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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":"532158d17799b1263662ab98cb53f01c0e41f4001d3ff7d790d30592026f081b","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-305352172c67c5f0","role":"sequence","name":"reversal of Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 1;\n   2: 1,  2;\n   3: 1,  6,   6;\n   4: 1, 10,  21,   12;\n   5: 1, 14,  48,   60,   25;\n   6: 1, 18,  87,  172,  150,    48;\n   7: 1, 22, 138,  380,  520,   348,    91;\n   8: 1, 26, 201,  716, 1355,  1410,   763,  168;\n   9: 1, 30, 276, 1212, 2955,  4230,  3542, 1608,  306;\n  10: 1, 34, 363, 1900, 5700, 10476, 12019, 8408, 3285, 550;\n","recurrence_sentence":"The row polynomials satisfy (n^2-n+(2*n^3-3*n^2+n)*x+(6*n^3-20*n^2+22*n-8)*x^2+(5*n^3-20*n^2+25*n-10)*x^3)*P(n,x) = (n^2-n+(2*n^3+5*n^2-13*n)*x+(14*n^3-3*n^2-43*n+2)*x^2+(39*n^3-51*n^2-58*n+16)*x^3+(50*n^3-105*n^2-5*n+30)*x^4+(25*n^3-75*n^2+50*n)*x^5)*P(n-1,x) - (n*x+(2*n^2+9*n-1)*x^2+(18*n^2+23*n-9)*x^3+(51*n^2+10*n-23)*x^4+(60*n^2-30*n-15)*x^5+(25*n^2-25*n)*x^6)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((n^2 - n + (2*n^3 + 5*n^2 - 13*n)*x + (14*n^3 - 3*n^2 - 43*n + 2)*x^2 + (39*n^3 - 51*n^2 - 58*n + 16)*x^3 + (50*n^3 - 105*n^2 - 5*n + 30)*x^4 + (25*n^3 - 75*n^2 + 50*n)*x^5)*P[n - 1, x] - (n*x + (2*n^2 + 9*n - 1)*x^2 + (18*n^2 + 23*n - 9)*x^3 + (51*n^2 + 10*n - 23)*x^4 + (60*n^2 - 30*n - 15)*x^5 + (25*n^2 - 25*n)*x^6)*D[P[n - 1, x], x])/(n^2 - n + (2*n^3 - 3*n^2 + n)*x + (6*n^3 - 20*n^2 + 22*n - 8)*x^2 + (5*n^3 - 20*n^2 + 25*n - 10)*x^3)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((n^2 - n + (2*n^3 + 5*n^2 - 13*n)*x + (14*n^3 - 3*n^2 - 43*n + 2)*x^2 + (39*n^3 - 51*n^2 - 58*n + 16)*x^3 + (50*n^3 - 105*n^2 - 5*n + 30)*x^4 + (25*n^3 - 75*n^2 + 50*n)*x^5)*v[i - 1] - (n*x + (2*n^2 + 9*n - 1)*x^2 + (18*n^2 + 23*n - 9)*x^3 + (51*n^2 + 10*n - 23)*x^4 + (60*n^2 - 30*n - 15)*x^5 + (25*n^2 - 25*n)*x^6)*deriv(v[i - 1], x))/(n^2 - n + (2*n^3 - 3*n^2 + n)*x + (6*n^3 - 20*n^2 + 22*n - 8)*x^2 + (5*n^3 - 20*n^2 + 25*n - 10)*x^3)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-c72653d9ab0ee8be","steps":["reverse","\u03b8+1"],"edges":["e-2c112d9ec33e1fb9","e-6e8437c77bd63bc2"],"text":"A063967 \u2192 reverse \u2192 \u03b8+1"},"headline":"A063967 \u2192 reverse \u2192 \u03b8+1","chain":null}