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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":"a28c2f2919e5916672d9ea51e21259ec78a288453535362e1658bcb870b98bf3","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-f25cd955a876cde3","role":"sequence","name":"Pascal-(1,2,1) array read by antidiagonals.","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,  2;\n  2: 1,  8,   3;\n  3: 1, 14,  21,    4;\n  4: 1, 20,  66,   40,     5;\n  5: 1, 26, 138,  184,    65,     6;\n  6: 1, 32, 237,  544,   395,    96,    7;\n  7: 1, 38, 363, 1228,  1535,   726,  133,    8;\n  8: 1, 44, 516, 2344,  4430,  3516, 1204,  176,   9;\n  9: 1, 50, 696, 4000, 10430, 12516, 7000, 1856, 225, 10;\n","recurrence_sentence":"The row polynomials satisfy -(n+(3*n^2-2*n)*x-(3*n^2+5*n)*x^2)*P(n,x) = -(n+(3*n^2+10*n-10)*x+(15*n^2+7*n-28)*x^2-(6*n+10)*x^3)*P(n-1,x) + (x+(3*n+15)*x^2+(30*n+51)*x^3+(3*n+5)*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[(-(n + (3*n^2 + 10*n - 10)*x + (15*n^2 + 7*n - 28)*x^2 - (6*n + 10)*x^3)*P[n - 1, x] + (x + (3*n + 15)*x^2 + (30*n + 51)*x^3 + (3*n + 5)*x^4)*D[P[n - 1, x], x])/(-n - (3*n^2 - 2*n)*x + (3*n^2 + 5*n)*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] = (-(n + (3*n^2 + 10*n - 10)*x + (15*n^2 + 7*n - 28)*x^2 - (6*n + 10)*x^3)*v[i - 1] + (x + (3*n + 15)*x^2 + (30*n + 51)*x^3 + (3*n + 5)*x^4)*deriv(v[i - 1], x))/(-n - (3*n^2 - 2*n)*x + (3*n^2 + 5*n)*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-f25cd955a876cde3","steps":["\u03b8+1"],"edges":["e-f58bd164c0caee20"],"text":"A081577 \u2192 \u03b8+1"},"headline":"A081577 \u2192 \u03b8+1","chain":null}