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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":"086f853160ed1c54a998620a35d88699740ea939f9f9c3993b6fd97cc0ea31ee","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-f5286435ed391b4f","role":"sequence","name":"basis_map_A013610 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,       84,       108;\n  1:      41,      456,      1188,        864;\n  2:     153,     2232,      8748,      12960,        6480;\n  3:     571,    10308,     53892,     120960,      123120,       46656;\n  4:    2131,    45852,    299592,     902016,     1393200,     1073088,       326592;\n  5:    7953,   198672,   1557144,    5883840,    12214800,    14276736,      8817984,      2239488;\n  6:   29681,   844272,   7717032,   35090496,    91594800,   143980416,    134882496,     69424128,     15116544;\n  7:  110771,  3534588,  36909432,  196214400,   617343120,  1221127488,   1538574912,   1200365568,    529079040,   100776960;\n  8:  413403, 14623332, 171731988, 1045042560,  3849385680,  9185400000,  14567309568,  15282266112,  10203667200,  3930301440,   665127936;\n  9: 1542841, 59919576, 781628508, 5357811744, 22618018800, 63236049408, 121028463360, 159818821632, 143440886016, 83644876800, 28600501248, 4353564672;\n","recurrence_sentence":"The row polynomials satisfy -(4*n+6+(10*n+15)*x-(6*n^3+27*n^2+39*n+18)*x^2)*P(n,x) = -(8*n+14+(6*n^2+62*n+95)*x-(12*n^3+57*n^2+27*n-78)*x^2-(18*n^3+108*n^2+198*n+108)*x^3)*P(n-1,x) - (2+18*x-(6*n^2+24*n-28)*x^2-(24*n^2+96*n+42)*x^3-(18*n^2+72*n+54)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 11 + 84*x + 108*x^2.","mathematica":"P[0, x_] := 11 + 84*x + 108*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(8*n + 14 + (6*n^2 + 62*n + 95)*x - (12*n^3 + 57*n^2 + 27*n - 78)*x^2 - (18*n^3 + 108*n^2 + 198*n + 108)*x^3)*P[n - 1, x] - (2 + 18*x - (6*n^2 + 24*n - 28)*x^2 - (24*n^2 + 96*n + 42)*x^3 - (18*n^2 + 72*n + 54)*x^4)*D[P[n - 1, x], x])/(-4*n - 6 - (10*n + 15)*x + (6*n^3 + 27*n^2 + 39*n + 18)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 11 + 84*x + 108*x^2; for(n = 1, N, my(i = n + 1); v[i] = (-(8*n + 14 + (6*n^2 + 62*n + 95)*x - (12*n^3 + 57*n^2 + 27*n - 78)*x^2 - (18*n^3 + 108*n^2 + 198*n + 108)*x^3)*v[i - 1] - (2 + 18*x - (6*n^2 + 24*n - 28)*x^2 - (24*n^2 + 96*n + 42)*x^3 - (18*n^2 + 72*n + 54)*x^4)*deriv(v[i - 1], x))/(-4*n - 6 - (10*n + 15)*x + (6*n^3 + 27*n^2 + 39*n + 18)*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 A013610","\u03b8+1"],"edges":["e-8cd26e1d1ff01e6d","e-7be85cc8f1a31a93"],"text":"A123519 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"A123519 \u2192 basis A013610 \u2192 \u03b8+1","chain":null}