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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":"96d87cce0b33d3208ee5dc5f07b985aca1c7af1a5bd625a2acdeccf31e35802b","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-f8bb6517bc8332de","role":"sequence","name":"theta_plus_one of A Jacobsthal triangle.","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: 0,  8;\n  2: 0, 16,  27;\n  3: 0,  0, 108,  64;\n  4: 0,  0, 108, 384,  125;\n  5: 0,  0,   0, 768, 1000,   216;\n  6: 0,  0,   0, 512, 3000,  2160,   343;\n  7: 0,  0,   0,   0, 4000,  8640,  4116,   512;\n  8: 0,  0,   0,   0, 2000, 17280, 20580,  7168,   729;\n  9: 0,  0,   0,   0,    0, 17280, 54880, 43008, 11664, 1000;\n","recurrence_sentence":"The row polynomials satisfy (4*n^5+24*n^4+52*n^3+48*n^2+16*n-(6*n^4+24*n^3+18*n^2-4*n)*x+n^3*x^2)*P(n,x) = -(16*n^5+80*n^4+112*n^3-16*n^2-128*n-64-(40*n^4+216*n^3+400*n^2+352*n+144)*x+(22*n^3+102*n^2+156*n+72)*x^2-(2*n^2+4*n+2)*x^3)*P(n-1,x) + ((32*n^4+192*n^3+416*n^2+384*n+128)*x+(4*n^4-24*n^3-188*n^2-304*n-144)*x^2-(6*n^3+22*n^2+28*n+12)*x^3+(n^2+2*n+1)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(16*n^5 + 80*n^4 + 112*n^3 - 16*n^2 - 128*n - 64 - (40*n^4 + 216*n^3 + 400*n^2 + 352*n + 144)*x + (22*n^3 + 102*n^2 + 156*n + 72)*x^2 - (2*n^2 + 4*n + 2)*x^3)*P[n - 1, x] + ((32*n^4 + 192*n^3 + 416*n^2 + 384*n + 128)*x + (4*n^4 - 24*n^3 - 188*n^2 - 304*n - 144)*x^2 - (6*n^3 + 22*n^2 + 28*n + 12)*x^3 + (n^2 + 2*n + 1)*x^4)*D[P[n - 1, x], x])/(4*n^5 + 24*n^4 + 52*n^3 + 48*n^2 + 16*n - (6*n^4 + 24*n^3 + 18*n^2 - 4*n)*x + n^3*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(16*n^5 + 80*n^4 + 112*n^3 - 16*n^2 - 128*n - 64 - (40*n^4 + 216*n^3 + 400*n^2 + 352*n + 144)*x + (22*n^3 + 102*n^2 + 156*n + 72)*x^2 - (2*n^2 + 4*n + 2)*x^3)*v[i - 1] + ((32*n^4 + 192*n^3 + 416*n^2 + 384*n + 128)*x + (4*n^4 - 24*n^3 - 188*n^2 - 304*n - 144)*x^2 - (6*n^3 + 22*n^2 + 28*n + 12)*x^3 + (n^2 + 2*n + 1)*x^4)*deriv(v[i - 1], x))/(4*n^5 + 24*n^4 + 52*n^3 + 48*n^2 + 16*n - (6*n^4 + 24*n^3 + 18*n^2 - 4*n)*x + n^3*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-ba196dcd01800741","steps":["\u03b8+1","(\u03b8+1)\u00b2"],"edges":["e-ecb19626d7d3a316","e-511e4390cdf49cee"],"text":"A113953 \u2192 \u03b8+1 \u2192 (\u03b8+1)\u00b2"},"headline":"A113953 \u2192 \u03b8+1 \u2192 (\u03b8+1)\u00b2","chain":null}