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Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"certificate_hash":"03080d8216aca9b68c1c7b5e432d06b155e15831fd4ae859e2b8c52aa89c04d8","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-daa3b3e1ead00edc","role":"sequence","name":"Triangle read by rows, coefficients of the generalized Eulerian polynomials A_{n, 3}(x) in descending order.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:   1;\n   2:   2,       4;\n   3:   4,      52,          9;\n   4:   8,     372,        540,          16;\n   5:  16,    2180,      10179,        4016,          25;\n   6:  32,   11732,     130482,      170848,       25450,          36;\n   7:  64,   60708,    1387017,     4199008,     2224950,      147204,          49;\n   8: 128,  308404,   13281624,    78244592,    96758800,    24929964,      802424,         64;\n   9: 256, 1553284,  119437263,  1238478928,  2943966475,  1817136108,   253575637,    4193728,       81;\n  10: 512, 7792020, 1032385662, 17657593536, 72177503850, 86002366248, 29804754210, 2414861568, 21232854, 100;\n","recurrence_sentence":"The row polynomials satisfy (9*n^2-30*n+25)*P(n,x) = (18*n^2-42*n+20-(216*n^3-1080*n^2+1476*n-352)*x+(729*n^4-5022*n^3+12393*n^2-12960*n+4800)*x^2)*P(n-2,x) - ((108*n^2-252*n+75)*x-(216*n^3-540*n^2+405*n-310)*x^2-(324*n^4-3294*n^3+10017*n^2-11940*n+4825)*x^3)*d/dx P(n-2,x) - ((81*n^2-189*n+63)*x^2-(324*n^3-1296*n^2+1701*n-762)*x^3+(324*n^3-1377*n^2+1890*n-825)*x^4)*d^2/dx^2 P(n-2,x) + (9*n^2-39*n+40+(108*n^3-540*n^2+900*n-500)*x)*P(n-1,x) + ((54*n^2-153*n+96)*x-(108*n^3-432*n^2+540*n-200)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + 4*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 2 + 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[((18*n^2 - 42*n + 20 - (216*n^3 - 1080*n^2 + 1476*n - 352)*x + (729*n^4 - 5022*n^3 + 12393*n^2 - 12960*n + 4800)*x^2)*P[n - 2, x] - ((108*n^2 - 252*n + 75)*x - (216*n^3 - 540*n^2 + 405*n - 310)*x^2 - (324*n^4 - 3294*n^3 + 10017*n^2 - 11940*n + 4825)*x^3)*D[P[n - 2, x], x] - ((81*n^2 - 189*n + 63)*x^2 - (324*n^3 - 1296*n^2 + 1701*n - 762)*x^3 + (324*n^3 - 1377*n^2 + 1890*n - 825)*x^4)*D[P[n - 2, x], {x, 2}] + (9*n^2 - 39*n + 40 + (108*n^3 - 540*n^2 + 900*n - 500)*x)*P[n - 1, x] + ((54*n^2 - 153*n + 96)*x - (108*n^3 - 432*n^2 + 540*n - 200)*x^2)*D[P[n - 1, x], x])/(9*n^2 - 30*n + 25)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 4*x; for(n = 3, N, my(i = n); v[i] = ((18*n^2 - 42*n + 20 - (216*n^3 - 1080*n^2 + 1476*n - 352)*x + (729*n^4 - 5022*n^3 + 12393*n^2 - 12960*n + 4800)*x^2)*v[i - 2] - ((108*n^2 - 252*n + 75)*x - (216*n^3 - 540*n^2 + 405*n - 310)*x^2 - (324*n^4 - 3294*n^3 + 10017*n^2 - 11940*n + 4825)*x^3)*deriv(v[i - 2], x) - ((81*n^2 - 189*n + 63)*x^2 - (324*n^3 - 1296*n^2 + 1701*n - 762)*x^3 + (324*n^3 - 1377*n^2 + 1890*n - 825)*x^4)*deriv(deriv(v[i - 2], x), x) + (9*n^2 - 39*n + 40 + (108*n^3 - 540*n^2 + 900*n - 500)*x)*v[i - 1] + ((54*n^2 - 153*n + 96)*x - (108*n^3 - 432*n^2 + 540*n - 200)*x^2)*deriv(v[i - 1], x))/(9*n^2 - 30*n + 25)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-daa3b3e1ead00edc","steps":["(\u03b8+1)\u00b2"],"edges":["e-5948d7989cc0faaa"],"text":"A225117 \u2192 (\u03b8+1)\u00b2"},"headline":"A225117 \u2192 (\u03b8+1)\u00b2","chain":null}