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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":"a40699c07d64b2ab8e5e719178ff3b041c2bf4b25d40d04e2b1fcb24032eb125","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-0135f5d413e6cc58","role":"sequence","name":"basis_map_A007318 of Coefficient table for Chebyshev's U(2*n,x) polynomial expanded in decreasing powers of (-4*(1-x^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:    4,     12;\n  2:   11,     60,      45;\n  3:   29,    224,     315,     112;\n  4:   76,    756,    1539,    1056,     225;\n  5:  199,   2420,    6435,    6512,    2750,     396;\n  6:  521,   7488,   24570,   32864,   20800,    6084,     637;\n  7: 1364,  22620,   88290,  146880,  124125,   55188,   12005,    960;\n  8: 3571,  67116,  303705,  604384,  639625,  383112,  128282,  21760,  1377;\n  9: 9349, 196384, 1010781, 2341712, 2979200, 2248308, 1022238, 269952, 36936, 1900;\n","recurrence_sentence":"The row polynomials satisfy (200*n-100+(500*n^2-130*n-60)*x+(600*n^3-180*n^2-68*n+4)*x^2+(240*n^4+56*n^3-116*n^2-10*n+12)*x^3+(32*n^5+16*n^4-8*n^3-4*n^2)*x^4)*P(n,x) = (800*n-400+(2700*n^2+830*n-1090)*x+(3000*n^3+4380*n^2-1532*n-704)*x^2+(1120*n^4+4344*n^3+1696*n^2-2246*n+86)*x^3+(128*n^5+1264*n^4+1760*n^3-196*n^2-748*n+120)*x^4+(96*n^5+304*n^4+168*n^3-108*n^2-24*n-4)*x^5)*P(n-1,x) - (125+(750*n+450)*x+(1200*n^2+2750*n+510)*x^2+(600*n^3+3500*n^2+3810*n+144)*x^3+(80*n^4+1400*n^3+3712*n^2+2442*n-67)*x^4+(144*n^4+1056*n^3+1684*n^2+696*n-18)*x^5+(64*n^4+256*n^3+272*n^2+64*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 - 400 + (2700*n^2 + 830*n - 1090)*x + (3000*n^3 + 4380*n^2 - 1532*n - 704)*x^2 + (1120*n^4 + 4344*n^3 + 1696*n^2 - 2246*n + 86)*x^3 + (128*n^5 + 1264*n^4 + 1760*n^3 - 196*n^2 - 748*n + 120)*x^4 + (96*n^5 + 304*n^4 + 168*n^3 - 108*n^2 - 24*n - 4)*x^5)*P[n - 1, x] - (125 + (750*n + 450)*x + (1200*n^2 + 2750*n + 510)*x^2 + (600*n^3 + 3500*n^2 + 3810*n + 144)*x^3 + (80*n^4 + 1400*n^3 + 3712*n^2 + 2442*n - 67)*x^4 + (144*n^4 + 1056*n^3 + 1684*n^2 + 696*n - 18)*x^5 + (64*n^4 + 256*n^3 + 272*n^2 + 64*n + 8)*x^6)*D[P[n - 1, x], x])/(200*n - 100 + (500*n^2 - 130*n - 60)*x + (600*n^3 - 180*n^2 - 68*n + 4)*x^2 + (240*n^4 + 56*n^3 - 116*n^2 - 10*n + 12)*x^3 + (32*n^5 + 16*n^4 - 8*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 - 400 + (2700*n^2 + 830*n - 1090)*x + (3000*n^3 + 4380*n^2 - 1532*n - 704)*x^2 + (1120*n^4 + 4344*n^3 + 1696*n^2 - 2246*n + 86)*x^3 + (128*n^5 + 1264*n^4 + 1760*n^3 - 196*n^2 - 748*n + 120)*x^4 + (96*n^5 + 304*n^4 + 168*n^3 - 108*n^2 - 24*n - 4)*x^5)*v[i - 1] - (125 + (750*n + 450)*x + (1200*n^2 + 2750*n + 510)*x^2 + (600*n^3 + 3500*n^2 + 3810*n + 144)*x^3 + (80*n^4 + 1400*n^3 + 3712*n^2 + 2442*n - 67)*x^4 + (144*n^4 + 1056*n^3 + 1684*n^2 + 696*n - 18)*x^5 + (64*n^4 + 256*n^3 + 272*n^2 + 64*n + 8)*x^6)*deriv(v[i - 1], x))/(200*n - 100 + (500*n^2 - 130*n - 60)*x + (600*n^3 - 180*n^2 - 68*n + 4)*x^2 + (240*n^4 + 56*n^3 - 116*n^2 - 10*n + 12)*x^3 + (32*n^5 + 16*n^4 - 8*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-9496336e2b29e28e","steps":["x \u21a6 x+1","(\u03b8+1)\u00b2"],"edges":["e-557cf165357d4e1f","e-b05a145a5a5d7342"],"text":"A082985 \u2192 x \u21a6 x+1 \u2192 (\u03b8+1)\u00b2"},"headline":"A082985 \u2192 x \u21a6 x+1 \u2192 (\u03b8+1)\u00b2","chain":null}