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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":"60b1483f6922f9c66de1e5eb41399870b0a949ebc125fc342efc53f30565a4fe","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-b69b5201d4a2fd8f","role":"sequence","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*6^j.","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:            289,             816,              432;\n  1:           4913,           20808,            22032,             6912;\n  2:          83521,          471648,           749088,           470016,            103680;\n  3:        1419857,        10022520,         21224160,         19975680,           8812800,           1492992;\n  4:       24137569,       204459408,        541216080,        679173120,         449452800,         152285184,          20901888;\n  5:      410338673,      4055111592,      12880942704,      20205400320,       17828294400,        9060968448,        2487324672,        286654464;\n  6:     6975757441,     78785025216,     291968034624,     549586888704,      606162009600,      410763902976,      169138077696,      38985007104,       3869835264;\n  7:   118587876497,   1506763607256,    6381587042496,   14014465661952,    18548557493760,    15711719288832,     8626041962496,    2982353043456,     592084795392,     51597803520;\n  8:  2015993900449,  28461090359280,  135608724653040,  340351308933120,   525542462323200,   534198455820288,   366606783406080,  169000005795840,   50327207608320,   8771626598400,    681091006464;\n  9: 34271896307633, 532222389718536, 2817647945568720, 7955711846311680, 14039491493491200, 16649185206398976, 13711093699387392, 7900750270955520, 3137062607585280, 820147086950400, 127364018208768, 8916100448256;\n","recurrence_sentence":"The row polynomials satisfy (17+(12*n+24)*x)*P(n,x) = (289+(204*n+816)*x+(144*n+432)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 289 + 816*x + 432*x^2.","mathematica":"P[0, x_] := 289 + 816*x + 432*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((289 + (204*n + 816)*x + (144*n + 432)*x^2)*P[n - 1, x])/(17 + (12*n + 24)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 289 + 816*x + 432*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((289 + (204*n + 816)*x + (144*n + 432)*x^2)*v[i - 1])/(17 + (12*n + 24)*x)); 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-1f2dec04776afb2b","steps":["basis A013609","\u03b8+1"],"edges":["e-7304be3f66cc3caf","e-3ce913900d8faaf9"],"text":"A038320 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A038320 \u2192 basis A013609 \u2192 \u03b8+1","chain":null}