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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":"cbe8ea7178e3197cb6ea28560d2ee9f54171ccbe472f5d6108272d4228ab56fb","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-aaf95b093ddbf844","role":"sequence","name":"gamma_polynomial of Pascal-(1,5,1) array.","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;\n  2: 1, 10;\n  3: 1, 20;\n  4: 1, 30,   75;\n  5: 1, 40,  225;\n  6: 1, 50,  450,   500;\n  7: 1, 60,  750,  2000;\n  8: 1, 70, 1125,  5000,  3125;\n  9: 1, 80, 1575, 10000, 15625;\n","recurrence_sentence":"The row polynomials satisfy (n+5*n^2*x)*P(n,x) = (n+(5*n^2+20*n-20)*x+(50*n^2+50*n-100)*x^2)*P(n-1,x) - (x+(5*n+30)*x^2+(100*n+200)*x^3)*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 + (5*n^2 + 20*n - 20)*x + (50*n^2 + 50*n - 100)*x^2)*P[n - 1, x] - (x + (5*n + 30)*x^2 + (100*n + 200)*x^3)*D[P[n - 1, x], x])/(n + 5*n^2*x)]];\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 + (5*n^2 + 20*n - 20)*x + (50*n^2 + 50*n - 100)*x^2)*v[i - 1] - (x + (5*n + 30)*x^2 + (100*n + 200)*x^3)*deriv(v[i - 1], x))/(n + 5*n^2*x)); 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-6ad01dc6e705cdb6","steps":["\u03b3","\u03b8+1"],"edges":["e-90e54edd477f164c","e-33f8c14416d3481a"],"text":"A081580 \u2192 \u03b3 \u2192 \u03b8+1"},"headline":"A081580 \u2192 \u03b3 \u2192 \u03b8+1","chain":null}