1,2
%A A171633 _Roger L. Bagula_, Dec 13 2009