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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":"4aa6b6b539ccd21426cf7534f43821b96ae3228c310a389fc413e8443dca096b","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-726472edd5bb65cd","role":"sequence","name":"reversal of Triangle read by rows: T(n, k) = (k+1)*S2(n+1, k+1), for n >= k >= 0, and S2 = A048993 (Stirling2).","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:  1;\n   2:  2,   2;\n   3:  3,  12,     3;\n   4:  4,  36,    42,      4;\n   5:  5,  80,   225,    120,      5;\n   6:  6, 150,   780,   1080,    310,       6;\n   7:  7, 252,  2100,   5600,   4515,     756,      7;\n   8:  8, 392,  4788,  21000,  34020,   17388,   1778,      8;\n   9:  9, 576,  9702,  63504, 173775,  186480,  63525,   4080,    9;\n  10: 10, 810, 18000, 164640, 684810, 1275750, 954940, 223920, 9198, 10;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(1+(2*n-10)*x+(n^2-9*n+20)*x^2)*P(n-4,x) + (x^2+(n-4)*x^3)*d/dx P(n-4,x) + (4+(6*n-22)*x+(2*n^2-11*n+12)*x^2)*P(n-3,x) - (3*x^2+(2*n-4)*x^3)*d/dx P(n-3,x) - (6+(6*n-14)*x+(n^2-2*n)*x^2)*P(n-2,x) + (3*x^2+n*x^3)*d/dx P(n-2,x) + (4+(2*n-2)*x)*P(n-1,x) - x^2*d/dx P(n-1,x) for n >= 5, with P(1,x) = 1, P(2,x) = 2 + 2*x, P(3,x) = 3 + 12*x + 3*x^2, P(4,x) = 4 + 36*x + 42*x^2 + 4*x^3.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 2 + 2*x;\nP[3, x_] := 3 + 12*x + 3*x^2;\nP[4, x_] := 4 + 36*x + 42*x^2 + 4*x^3;\nP[n_, x_] := P[n, x] = Expand[-(1 + (2*n - 10)*x + (n^2 - 9*n + 20)*x^2)*P[n - 4, x] + (x^2 + (n - 4)*x^3)*D[P[n - 4, x], x] + (4 + (6*n - 22)*x + (2*n^2 - 11*n + 12)*x^2)*P[n - 3, x] - (3*x^2 + (2*n - 4)*x^3)*D[P[n - 3, x], x] - (6 + (6*n - 14)*x + (n^2 - 2*n)*x^2)*P[n - 2, x] + (3*x^2 + n*x^3)*D[P[n - 2, x], x] + (4 + (2*n - 2)*x)*P[n - 1, x] - x^2*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 2*x; v[3] = 3 + 12*x + 3*x^2; v[4] = 4 + 36*x + 42*x^2 + 4*x^3; for(n = 5, N, my(i = n); v[i] = -(1 + (2*n - 10)*x + (n^2 - 9*n + 20)*x^2)*v[i - 4] + (x^2 + (n - 4)*x^3)*deriv(v[i - 4], x) + (4 + (6*n - 22)*x + (2*n^2 - 11*n + 12)*x^2)*v[i - 3] - (3*x^2 + (2*n - 4)*x^3)*deriv(v[i - 3], x) - (6 + (6*n - 14)*x + (n^2 - 2*n)*x^2)*v[i - 2] + (3*x^2 + n*x^3)*deriv(v[i - 2], x) + (4 + (2*n - 2)*x)*v[i - 1] - x^2*deriv(v[i - 1], x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-3cb22164e2f83fe1","steps":["\u03b8+1","reverse","\u03b8+1"],"edges":["e-907e3c6c6275331d","e-6878e8dc6c11eb74","e-94f63deb315a5bf7"],"text":"A008277 \u2192 \u03b8+1 \u2192 reverse \u2192 \u03b8+1"},"headline":"A008277 \u2192 \u03b8+1 \u2192 reverse \u2192 \u03b8+1","chain":null}