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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":"35ed4e1fb45538a8f789207fdeeda1123422c52a552912badd3ce7387da83919","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-56eee146d6109546","role":"sequence","name":"Triangle T(n,k) read by rows: T(n,k) = (n-k+1)*T(n-1,k-1) + (3*k-2)*k*T(n-1,k), initialized by T(n,1) = T(n,n) = 1.","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,        4;\n  2: 1,       40,           9;\n  3: 1,      332,         369,           16;\n  4: 1,     2672,        9990,         1952,           25;\n  5: 1,    21396,      233838,       131360,         7725,           36;\n  6: 1,   171192,     5151303,      6917248,      1117875,        25704,          49;\n  7: 1,  1369564,   110488455,    322479280,    115894675,      7296804,       76489,        64;\n  8: 1, 10956544,  2341828188,  14077714720,  10052523250,   1368046512,    39968320,    211072,     81;\n  9: 1, 87652388, 49375609740, 592251339584, 785392586750, 203710632552, 12764039792, 193759232, 552501, 100;\n","recurrence_sentence":"The row polynomials satisfy n*P(n,x) = ((8*n^2-8)*x+(9*n^3-2*n^2-7*n)*x^2)*P(n-2,x) + ((16*n+16)*x+(19*n^2+5*n-10)*x^2+(4*n^3-14*n^2-13*n)*x^3)*d/dx P(n-2,x) + ((26*n+26)*x^2+(43*n^2+31*n-2)*x^3-(4*n^2+3*n)*x^4)*d^2/dx^2 P(n-2,x) + ((6*n+6)*x^3+(12*n^2+9*n)*x^4)*d^3/dx^3 P(n-2,x) + (n+4*n^2*x)*P(n-1,x) + ((5*n-2)*x-(4*n^2+4*n)*x^2)*d/dx P(n-1,x) + 3*n*x^2*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[(((8*n^2 - 8)*x + (9*n^3 - 2*n^2 - 7*n)*x^2)*P[n - 2, x] + ((16*n + 16)*x + (19*n^2 + 5*n - 10)*x^2 + (4*n^3 - 14*n^2 - 13*n)*x^3)*D[P[n - 2, x], x] + ((26*n + 26)*x^2 + (43*n^2 + 31*n - 2)*x^3 - (4*n^2 + 3*n)*x^4)*D[P[n - 2, x], {x, 2}] + ((6*n + 6)*x^3 + (12*n^2 + 9*n)*x^4)*D[P[n - 2, x], {x, 3}] + (n + 4*n^2*x)*P[n - 1, x] + ((5*n - 2)*x - (4*n^2 + 4*n)*x^2)*D[P[n - 1, x], x] + 3*n*x^2*D[P[n - 1, x], {x, 2}])/(n)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (((8*n^2 - 8)*x + (9*n^3 - 2*n^2 - 7*n)*x^2)*v[i - 2] + ((16*n + 16)*x + (19*n^2 + 5*n - 10)*x^2 + (4*n^3 - 14*n^2 - 13*n)*x^3)*deriv(v[i - 2], x) + ((26*n + 26)*x^2 + (43*n^2 + 31*n - 2)*x^3 - (4*n^2 + 3*n)*x^4)*deriv(deriv(v[i - 2], x), x) + ((6*n + 6)*x^3 + (12*n^2 + 9*n)*x^4)*deriv(deriv(deriv(v[i - 2], x), x), x) + (n + 4*n^2*x)*v[i - 1] + ((5*n - 2)*x - (4*n^2 + 4*n)*x^2)*deriv(v[i - 1], x) + 3*n*x^2*deriv(deriv(v[i - 1], x), x))/(n)); 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-56eee146d6109546","steps":["(\u03b8+1)\u00b2"],"edges":["e-9cbed4e7d44e038d"],"text":"A166972 \u2192 (\u03b8+1)\u00b2"},"headline":"A166972 \u2192 (\u03b8+1)\u00b2","chain":null}