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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":"377d23ec7b297d443766134d18bdb9734d0ed78e0cec455befc7f753721e011b","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-47e3a1a59edf16da","role":"sequence","name":"A triangle of numbers related to triangle A030528; array a(n,m), read by rows (1 <= m <= n).","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: 0, 12,   9;\n  1: 0, 12,  54,   16;\n  2: 0,  0, 135,  160,    25;\n  3: 0,  0, 135,  720,   375,     36;\n  4: 0,  0,   0, 1680,  2625,    756,      49;\n  5: 0,  0,   0, 1680, 10500,   7560,    1372,      64;\n  6: 0,  0,   0,    0, 23625,  45360,   18522,    2304,      81;\n  7: 0,  0,   0,    0, 23625, 170100,  154350,   40320,    3645,    100;\n  8: 0,  0,   0,    0,     0, 374220,  848925,  443520,   80190,   5500,  121;\n  9: 0,  0,   0,    0,     0, 374220, 3056130, 3326400, 1122660, 148500, 7986, 144;\n","recurrence_sentence":"The row polynomials satisfy (n^4+10*n^3+37*n^2+60*n+36-(4*n^3+34*n^2+90*n+76)*x+(4*n^2+16*n+16)*x^2)*P(n,x) = -(n^5+10*n^4+37*n^3+60*n^2+36*n-(5*n^4+52*n^3+191*n^2+288*n+144)*x+(8*n^3+70*n^2+206*n+204)*x^2-(4*n^2+24*n+36)*x^3)*P(n-1,x) + ((2*n^4+20*n^3+74*n^2+120*n+72)*x-(8*n^3+76*n^2+236*n+240)*x^2+(8*n^2+48*n+72)*x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 12*x + 9*x^2.","mathematica":"P[0, x_] := 12*x + 9*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^5 + 10*n^4 + 37*n^3 + 60*n^2 + 36*n - (5*n^4 + 52*n^3 + 191*n^2 + 288*n + 144)*x + (8*n^3 + 70*n^2 + 206*n + 204)*x^2 - (4*n^2 + 24*n + 36)*x^3)*P[n - 1, x] + ((2*n^4 + 20*n^3 + 74*n^2 + 120*n + 72)*x - (8*n^3 + 76*n^2 + 236*n + 240)*x^2 + (8*n^2 + 48*n + 72)*x^3)*D[P[n - 1, x], x])/(n^4 + 10*n^3 + 37*n^2 + 60*n + 36 - (4*n^3 + 34*n^2 + 90*n + 76)*x + (4*n^2 + 16*n + 16)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 12*x + 9*x^2; for(n = 1, N, my(i = n + 1); v[i] = (-(n^5 + 10*n^4 + 37*n^3 + 60*n^2 + 36*n - (5*n^4 + 52*n^3 + 191*n^2 + 288*n + 144)*x + (8*n^3 + 70*n^2 + 206*n + 204)*x^2 - (4*n^2 + 24*n + 36)*x^3)*v[i - 1] + ((2*n^4 + 20*n^3 + 74*n^2 + 120*n + 72)*x - (8*n^3 + 76*n^2 + 236*n + 240)*x^2 + (8*n^2 + 48*n + 72)*x^3)*deriv(v[i - 1], x))/(n^4 + 10*n^3 + 37*n^2 + 60*n + 36 - (4*n^3 + 34*n^2 + 90*n + 76)*x + (4*n^2 + 16*n + 16)*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-47e3a1a59edf16da","steps":["(\u03b8+1)\u00b2"],"edges":["e-d8894e533519ca8e"],"text":"A049403 \u2192 (\u03b8+1)\u00b2"},"headline":"A049403 \u2192 (\u03b8+1)\u00b2","chain":null}