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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":"f1ed9d8b02754d0c8f074679d3bec61476b679facda42c04fbecad94751abeeb","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-6fa6977267f2e73d","role":"sequence","name":"Triangle read by rows, T(n,k) = C(n,k)*C(k,n-k).","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: 0, 4;\n  2: 0, 8,  9;\n  3: 0, 0, 54,  16;\n  4: 0, 0, 54, 192,   25;\n  5: 0, 0,  0, 480,  500,    36;\n  6: 0, 0,  0, 320, 2250,  1080,    49;\n  7: 0, 0,  0,   0, 3500,  7560,  2058,    64;\n  8: 0, 0,  0,   0, 1750, 20160, 20580,  3584,   81;\n  9: 0, 0,  0,   0,    0, 22680, 82320, 48384, 5832, 100;\n","recurrence_sentence":"The row polynomials satisfy (n^5+6*n^4+13*n^3+12*n^2+4*n-(2*n^5+8*n^4+4*n^3-4*n^2)*x+n^5*x^2)*P(n,x) = -(4*n^5+20*n^4+28*n^3-4*n^2-32*n-16-(11*n^5+60*n^4+99*n^3+42*n^2-36*n-32)*x+(10*n^5+44*n^4+45*n^3-29*n^2-38*n+8)*x^2-(3*n^5+6*n^4-n^3-4*n^2)*x^3)*P(n-1,x) + ((8*n^4+48*n^3+104*n^2+96*n+32)*x-(18*n^4+100*n^3+170*n^2+88*n-8)*x^2+(12*n^4+46*n^3+52*n^2+16*n-4)*x^3-(2*n^4+6*n^3+4*n^2)*x^4)*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[(-(4*n^5 + 20*n^4 + 28*n^3 - 4*n^2 - 32*n - 16 - (11*n^5 + 60*n^4 + 99*n^3 + 42*n^2 - 36*n - 32)*x + (10*n^5 + 44*n^4 + 45*n^3 - 29*n^2 - 38*n + 8)*x^2 - (3*n^5 + 6*n^4 - n^3 - 4*n^2)*x^3)*P[n - 1, x] + ((8*n^4 + 48*n^3 + 104*n^2 + 96*n + 32)*x - (18*n^4 + 100*n^3 + 170*n^2 + 88*n - 8)*x^2 + (12*n^4 + 46*n^3 + 52*n^2 + 16*n - 4)*x^3 - (2*n^4 + 6*n^3 + 4*n^2)*x^4)*D[P[n - 1, x], x])/(n^5 + 6*n^4 + 13*n^3 + 12*n^2 + 4*n - (2*n^5 + 8*n^4 + 4*n^3 - 4*n^2)*x + n^5*x^2)]];\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] = (-(4*n^5 + 20*n^4 + 28*n^3 - 4*n^2 - 32*n - 16 - (11*n^5 + 60*n^4 + 99*n^3 + 42*n^2 - 36*n - 32)*x + (10*n^5 + 44*n^4 + 45*n^3 - 29*n^2 - 38*n + 8)*x^2 - (3*n^5 + 6*n^4 - n^3 - 4*n^2)*x^3)*v[i - 1] + ((8*n^4 + 48*n^3 + 104*n^2 + 96*n + 32)*x - (18*n^4 + 100*n^3 + 170*n^2 + 88*n - 8)*x^2 + (12*n^4 + 46*n^3 + 52*n^2 + 16*n - 4)*x^3 - (2*n^4 + 6*n^3 + 4*n^2)*x^4)*deriv(v[i - 1], x))/(n^5 + 6*n^4 + 13*n^3 + 12*n^2 + 4*n - (2*n^5 + 8*n^4 + 4*n^3 - 4*n^2)*x + n^5*x^2)); 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-6fa6977267f2e73d","steps":["(\u03b8+1)\u00b2"],"edges":["e-305056759dfbb594"],"text":"A105868 \u2192 (\u03b8+1)\u00b2"},"headline":"A105868 \u2192 (\u03b8+1)\u00b2","chain":null}