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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":"ba7fd746ff651ed791f46fff1ae58c8afb21a3005dbcadd17352068eed59b775","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-7a809bc95dc4396c","role":"sequence","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*1^j.","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:          13,           4;\n  2:         169,         104,           9;\n  3:        2197,        2028,         351,         16;\n  4:       28561,       35152,        9126,        832,         25;\n  5:      371293,      571220,      197730,      27040,       1625,        36;\n  6:     4826809,     8911032,     3855735,     703040,      63375,      2808,      49;\n  7:    62748517,   135150652,    70174377,   15994160,    1922375,    127764,    4459,     64;\n  8:   815730721,  2007952544,  1216355868,  332678528,   49981750,   4429152,  231868,   6656,   81;\n  9: 10604499373, 29366305956, 20330519508, 6487231296, 1169572950, 129552696, 9042852, 389376, 9477, 100;\n","recurrence_sentence":"The row polynomials satisfy (169+(39*n-13)*x+n^2*x^2)*P(n,x) = (2197+(507*n+507)*x+(13*n^2+65*n+39)*x^2+(n^2+2*n+1)*x^3)*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[((2197 + (507*n + 507)*x + (13*n^2 + 65*n + 39)*x^2 + (n^2 + 2*n + 1)*x^3)*P[n - 1, x])/(169 + (39*n - 13)*x + n^2*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] = ((2197 + (507*n + 507)*x + (13*n^2 + 65*n + 39)*x^2 + (n^2 + 2*n + 1)*x^3)*v[i - 1])/(169 + (39*n - 13)*x + n^2*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-81e6fb4f9f226d77","steps":["x \u21a6 x+1","(\u03b8+1)\u00b2"],"edges":["e-4bd827707ee8c2a0","e-403fc37367432712"],"text":"A038327 \u2192 x \u21a6 x+1 \u2192 (\u03b8+1)\u00b2"},"headline":"A038327 \u2192 x \u21a6 x+1 \u2192 (\u03b8+1)\u00b2","chain":null}