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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":"062ce31183651f7af80717ff17b919847a87f9ec591b51fe3c245fb031ed096d","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-9c403911703f1b61","role":"sequence","name":"basis_map_A013610 of Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.","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:    1,    12;\n   3:    3,    48,      81;\n   4:    7,   156,     567,     432;\n   5:   17,   480,    2592,    4320,     2025;\n   6:   41,  1404,   10044,   25920,    26325,     8748;\n   7:   99,  3984,   35397,  123552,   196425,   139968,    35721;\n   8:  239, 11052,  117207,  515376,  1119825,  1250964,   678699,   139968;\n   9:  577, 30144,  371304, 1968192,  5431050,  8328096,  7072758,  3079296,   531441;\n  10: 1393, 81132, 1137888, 7057152, 23623650, 46206936, 53938710, 36671616, 13286025, 1968300;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(1+12*x-81*x^2)*P(n-4,x) + (12*x^2+36*x^3)*d/dx P(n-4,x) - (4+36*x-162*x^2)*P(n-3,x) + (36*x^2+72*x^3)*d/dx P(n-3,x) - (2+12*x-81*x^2)*P(n-2,x) + (12*x^2+36*x^3)*d/dx P(n-2,x) + (4+12*x)*P(n-1,x) - 12*x^2*d/dx P(n-1,x) for n >= 5, with P(1,x) = 1, P(2,x) = 1 + 12*x, P(3,x) = 3 + 48*x + 81*x^2, P(4,x) = 7 + 156*x + 567*x^2 + 432*x^3.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 1 + 12*x;\nP[3, x_] := 3 + 48*x + 81*x^2;\nP[4, x_] := 7 + 156*x + 567*x^2 + 432*x^3;\nP[n_, x_] := P[n, x] = Expand[-(1 + 12*x - 81*x^2)*P[n - 4, x] + (12*x^2 + 36*x^3)*D[P[n - 4, x], x] - (4 + 36*x - 162*x^2)*P[n - 3, x] + (36*x^2 + 72*x^3)*D[P[n - 3, x], x] - (2 + 12*x - 81*x^2)*P[n - 2, x] + (12*x^2 + 36*x^3)*D[P[n - 2, x], x] + (4 + 12*x)*P[n - 1, x] - 12*x^2*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 12*x; v[3] = 3 + 48*x + 81*x^2; v[4] = 7 + 156*x + 567*x^2 + 432*x^3; for(n = 5, N, my(i = n); v[i] = -(1 + 12*x - 81*x^2)*v[i - 4] + (12*x^2 + 36*x^3)*deriv(v[i - 4], x) - (4 + 36*x - 162*x^2)*v[i - 3] + (36*x^2 + 72*x^3)*deriv(v[i - 3], x) - (2 + 12*x - 81*x^2)*v[i - 2] + (12*x^2 + 36*x^3)*deriv(v[i - 2], x) + (4 + 12*x)*v[i - 1] - 12*x^2*deriv(v[i - 1], x)); 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-b0fc620949fb5990","steps":["basis A013610","(\u03b8+1)\u00b2"],"edges":["e-7f7d545e75881e09","e-2923eb18e59376a5"],"text":"A122542 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2"},"headline":"A122542 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2","chain":null}