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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":"3a7b651b0ec32fe2803866f6299959f0511f7abc88382d48b020eb2bc31e34cd","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-c02d1829f7ed6fe0","role":"sequence","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*11^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:             400,             1760,              1089;\n  1:            8000,            52800,             65340,             21296;\n  2:          160000,          1408000,           2613600,           1703680,             366025;\n  3:         3200000,         35200000,          87120000,          85184000,           36602500,            5797836;\n  4:        64000000,        844800000,        2613600000,        3407360000,         2196150000,          695740320,           86806489;\n  5:      1280000000,      19712000000,       73180800000,      119257600000,       102487000000,        48701822400,        12152908460,        1247178944;\n  6:     25600000000,     450560000000,     1951488000000,     3816243200000,      4099480000000,      2597430528000,       972232676800,      199548631040,       17363069361;\n  7:    512000000000,   10137600000000,    50181120000000,   114487296000000,    147581280000000,    116884373760000,     58333960608000,    17959376793600,     3125352484980,     235794769100;\n  8:  10240000000000,  225280000000000,  1254528000000000,  3271065600000000,   4919376000000000,   4675374950400000,   2916698030400000,  1197291786240000,   312535248498000,   47158953820000,   3138428376721;\n  9: 204800000000000, 4956160000000000, 30666240000000000, 89954304000000000, 154608960000000000, 171430414848000000, 128334713337600000, 65851048243200000, 22919251556520000, 5187484920200000, 690454242878620, 41084880567984;\n","recurrence_sentence":"The row polynomials satisfy (400+(660*n+1100)*x+(121*n^2+484*n+484)*x^2)*P(n,x) = (8000+(13200*n+39600)*x+(2420*n^2+21780*n+41140)*x^2+(1331*n^2+7986*n+11979)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 400 + 1760*x + 1089*x^2.","mathematica":"P[0, x_] := 400 + 1760*x + 1089*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((8000 + (13200*n + 39600)*x + (2420*n^2 + 21780*n + 41140)*x^2 + (1331*n^2 + 7986*n + 11979)*x^3)*P[n - 1, x])/(400 + (660*n + 1100)*x + (121*n^2 + 484*n + 484)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 400 + 1760*x + 1089*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((8000 + (13200*n + 39600)*x + (2420*n^2 + 21780*n + 41140)*x^2 + (1331*n^2 + 7986*n + 11979)*x^3)*v[i - 1])/(400 + (660*n + 1100)*x + (121*n^2 + 484*n + 484)*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-cfc5b48d32898a3e","steps":["x \u21a6 x+1","(\u03b8+1)\u00b2"],"edges":["e-1af17fa5254d2e99","e-efa20488259f353a"],"text":"A038301 \u2192 x \u21a6 x+1 \u2192 (\u03b8+1)\u00b2"},"headline":"A038301 \u2192 x \u21a6 x+1 \u2192 (\u03b8+1)\u00b2","chain":null}