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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":"48fb81d9efe5519590f01c02112ae1ca9fe140b8060167e3c97e6d90356ef36c","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-313c8ccf28f71545","role":"sequence","name":"Triangle read by rows, Lah numbers with level 2.","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: 0,            4;\n   3: 0,           16,             9;\n   4: 0,          144,           180,            16;\n   5: 0,         2304,          4824,           896,           25;\n   6: 0,        57600,        178848,         52480,         3000,          36;\n   7: 0,      2073600,       8893152,       3676672,       325000,        7920,         49;\n   8: 0,    101606400,     573827328,     313620480,     38244800,     1426320,      17836,       64;\n   9: 0,   6502809600,   46708627968,   32382185472,   5117652800,   260462592,    4955664,    35840,    81;\n  10: 0, 526727577600, 4685494118400, 4001282002944, 787539168000, 51387598080, 1325828672, 14536704, 66096, 100;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^4-14*n^3+73*n^2-168*n+144+(4*n^2-24*n+32)*x-9*x^2)*P(n-2,x) - ((4*n^3-37*n^2+112*n-112)*x-(4*n^2-24*n+41)*x^2-4*x^3)*d/dx P(n-2,x) - ((5*n^2-30*n+40)*x^2-(8*n-15)*x^3)*d^2/dx^2 P(n-2,x) - ((2*n-8)*x^3-4*x^4)*d^3/dx^3 P(n-2,x) + (2*n^2-12*n+20+4*x)*P(n-1,x) + ((4*n-11)*x-4*x^2)*d/dx P(n-1,x) + x^2*d^2/dx^2 P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 4*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 4*x;\nP[n_, x_] := P[n, x] = Expand[-(n^4 - 14*n^3 + 73*n^2 - 168*n + 144 + (4*n^2 - 24*n + 32)*x - 9*x^2)*P[n - 2, x] - ((4*n^3 - 37*n^2 + 112*n - 112)*x - (4*n^2 - 24*n + 41)*x^2 - 4*x^3)*D[P[n - 2, x], x] - ((5*n^2 - 30*n + 40)*x^2 - (8*n - 15)*x^3)*D[P[n - 2, x], {x, 2}] - ((2*n - 8)*x^3 - 4*x^4)*D[P[n - 2, x], {x, 3}] + (2*n^2 - 12*n + 20 + 4*x)*P[n - 1, x] + ((4*n - 11)*x - 4*x^2)*D[P[n - 1, x], x] + x^2*D[P[n - 1, x], {x, 2}]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 4*x; for(n = 3, N, my(i = n); v[i] = -(n^4 - 14*n^3 + 73*n^2 - 168*n + 144 + (4*n^2 - 24*n + 32)*x - 9*x^2)*v[i - 2] - ((4*n^3 - 37*n^2 + 112*n - 112)*x - (4*n^2 - 24*n + 41)*x^2 - 4*x^3)*deriv(v[i - 2], x) - ((5*n^2 - 30*n + 40)*x^2 - (8*n - 15)*x^3)*deriv(deriv(v[i - 2], x), x) - ((2*n - 8)*x^3 - 4*x^4)*deriv(deriv(deriv(v[i - 2], x), x), x) + (2*n^2 - 12*n + 20 + 4*x)*v[i - 1] + ((4*n - 11)*x - 4*x^2)*deriv(v[i - 1], x) + x^2*deriv(deriv(v[i - 1], x), x)); 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-313c8ccf28f71545","steps":["(\u03b8+1)\u00b2"],"edges":["e-bec6d3eaef62b80a"],"text":"A390433 \u2192 (\u03b8+1)\u00b2"},"headline":"A390433 \u2192 (\u03b8+1)\u00b2","chain":null}