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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":"97ef550e9f0696dcfb5a01a374aebab724489321c68d9dfc555959d697faef41","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-06404fc73ac5a6a1","role":"sequence","name":"reversal of Triangle read by rows: 3-Stirling numbers of the second kind.","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: 1,  12;\n   3: 1,  28,    81;\n   4: 1,  48,   333,    432;\n   5: 1,  72,   873,   2800,    2025;\n   6: 1, 100,  1845,  10560,   19525,     8748;\n   7: 1, 132,  3420,  30240,  102025,   121212,    35721;\n   8: 1, 168,  5796,  72800,  385525,   855792,   695653,   139968;\n   9: 1, 208,  9198, 155232, 1181775,  4186728,  6519793,  3774400,   531441;\n  10: 1, 252, 13878, 302400, 3122175, 16099020, 40711405, 46352640, 19639341, 1968300;\n","recurrence_sentence":"The row polynomials satisfy (n^2+2*n+1)*P(n,x) = -(n^2+3*n+2+(4*n^3+12*n^2+12*n)*x-(9*n^4+34*n^3+41*n^2+16*n)*x^2)*P(n-2,x) - (x-(4*n^3+16*n^2+20*n+13)*x^2-(4*n^4+2*n^3-31*n^2-52*n-23)*x^3)*d/dx P(n-2,x) + (x^3-(4*n^3+15*n^2+18*n+7)*x^4)*d^2/dx^2 P(n-2,x) + (2*n^2+5*n+3+(4*n^3+12*n^2+12*n+4)*x)*P(n-1,x) + (x-(4*n^3+16*n^2+20*n+8)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 12*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 1 + 12*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^2 + 3*n + 2 + (4*n^3 + 12*n^2 + 12*n)*x - (9*n^4 + 34*n^3 + 41*n^2 + 16*n)*x^2)*P[n - 2, x] - (x - (4*n^3 + 16*n^2 + 20*n + 13)*x^2 - (4*n^4 + 2*n^3 - 31*n^2 - 52*n - 23)*x^3)*D[P[n - 2, x], x] + (x^3 - (4*n^3 + 15*n^2 + 18*n + 7)*x^4)*D[P[n - 2, x], {x, 2}] + (2*n^2 + 5*n + 3 + (4*n^3 + 12*n^2 + 12*n + 4)*x)*P[n - 1, x] + (x - (4*n^3 + 16*n^2 + 20*n + 8)*x^2)*D[P[n - 1, x], x])/(n^2 + 2*n + 1)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 12*x; for(n = 3, N, my(i = n); v[i] = (-(n^2 + 3*n + 2 + (4*n^3 + 12*n^2 + 12*n)*x - (9*n^4 + 34*n^3 + 41*n^2 + 16*n)*x^2)*v[i - 2] - (x - (4*n^3 + 16*n^2 + 20*n + 13)*x^2 - (4*n^4 + 2*n^3 - 31*n^2 - 52*n - 23)*x^3)*deriv(v[i - 2], x) + (x^3 - (4*n^3 + 15*n^2 + 18*n + 7)*x^4)*deriv(deriv(v[i - 2], x), x) + (2*n^2 + 5*n + 3 + (4*n^3 + 12*n^2 + 12*n + 4)*x)*v[i - 1] + (x - (4*n^3 + 16*n^2 + 20*n + 8)*x^2)*deriv(v[i - 1], x))/(n^2 + 2*n + 1)); 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-a2b8ff03da0c14a3","steps":["reverse","(\u03b8+1)\u00b2"],"edges":["e-0c475173369de966","e-257269f00533e315"],"text":"A143495 \u2192 reverse \u2192 (\u03b8+1)\u00b2"},"headline":"A143495 \u2192 reverse \u2192 (\u03b8+1)\u00b2","chain":null}