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f(k) x^k maps every real-rooted polynomial to a real-rooted polynomial or zero (Laguerre; see T. Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. 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":"a322bcab864e2c44322b8120a21dced803a57d30e8f245c64d012d88ebd9b81b","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-6b8ac0ed9d98a614","role":"sequence","name":"basis_map_A007318 of Triangle of coefficients of (x+1)*(x+3)*(x+6)*...*(x+n(n+1)/2).","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, 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":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:           1;\n  1:           2,            2;\n  2:           8,           20,            9;\n  3:          56,          236,          243,            72;\n  4:         616,         3716,         6213,          4032,           900;\n  5:        9856,        77936,       183018,        188772,         90000,         16200;\n  6:      216832,      2128544,      6481380,       9277488,       6935265,       2624400,        396900;\n  7:     6288128,     73870368,    277358868,     511018672,     525834765,     309132504,      97240500,      12700800;\n  8:   232660736,   3185948832,  14251277988,   32220916528,   42451726545,   34153964496,   16581463668,    4470681600,    514382400;\n  9: 10702393856, 167493112512, 870610333608, 2337238839568, 3765205975770, 3863475600246, 2555830464768, 1058412342240, 249989846400, 25719120000;\n","recurrence_sentence":"The row polynomials satisfy (4*n-4)*P(n,x) = -(n^5-n^4+3*n^3+n^2+4+(2*n^5-2*n^4+2*n^3+2*n^2-4*n)*x+(n^5-n^4-n^3+n^2)*x^2)*P(n-2,x) + (4*n^3+4*n+(4*n^3-4*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^5 - n^4 + 3*n^3 + n^2 + 4 + (2*n^5 - 2*n^4 + 2*n^3 + 2*n^2 - 4*n)*x + (n^5 - n^4 - n^3 + n^2)*x^2)*P[n - 2, x] + (4*n^3 + 4*n + (4*n^3 - 4*n)*x)*P[n - 1, x])/(4*n - 4)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^5 - n^4 + 3*n^3 + n^2 + 4 + (2*n^5 - 2*n^4 + 2*n^3 + 2*n^2 - 4*n)*x + (n^5 - n^4 - n^3 + n^2)*x^2)*v[i - 2] + (4*n^3 + 4*n + (4*n^3 - 4*n)*x)*v[i - 1])/(4*n - 4)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-ecd77b030b8fab94","steps":["basis A007318","\u03b8+1"],"edges":["e-8bd02eafb0f0a53d","e-19b1ee0a25a0d011"],"text":"A128813 \u2192 basis A007318 \u2192 \u03b8+1"},"headline":"A128813 \u2192 basis A007318 \u2192 \u03b8+1","chain":null}