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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":"b0dc109062823586d57c4385e0391a3fbbb03797573540f0277cd300be1dd19a","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-65034f179833a7f6","role":"sequence","name":"Stirling2 triangle with scaled diagonals (powers of 10).","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:          100,            120,               9;\n  1:         1000,           2800,             540,              16;\n  2:        10000,          60000,           22500,            1600,              25;\n  3:       100000,        1240000,          810000,          104000,            3750,             36;\n  4:      1000000,       25200000,        27090000,         5600000,          350000,           7560,            49;\n  5:     10000000,      508000000,       869400000,       272160000,        26250000,         957600,         13720,           64;\n  6:    100000000,    10200000000,     27225000000,     12432000000,      1737750000,       95256000,       2263800,        23040,         81;\n  7:   1000000000,   204400000000,    839700000000,    545680000000,    106312500000,     8217720000,     288120000,      4800000,      36450,      100;\n  8:  10000000000,  4092000000000,  25650900000000,  23320000000000,   6168250000000,   646153200000,   31353630000,    760320000,    9355500,    55000,   121;\n  9: 100000000000, 81880000000000, 778734000000000, 978401600000000, 344850000000000, 47651472000000, 3074240400000, 101777280000, 1804275000, 17050000, 79860, 144;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(40*x-9*x^2)*P(n-2,x) - (200*x-40*x^2-4*x^3)*d/dx P(n-2,x) - (100*x^2-40*x^3)*d^2/dx^2 P(n-2,x) + (10+4*x)*P(n-1,x) + (20*x-4*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 100 + 120*x + 9*x^2, P(1,x) = 1000 + 2800*x + 540*x^2 + 16*x^3.","mathematica":"P[0, x_] := 100 + 120*x + 9*x^2;\nP[1, x_] := 1000 + 2800*x + 540*x^2 + 16*x^3;\nP[n_, x_] := P[n, x] = Expand[-(40*x - 9*x^2)*P[n - 2, x] - (200*x - 40*x^2 - 4*x^3)*D[P[n - 2, x], x] - (100*x^2 - 40*x^3)*D[P[n - 2, x], {x, 2}] + (10 + 4*x)*P[n - 1, x] + (20*x - 4*x^2)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 100 + 120*x + 9*x^2; v[2] = 1000 + 2800*x + 540*x^2 + 16*x^3; for(n = 2, N, my(i = n + 1); v[i] = -(40*x - 9*x^2)*v[i - 2] - (200*x - 40*x^2 - 4*x^3)*deriv(v[i - 2], x) - (100*x^2 - 40*x^3)*deriv(deriv(v[i - 2], x), x) + (10 + 4*x)*v[i - 1] + (20*x - 4*x^2)*deriv(v[i - 1], 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-65034f179833a7f6","steps":["(\u03b8+1)\u00b2"],"edges":["e-f3ea360b46818487"],"text":"A075505 \u2192 (\u03b8+1)\u00b2"},"headline":"A075505 \u2192 (\u03b8+1)\u00b2","chain":null}