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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":"e0c6a75a84ba23425e62ccb3bad0169805c387bdc2c4c4d1978b90f4e6d227e4","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-d0986a7acf60271a","role":"sequence","name":"shift_x_plus_one of Triangle T(n,k), read by rows given by [1,0,1,0,0,0,0,0,0,...] DELTA [0,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.","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:    1;\n  2:    2,     4;\n  3:    5,    20,     9;\n  4:   13,    76,    72,     16;\n  5:   34,   260,   378,    176,    25;\n  6:   89,   840,  1647,   1184,   350,    36;\n  7:  233,  2616,  6453,   6304,  2875,   612,    49;\n  8:  610,  7940, 23598,  29200, 18125,  5940,   980,   64;\n  9: 1597, 23644, 82206, 123248, 97125, 43308, 10976, 1472, 81;\n","recurrence_sentence":"The row polynomials satisfy (200*n-300+(280*n-420)*x-(100*n^3-450*n^2+562*n-168)*x^2-(60*n^3-270*n^2+406*n-204)*x^3+(2*n^5-15*n^4+44*n^3-63*n^2+44*n-12)*x^4)*P(n,x) = (300*n-400+(175*n^2+295*n-610)*x-(150*n^3-710*n^2+388*n+216)*x^2-(10*n^4+115*n^3-510*n^2+423*n-46)*x^3+(3*n^5-22*n^4+11*n^3+68*n^2-76*n+16)*x^4+(n^5-5*n^4+8*n^3-4*n^2)*x^5)*P(n-1,x) + (125+600*x-(175*n^2-350*n-760)*x^2-(360*n^2-720*n-244)*x^3+(5*n^4-20*n^3-195*n^2+430*n-57)*x^4+(6*n^4-24*n^3-6*n^2+60*n-16)*x^5+(n^4-4*n^3+4*n^2)*x^6)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((300*n - 400 + (175*n^2 + 295*n - 610)*x - (150*n^3 - 710*n^2 + 388*n + 216)*x^2 - (10*n^4 + 115*n^3 - 510*n^2 + 423*n - 46)*x^3 + (3*n^5 - 22*n^4 + 11*n^3 + 68*n^2 - 76*n + 16)*x^4 + (n^5 - 5*n^4 + 8*n^3 - 4*n^2)*x^5)*P[n - 1, x] + (125 + 600*x - (175*n^2 - 350*n - 760)*x^2 - (360*n^2 - 720*n - 244)*x^3 + (5*n^4 - 20*n^3 - 195*n^2 + 430*n - 57)*x^4 + (6*n^4 - 24*n^3 - 6*n^2 + 60*n - 16)*x^5 + (n^4 - 4*n^3 + 4*n^2)*x^6)*D[P[n - 1, x], x])/(200*n - 300 + (280*n - 420)*x - (100*n^3 - 450*n^2 + 562*n - 168)*x^2 - (60*n^3 - 270*n^2 + 406*n - 204)*x^3 + (2*n^5 - 15*n^4 + 44*n^3 - 63*n^2 + 44*n - 12)*x^4)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((300*n - 400 + (175*n^2 + 295*n - 610)*x - (150*n^3 - 710*n^2 + 388*n + 216)*x^2 - (10*n^4 + 115*n^3 - 510*n^2 + 423*n - 46)*x^3 + (3*n^5 - 22*n^4 + 11*n^3 + 68*n^2 - 76*n + 16)*x^4 + (n^5 - 5*n^4 + 8*n^3 - 4*n^2)*x^5)*v[i - 1] + (125 + 600*x - (175*n^2 - 350*n - 760)*x^2 - (360*n^2 - 720*n - 244)*x^3 + (5*n^4 - 20*n^3 - 195*n^2 + 430*n - 57)*x^4 + (6*n^4 - 24*n^3 - 6*n^2 + 60*n - 16)*x^5 + (n^4 - 4*n^3 + 4*n^2)*x^6)*deriv(v[i - 1], x))/(200*n - 300 + (280*n - 420)*x - (100*n^3 - 450*n^2 + 562*n - 168)*x^2 - (60*n^3 - 270*n^2 + 406*n - 204)*x^3 + (2*n^5 - 15*n^4 + 44*n^3 - 63*n^2 + 44*n - 12)*x^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-01fe968f3e05a927","steps":["x \u21a6 x+1","(\u03b8+1)\u00b2"],"edges":["e-17055ffc05dec6ce","e-9a9fe08936a394da"],"text":"A165253 \u2192 x \u21a6 x+1 \u2192 (\u03b8+1)\u00b2"},"headline":"A165253 \u2192 x \u21a6 x+1 \u2192 (\u03b8+1)\u00b2","chain":null}