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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":"ffc6535df8c69224a762e7ce65855b343a22b525c89eefc3f9010599e096e509","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-86fb6463ff8368f2","role":"sequence","name":"Triangle of third-order Eulerian numbers: 3-Stirling permutations enumerated by ascents.","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,    12;\n  2: 1,    48,     135;\n  3: 1,   132,    1269,      1680;\n  4: 1,   312,    7074,     29280,      23625;\n  5: 1,   684,   31050,    280608,     667125,      374220;\n  6: 1,  1440,  119313,   2005632,    9912375,    15694560,      6621615;\n  7: 1,  2964,  422739,  12052656,  105970275,   336822300,    388143945,    129729600;\n  8: 1,  6024, 1421604,  64744416,  925329150,  5072877720,  11427606540,  10163180160,   2791213425;\n  9: 1, 12156, 4617216, 322160064, 7054561350, 61084167768, 231897750840, 394748363520, 282376418625, 65472907500;\n","recurrence_sentence":"The row polynomials satisfy (3*n-2)*P(n,x) = -((36*n^2-24*n-12)*x-(243*n^3-342*n^2+57*n+42)*x^2)*P(n-2,x) - ((6*n+2)*x-(36*n^2-1)*x^2-(108*n^3-270*n^2+75*n+38)*x^3)*d/dx P(n-2,x) - ((3*n+1)*x^2-(36*n^2-12*n-5)*x^3+(36*n^2-15*n-6)*x^4)*d^2/dx^2 P(n-2,x) + (3*n-2+(36*n^2-24*n)*x)*P(n-1,x) + ((6*n-1)*x-(36*n^2-12*n-8)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 12*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 12*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-((36*n^2 - 24*n - 12)*x - (243*n^3 - 342*n^2 + 57*n + 42)*x^2)*P[n - 2, x] - ((6*n + 2)*x - (36*n^2 - 1)*x^2 - (108*n^3 - 270*n^2 + 75*n + 38)*x^3)*D[P[n - 2, x], x] - ((3*n + 1)*x^2 - (36*n^2 - 12*n - 5)*x^3 + (36*n^2 - 15*n - 6)*x^4)*D[P[n - 2, x], {x, 2}] + (3*n - 2 + (36*n^2 - 24*n)*x)*P[n - 1, x] + ((6*n - 1)*x - (36*n^2 - 12*n - 8)*x^2)*D[P[n - 1, x], x])/(3*n - 2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 12*x; for(n = 2, N, my(i = n + 1); v[i] = (-((36*n^2 - 24*n - 12)*x - (243*n^3 - 342*n^2 + 57*n + 42)*x^2)*v[i - 2] - ((6*n + 2)*x - (36*n^2 - 1)*x^2 - (108*n^3 - 270*n^2 + 75*n + 38)*x^3)*deriv(v[i - 2], x) - ((3*n + 1)*x^2 - (36*n^2 - 12*n - 5)*x^3 + (36*n^2 - 15*n - 6)*x^4)*deriv(deriv(v[i - 2], x), x) + (3*n - 2 + (36*n^2 - 24*n)*x)*v[i - 1] + ((6*n - 1)*x - (36*n^2 - 12*n - 8)*x^2)*deriv(v[i - 1], x))/(3*n - 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-86fb6463ff8368f2","steps":["(\u03b8+1)\u00b2"],"edges":["e-33549f372da17d84"],"text":"A219512 \u2192 (\u03b8+1)\u00b2"},"headline":"A219512 \u2192 (\u03b8+1)\u00b2","chain":null}