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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":"a8c37ca212d3a116dfc1a980bb28da954c0de6ade3da807086b4fc396dd33bbe","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-d4c5acf24bc40277","role":"sequence","name":"Triangle of coefficients of polynomials v(n,x) jointly generated with A208508; see the Formula section.","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:  3;\n  2:  5,   4;\n  3:  7,  20;\n  4:  9,  56,    9;\n  5: 11, 120,   63;\n  6: 13, 220,  243,   16;\n  7: 15, 364,  693,  144;\n  8: 17, 560, 1638,  704,  25;\n  9: 19, 816, 3402, 2496, 275;\n","recurrence_sentence":"The row polynomials satisfy -(16*n-8-(40*n^3-52*n^2+8*n+28)*x+(4*n^5-2*n^4+6*n^3+26*n^2+2*n-12)*x^2-(n^5+3*n^4+3*n^3+n^2)*x^3)*P(n,x) = -(16*n+8-(40*n^3-12*n^2-88*n+132)*x+(4*n^5+2*n^4-66*n^3+78*n^2+194*n-140)*x^2+(3*n^5+5*n^4-n^3+3*n^2-6*n-28)*x^3-(n^5+5*n^4+7*n^3-n^2-8*n-4)*x^4)*P(n-1,x) - (32*x-(80*n^2+96*n-8)*x^2+(8*n^4+32*n^3+132*n^2+148*n-8)*x^3-(10*n^4+44*n^3+78*n^2+76*n+40)*x^4+(2*n^4+12*n^3+26*n^2+24*n+8)*x^5)*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[(-(16*n + 8 - (40*n^3 - 12*n^2 - 88*n + 132)*x + (4*n^5 + 2*n^4 - 66*n^3 + 78*n^2 + 194*n - 140)*x^2 + (3*n^5 + 5*n^4 - n^3 + 3*n^2 - 6*n - 28)*x^3 - (n^5 + 5*n^4 + 7*n^3 - n^2 - 8*n - 4)*x^4)*P[n - 1, x] - (32*x - (80*n^2 + 96*n - 8)*x^2 + (8*n^4 + 32*n^3 + 132*n^2 + 148*n - 8)*x^3 - (10*n^4 + 44*n^3 + 78*n^2 + 76*n + 40)*x^4 + (2*n^4 + 12*n^3 + 26*n^2 + 24*n + 8)*x^5)*D[P[n - 1, x], x])/(-16*n + 8 + (40*n^3 - 52*n^2 + 8*n + 28)*x - (4*n^5 - 2*n^4 + 6*n^3 + 26*n^2 + 2*n - 12)*x^2 + (n^5 + 3*n^4 + 3*n^3 + n^2)*x^3)]];\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] = (-(16*n + 8 - (40*n^3 - 12*n^2 - 88*n + 132)*x + (4*n^5 + 2*n^4 - 66*n^3 + 78*n^2 + 194*n - 140)*x^2 + (3*n^5 + 5*n^4 - n^3 + 3*n^2 - 6*n - 28)*x^3 - (n^5 + 5*n^4 + 7*n^3 - n^2 - 8*n - 4)*x^4)*v[i - 1] - (32*x - (80*n^2 + 96*n - 8)*x^2 + (8*n^4 + 32*n^3 + 132*n^2 + 148*n - 8)*x^3 - (10*n^4 + 44*n^3 + 78*n^2 + 76*n + 40)*x^4 + (2*n^4 + 12*n^3 + 26*n^2 + 24*n + 8)*x^5)*deriv(v[i - 1], x))/(-16*n + 8 + (40*n^3 - 52*n^2 + 8*n + 28)*x - (4*n^5 - 2*n^4 + 6*n^3 + 26*n^2 + 2*n - 12)*x^2 + (n^5 + 3*n^4 + 3*n^3 + n^2)*x^3)); 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-d4c5acf24bc40277","steps":["(\u03b8+1)\u00b2"],"edges":["e-ef5be206ca887344"],"text":"A208509 \u2192 (\u03b8+1)\u00b2"},"headline":"A208509 \u2192 (\u03b8+1)\u00b2","chain":null}