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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":"900d917837f2c8ac1a5d0eef5c360246fd6ac1b5a3950f4d122a53ec088715eb","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-47b361d1e904fbeb","role":"sequence","name":"basis_map_A007318 of Triangle read by rows: T(n,k) number of tilings of a 2n X 3 grid by dominoes, 2k of which are in a vertical position (0<=k<=n).","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:      11,       28,       12;\n  1:      41,      152,      132,        32;\n  2:     153,      744,      972,       480,        80;\n  3:     571,     3436,     5988,      4480,      1520,       192;\n  4:    2131,    15284,    33288,     33408,     17200,      4416,       448;\n  5:    7953,    66224,   173016,    217920,    150800,     58752,     12096,     1024;\n  6:   29681,   281424,   857448,   1299648,   1130800,    592512,    185024,    31744,     2304;\n  7:  110771,  1178196,  4101048,   7267200,   7621520,   5025216,   2110528,   548864,    80640,    5120;\n  8:  413403,  4874444, 19081332,  38705280,  47523280,  37800000,  19982592,  6987776,  1555200,  199680,  11264;\n  9: 1542841, 19973192, 86847612, 198437472, 279234800, 260230656, 166019840, 73076736, 21862656, 4249600, 484352, 24576;\n","recurrence_sentence":"The row polynomials satisfy -(12*n+18+(10*n+15)*x-(2*n^3+9*n^2+13*n+6)*x^2)*P(n,x) = -(24*n+42+(6*n^2+62*n+95)*x-(4*n^3+19*n^2+9*n-26)*x^2-(2*n^3+12*n^2+22*n+12)*x^3)*P(n-1,x) - (18+54*x-(6*n^2+24*n-28)*x^2-(8*n^2+32*n+14)*x^3-(2*n^2+8*n+6)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 11 + 28*x + 12*x^2.","mathematica":"P[0, x_] := 11 + 28*x + 12*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(24*n + 42 + (6*n^2 + 62*n + 95)*x - (4*n^3 + 19*n^2 + 9*n - 26)*x^2 - (2*n^3 + 12*n^2 + 22*n + 12)*x^3)*P[n - 1, x] - (18 + 54*x - (6*n^2 + 24*n - 28)*x^2 - (8*n^2 + 32*n + 14)*x^3 - (2*n^2 + 8*n + 6)*x^4)*D[P[n - 1, x], x])/(-12*n - 18 - (10*n + 15)*x + (2*n^3 + 9*n^2 + 13*n + 6)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 11 + 28*x + 12*x^2; for(n = 1, N, my(i = n + 1); v[i] = (-(24*n + 42 + (6*n^2 + 62*n + 95)*x - (4*n^3 + 19*n^2 + 9*n - 26)*x^2 - (2*n^3 + 12*n^2 + 22*n + 12)*x^3)*v[i - 1] - (18 + 54*x - (6*n^2 + 24*n - 28)*x^2 - (8*n^2 + 32*n + 14)*x^3 - (2*n^2 + 8*n + 6)*x^4)*deriv(v[i - 1], x))/(-12*n - 18 - (10*n + 15)*x + (2*n^3 + 9*n^2 + 13*n + 6)*x^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-8eb9bb3a09f0b7fd","steps":["basis A007318","\u03b8+1"],"edges":["e-1acefd6886502727","e-daf5edc3a384fecc"],"text":"A123519 \u2192 basis A007318 \u2192 \u03b8+1"},"headline":"A123519 \u2192 basis A007318 \u2192 \u03b8+1","chain":null}