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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":"1d41b9e647d8feb761d9e3b93864605c7d52275eea68b99cf1fe0bfd9317d6b4","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-ea2139b8dbf6bf8d","role":"sequence","name":"Interpolation operator: Triangle with an even number of zeros in each line followed by 1 or 2 ones.","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, 4;\n  2: 0, 0, 9;\n  3: 0, 0, 9, 16;\n  4: 0, 0, 0,  0, 25;\n  5: 0, 0, 0,  0, 25, 36;\n  6: 0, 0, 0,  0,  0,  0, 49;\n  7: 0, 0, 0,  0,  0,  0, 49, 64;\n  8: 0, 0, 0,  0,  0,  0,  0,  0, 81;\n  9: 0, 0, 0,  0,  0,  0,  0,  0, 81, 100;\n","recurrence_sentence":"The row polynomials satisfy (n^4-2*n^3+n^2)*P(n,x) = -(n^5-4*n^4+5*n^3-2*n^2+(2*n^5-3*n^4-2*n^3+4*n^2+n-2)*x)*P(n-1,x) + ((n^4-2*n^3+n^2)*x+(2*n^4-2*n^2+1)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^5 - 4*n^4 + 5*n^3 - 2*n^2 + (2*n^5 - 3*n^4 - 2*n^3 + 4*n^2 + n - 2)*x)*P[n - 1, x] + ((n^4 - 2*n^3 + n^2)*x + (2*n^4 - 2*n^2 + 1)*x^2)*D[P[n - 1, x], x])/(n^4 - 2*n^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 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^5 - 4*n^4 + 5*n^3 - 2*n^2 + (2*n^5 - 3*n^4 - 2*n^3 + 4*n^2 + n - 2)*x)*v[i - 1] + ((n^4 - 2*n^3 + n^2)*x + (2*n^4 - 2*n^2 + 1)*x^2)*deriv(v[i - 1], x))/(n^4 - 2*n^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-ea2139b8dbf6bf8d","steps":["(\u03b8+1)\u00b2"],"edges":["e-3e094d4cbb0f0f62"],"text":"A133080 \u2192 (\u03b8+1)\u00b2"},"headline":"A133080 \u2192 (\u03b8+1)\u00b2","chain":null}