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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":"d04aec80e6ff165dc3f73827135e2aa551489b469a6c6823e8dbbac9da0ed6d6","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-fac4d4a000d744e8","role":"sequence","name":"Bisection of Fibonacci triangle A037027: even-indexed members of column sequences of A037027 (not counting leading zeros).","uses":{"claim":"real_rooted","from_index":0,"region":{}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)"}]}],"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,     4;\n  2:    5,    20,      9;\n  3:   13,    80,     81,     16;\n  4:   34,   284,    459,    224,     25;\n  5:   89,   940,   2097,   1680,    500,     36;\n  6:  233,  2976,   8478,   9504,   4750,    972,     49;\n  7:  610,  9140,  31698,  45760,  32375,  11340,   1715,    64;\n  8: 1597, 27460, 112257, 198432, 182125,  91224,  24010,  2816,   81;\n  9: 4181, 81136, 382023, 799408, 903050, 590652, 224322, 46464, 4374, 100;\n","recurrence_sentence":"The row polynomials satisfy (200*n-100+(500*n^2-330*n+40)*x+(600*n^3-740*n^2+308*n-44)*x^2+(240*n^4-280*n^3+28*n^2+34*n-4)*x^3+(32*n^5-48*n^4+24*n^3-4*n^2)*x^4)*P(n,x) = (800*n-600+(2700*n^2-1170*n-710)*x+(3000*n^3+700*n^2-2788*n+176)*x^2+(1120*n^4+2440*n^3-2288*n^2-1306*n+566)*x^3+(128*n^5+976*n^4-160*n^3-956*n^2-116*n+208)*x^4+(96*n^5+80*n^4-152*n^3-52*n^2+56*n-12)*x^5)*P(n-1,x) - (125+(750*n+150)*x+(1200*n^2+1650*n-240)*x^2+(600*n^3+2660*n^2+1110*n-524)*x^3+(80*n^4+1240*n^3+1880*n^2+146*n-325)*x^4+(144*n^4+768*n^3+436*n^2-96*n-58)*x^5+(64*n^4+128*n^3+16*n^2-32*n+8)*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[((800*n - 600 + (2700*n^2 - 1170*n - 710)*x + (3000*n^3 + 700*n^2 - 2788*n + 176)*x^2 + (1120*n^4 + 2440*n^3 - 2288*n^2 - 1306*n + 566)*x^3 + (128*n^5 + 976*n^4 - 160*n^3 - 956*n^2 - 116*n + 208)*x^4 + (96*n^5 + 80*n^4 - 152*n^3 - 52*n^2 + 56*n - 12)*x^5)*P[n - 1, x] - (125 + (750*n + 150)*x + (1200*n^2 + 1650*n - 240)*x^2 + (600*n^3 + 2660*n^2 + 1110*n - 524)*x^3 + (80*n^4 + 1240*n^3 + 1880*n^2 + 146*n - 325)*x^4 + (144*n^4 + 768*n^3 + 436*n^2 - 96*n - 58)*x^5 + (64*n^4 + 128*n^3 + 16*n^2 - 32*n + 8)*x^6)*D[P[n - 1, x], x])/(200*n - 100 + (500*n^2 - 330*n + 40)*x + (600*n^3 - 740*n^2 + 308*n - 44)*x^2 + (240*n^4 - 280*n^3 + 28*n^2 + 34*n - 4)*x^3 + (32*n^5 - 48*n^4 + 24*n^3 - 4*n^2)*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] = ((800*n - 600 + (2700*n^2 - 1170*n - 710)*x + (3000*n^3 + 700*n^2 - 2788*n + 176)*x^2 + (1120*n^4 + 2440*n^3 - 2288*n^2 - 1306*n + 566)*x^3 + (128*n^5 + 976*n^4 - 160*n^3 - 956*n^2 - 116*n + 208)*x^4 + (96*n^5 + 80*n^4 - 152*n^3 - 52*n^2 + 56*n - 12)*x^5)*v[i - 1] - (125 + (750*n + 150)*x + (1200*n^2 + 1650*n - 240)*x^2 + (600*n^3 + 2660*n^2 + 1110*n - 524)*x^3 + (80*n^4 + 1240*n^3 + 1880*n^2 + 146*n - 325)*x^4 + (144*n^4 + 768*n^3 + 436*n^2 - 96*n - 58)*x^5 + (64*n^4 + 128*n^3 + 16*n^2 - 32*n + 8)*x^6)*deriv(v[i - 1], x))/(200*n - 100 + (500*n^2 - 330*n + 40)*x + (600*n^3 - 740*n^2 + 308*n - 44)*x^2 + (240*n^4 - 280*n^3 + 28*n^2 + 34*n - 4)*x^3 + (32*n^5 - 48*n^4 + 24*n^3 - 4*n^2)*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-fac4d4a000d744e8","steps":["(\u03b8+1)\u00b2"],"edges":["e-dda6d332e97688a5"],"text":"A060920 \u2192 (\u03b8+1)\u00b2"},"headline":"A060920 \u2192 (\u03b8+1)\u00b2","chain":null}