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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":"bebe1e52032bdbb3e17eb6b72642e3d8b2545c5fe3f82654bea47dd3f7f5116f","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-f3a4ae052af3da90","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) = 3*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,       160,          75;\n  3:     125,      1830,        2745,          500;\n  4:     625,     18140,       60390,        36280,          3125;\n  5:    3125,    166370,     1045230,      1393640,        415925,         18750;\n  6:   15625,   1455960,    15739965,     39021920,      26233275,       4367880,       109375;\n  7:   78125,  12366430,   216818415,    903079420,    1128849275,     433636830,     43282505,      625000;\n  8:  390625, 102993940,  2811644400,  18424936280,   38380875350,   27637404420,   6560503600,   411975760,    3515625;\n  9: 1953125, 846607770, 34944852060, 344172869520, 1113098287950, 1335717945540, 602302521660, 93186272160, 3809734965, 19531250;\n","recurrence_sentence":"The row polynomials satisfy (3*n+2)*P(n,x) = -(30*n+50+(36*n^2+48*n-20)*x+(27*n^3+54*n^2+9*n-10)*x^2)*P(n-2,x) - ((18*n+30)*x+(27*n^2+45*n)*x^2-(27*n^2+63*n+30)*x^3)*d/dx P(n-2,x) + (21*n+20+(18*n^2+24*n+8)*x)*P(n-1,x) + ((9*n+6)*x-(9*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 + 50 + (36*n^2 + 48*n - 20)*x + (27*n^3 + 54*n^2 + 9*n - 10)*x^2)*P[n - 2, x] - ((18*n + 30)*x + (27*n^2 + 45*n)*x^2 - (27*n^2 + 63*n + 30)*x^3)*D[P[n - 2, x], x] + (21*n + 20 + (18*n^2 + 24*n + 8)*x)*P[n - 1, x] + ((9*n + 6)*x - (9*n + 6)*x^2)*D[P[n - 1, x], x])/(3*n + 2)]];\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 + 50 + (36*n^2 + 48*n - 20)*x + (27*n^3 + 54*n^2 + 9*n - 10)*x^2)*v[i - 2] - ((18*n + 30)*x + (27*n^2 + 45*n)*x^2 - (27*n^2 + 63*n + 30)*x^3)*deriv(v[i - 2], x) + (21*n + 20 + (18*n^2 + 24*n + 8)*x)*v[i - 1] + ((9*n + 6)*x - (9*n + 6)*x^2)*deriv(v[i - 1], x))/(3*n + 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-f3a4ae052af3da90","steps":["\u03b8+1"],"edges":["e-0243f812b4db69f4"],"text":"A257624 \u2192 \u03b8+1"},"headline":"A257624 \u2192 \u03b8+1","chain":null}