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f(k) x^k maps every real-rooted polynomial to a real-rooted polynomial or zero (Laguerre; see T. 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":"f52c8ef2cf2b3fe0a836b333e881adbdf5878ae8383781aafb544ec26b0d3486","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-3f0a1785078e99be","role":"sequence","name":"Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 2*x + 5.","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:       5,        10;\n  2:      25,       140,         75;\n  3:     125,      1430,       2145,         500;\n  4:     625,     12760,      38610,       25520,         3125;\n  5:    3125,    105570,     558030,      744040,       263925,        18750;\n  6:   15625,    832740,    7080885,    16368880,     11801475,      2498220,       109375;\n  7:   78125,   6360430,   82464615,   302793020,    378491275,    164929230,     22261505,      625000;\n  8:  390625,  47491760,  904372500,  4980015520,   9840773150,   7470023280,   2110202500,   189967040,    3515625;\n  9: 1953125, 348848570, 9492867660, 75279280720, 221305341950, 265566410340, 131738741260, 25314313760, 1569818565, 19531250;\n","recurrence_sentence":"The row polynomials satisfy (2*n+3)*P(n,x) = -(30*n+75+(24*n^2+72*n+30)*x+(8*n^3+36*n^2+46*n+15)*x^2)*P(n-2,x) - ((12*n+30)*x+(8*n^2+20*n)*x^2-(8*n^2+32*n+30)*x^3)*d/dx P(n-2,x) + (16*n+30+(8*n^2+24*n+18)*x)*P(n-1,x) + ((4*n+6)*x-(4*n+6)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 5 + 10*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 5 + 10*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(30*n + 75 + (24*n^2 + 72*n + 30)*x + (8*n^3 + 36*n^2 + 46*n + 15)*x^2)*P[n - 2, x] - ((12*n + 30)*x + (8*n^2 + 20*n)*x^2 - (8*n^2 + 32*n + 30)*x^3)*D[P[n - 2, x], x] + (16*n + 30 + (8*n^2 + 24*n + 18)*x)*P[n - 1, x] + ((4*n + 6)*x - (4*n + 6)*x^2)*D[P[n - 1, x], x])/(2*n + 3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 5 + 10*x; for(n = 2, N, my(i = n + 1); v[i] = (-(30*n + 75 + (24*n^2 + 72*n + 30)*x + (8*n^3 + 36*n^2 + 46*n + 15)*x^2)*v[i - 2] - ((12*n + 30)*x + (8*n^2 + 20*n)*x^2 - (8*n^2 + 32*n + 30)*x^3)*deriv(v[i - 2], x) + (16*n + 30 + (8*n^2 + 24*n + 18)*x)*v[i - 1] + ((4*n + 6)*x - (4*n + 6)*x^2)*deriv(v[i - 1], x))/(2*n + 3)); 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-3f0a1785078e99be","steps":["\u03b8+1"],"edges":["e-c6f38f79c7d5b436"],"text":"A257615 \u2192 \u03b8+1"},"headline":"A257615 \u2192 \u03b8+1","chain":null}