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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":"831c0b8a8f397d1d33d14d3ef9c62e0a265afab5c04f80bdee0bd0b8260efdf8","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-ee73bfb592710e97","role":"sequence","name":"Triangle read by rows, (2, 2)-Lah numbers.","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:               16,                 4;\n  2:              400,               208,                9;\n  3:            14400,             11792,             1044,               16;\n  4:           705600,            812288,           111096,             3456,             25;\n  5:         45158400,          68617728,         12937248,           615680,           9000,            36;\n  6:       3657830400,        7042406400,       1719796896,        111657472,        2483000,         20016,          49;\n  7:     365783040000,      866762496000,     263496167424,      21927529472,      661132800,       8079120,       39788,       64;\n  8:   44259747840000,   126276931584000,   46481067910656,    4766233407488,   183016644800,    3020285952,    22495312,    72704,     81;\n  9: 6373403688960000, 21517840429056000, 9394412406552576, 1155035526303744, 54299500768000, 1136406608640, 11399425856, 55627776, 124416, 100;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^4+6*n^3+13*n^2+12*n+4+(4*n^2+16*n+12)*x-9*x^2)*P(n-2,x) - ((4*n^3+23*n^2+42*n+23)*x-(4*n^2+16*n+21)*x^2-4*x^3)*d/dx P(n-2,x) - ((5*n^2+20*n+15)*x^2-(8*n+25)*x^3)*d^2/dx^2 P(n-2,x) - ((2*n+2)*x^3-4*x^4)*d^3/dx^3 P(n-2,x) + (2*n^2+8*n+10+4*x)*P(n-1,x) + ((4*n+9)*x-4*x^2)*d/dx P(n-1,x) + x^2*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 16 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 16 + 4*x;\nP[n_, x_] := P[n, x] = Expand[-(n^4 + 6*n^3 + 13*n^2 + 12*n + 4 + (4*n^2 + 16*n + 12)*x - 9*x^2)*P[n - 2, x] - ((4*n^3 + 23*n^2 + 42*n + 23)*x - (4*n^2 + 16*n + 21)*x^2 - 4*x^3)*D[P[n - 2, x], x] - ((5*n^2 + 20*n + 15)*x^2 - (8*n + 25)*x^3)*D[P[n - 2, x], {x, 2}] - ((2*n + 2)*x^3 - 4*x^4)*D[P[n - 2, x], {x, 3}] + (2*n^2 + 8*n + 10 + 4*x)*P[n - 1, x] + ((4*n + 9)*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, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 16 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^4 + 6*n^3 + 13*n^2 + 12*n + 4 + (4*n^2 + 16*n + 12)*x - 9*x^2)*v[i - 2] - ((4*n^3 + 23*n^2 + 42*n + 23)*x - (4*n^2 + 16*n + 21)*x^2 - 4*x^3)*deriv(v[i - 2], x) - ((5*n^2 + 20*n + 15)*x^2 - (8*n + 25)*x^3)*deriv(deriv(v[i - 2], x), x) - ((2*n + 2)*x^3 - 4*x^4)*deriv(deriv(deriv(v[i - 2], x), x), x) + (2*n^2 + 8*n + 10 + 4*x)*v[i - 1] + ((4*n + 9)*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 - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-ee73bfb592710e97","steps":["(\u03b8+1)\u00b2"],"edges":["e-dec0bfc95bd131b2"],"text":"A371081 \u2192 (\u03b8+1)\u00b2"},"headline":"A371081 \u2192 (\u03b8+1)\u00b2","chain":null}