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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":"d6be1eefe7a7a6cc51bb6552e3a86ab17a7926941ad069fa4c49b55d344ee1d3","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-930be05ae3b321d9","role":"sequence","name":"Triangle read by rows: T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(n) = 4*n + 3.","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:     3,         6;\n  2:     9,        84,          27;\n  3:    27,       786,        1179,          108;\n  4:    81,      6312,       25938,        12624,          405;\n  5:   243,     47262,      427338,       569784,       118155,         1458;\n  6:   729,    342012,     6047685,     17093520,     10079475,      1026036,         5103;\n  7:  2187,   2433450,    78323949,    409610820,    512013525,    156647898,      8517075,       17496;\n  8:  6561,  17169744,   960118164,   8546096736,  19456513950,  12819145104,   2240275716,    68678976,     59049;\n  9: 19683, 120647478, 11359692900, 162755704944, 615374046210, 738448855452, 284822483652, 30292514400, 542913651, 196830;\n","recurrence_sentence":"The row polynomials satisfy (4*n-1)*P(n,x) = (12*n+9+(32*n^2-16*n-30)*x-(64*n^3-48*n^2-52*n+15)*x^2)*P(n-2,x) + ((16*n+12)*x-(64*n^2+48*n)*x^2+(64*n^2+32*n-12)*x^3)*d/dx P(n-2,x) + (8*n-6+(32*n^2-16*n+2)*x)*P(n-1,x) + ((16*n-4)*x-(16*n-4)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 6*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + 6*x;\nP[n_, x_] := P[n, x] = Expand[Together[((12*n + 9 + (32*n^2 - 16*n - 30)*x - (64*n^3 - 48*n^2 - 52*n + 15)*x^2)*P[n - 2, x] + ((16*n + 12)*x - (64*n^2 + 48*n)*x^2 + (64*n^2 + 32*n - 12)*x^3)*D[P[n - 2, x], x] + (8*n - 6 + (32*n^2 - 16*n + 2)*x)*P[n - 1, x] + ((16*n - 4)*x - (16*n - 4)*x^2)*D[P[n - 1, x], x])/(4*n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = ((12*n + 9 + (32*n^2 - 16*n - 30)*x - (64*n^3 - 48*n^2 - 52*n + 15)*x^2)*v[i - 2] + ((16*n + 12)*x - (64*n^2 + 48*n)*x^2 + (64*n^2 + 32*n - 12)*x^3)*deriv(v[i - 2], x) + (8*n - 6 + (32*n^2 - 16*n + 2)*x)*v[i - 1] + ((16*n - 4)*x - (16*n - 4)*x^2)*deriv(v[i - 1], x))/(4*n - 1)); 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-930be05ae3b321d9","steps":["\u03b8+1"],"edges":["e-536607f87ca8d101"],"text":"A257621 \u2192 \u03b8+1"},"headline":"A257621 \u2192 \u03b8+1","chain":null}