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(f(k))_(k >= 0) is a multiplier sequence, so x^k -> 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":"10b41d3249d7ebbf59ab1c4aaf63afa452ecc3af0d7ca89458badf07c2f80959","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-b362ab32ef269db1","role":"sequence","name":"basis_map_A013610 of Triangle read by rows: T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:    1;\n   2:    4,     12;\n   3:   11,     84,      81;\n   4:   29,    372,     810,      432;\n   5:   76,   1380,    4860,     5616,     2025;\n   6:  199,   4680,   23085,    42336,    32400,     8748;\n   7:  521,  15048,   95985,   244512,   293625,   166212,    35721;\n   8: 1364,  46716,  366444,  1203120,  1994625,  1758348,   785862,   139968;\n   9: 3571, 141468, 1318680,  5319216, 11329875, 13725612,  9501786,  3499200,   531441;\n  10: 9349, 420540, 4544505, 21787488, 56928825, 88363548, 83765745, 47589120, 14880348, 1968300;\n","recurrence_sentence":"The row polynomials satisfy (600*n-900+(720*n-1080)*x-(2700*n^3-12150*n^2+17766*n-8424)*x^2-(2916*n^3-13122*n^2+19602*n-9720)*x^3+(486*n^5-3645*n^4+10692*n^3-15309*n^2+10692*n-2916)*x^4)*P(n,x) = (900*n-600+(1575*n^2-45*n-1170)*x-(4050*n^3-14850*n^2+12204*n-1296)*x^2-(810*n^4+6399*n^3-25758*n^2+22599*n-3726)*x^3+(729*n^5-4860*n^4+4617*n^3+8262*n^2-10692*n+1944)*x^4+(729*n^5-3645*n^4+5832*n^3-2916*n^2)*x^5)*P(n-1,x) + (125+900*x-(1575*n^2-3150*n-1440)*x^2-(8100*n^2-16200*n+1728)*x^3+(405*n^4-1620*n^3-9963*n^2+23166*n-4617)*x^4+(1458*n^4-5832*n^3+1458*n^2+8748*n-1944)*x^5+(729*n^4-2916*n^3+2916*n^2)*x^6)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((900*n - 600 + (1575*n^2 - 45*n - 1170)*x - (4050*n^3 - 14850*n^2 + 12204*n - 1296)*x^2 - (810*n^4 + 6399*n^3 - 25758*n^2 + 22599*n - 3726)*x^3 + (729*n^5 - 4860*n^4 + 4617*n^3 + 8262*n^2 - 10692*n + 1944)*x^4 + (729*n^5 - 3645*n^4 + 5832*n^3 - 2916*n^2)*x^5)*P[n - 1, x] + (125 + 900*x - (1575*n^2 - 3150*n - 1440)*x^2 - (8100*n^2 - 16200*n + 1728)*x^3 + (405*n^4 - 1620*n^3 - 9963*n^2 + 23166*n - 4617)*x^4 + (1458*n^4 - 5832*n^3 + 1458*n^2 + 8748*n - 1944)*x^5 + (729*n^4 - 2916*n^3 + 2916*n^2)*x^6)*D[P[n - 1, x], x])/(600*n - 900 + (720*n - 1080)*x - (2700*n^3 - 12150*n^2 + 17766*n - 8424)*x^2 - (2916*n^3 - 13122*n^2 + 19602*n - 9720)*x^3 + (486*n^5 - 3645*n^4 + 10692*n^3 - 15309*n^2 + 10692*n - 2916)*x^4)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((900*n - 600 + (1575*n^2 - 45*n - 1170)*x - (4050*n^3 - 14850*n^2 + 12204*n - 1296)*x^2 - (810*n^4 + 6399*n^3 - 25758*n^2 + 22599*n - 3726)*x^3 + (729*n^5 - 4860*n^4 + 4617*n^3 + 8262*n^2 - 10692*n + 1944)*x^4 + (729*n^5 - 3645*n^4 + 5832*n^3 - 2916*n^2)*x^5)*v[i - 1] + (125 + 900*x - (1575*n^2 - 3150*n - 1440)*x^2 - (8100*n^2 - 16200*n + 1728)*x^3 + (405*n^4 - 1620*n^3 - 9963*n^2 + 23166*n - 4617)*x^4 + (1458*n^4 - 5832*n^3 + 1458*n^2 + 8748*n - 1944)*x^5 + (729*n^4 - 2916*n^3 + 2916*n^2)*x^6)*deriv(v[i - 1], x))/(600*n - 900 + (720*n - 1080)*x - (2700*n^3 - 12150*n^2 + 17766*n - 8424)*x^2 - (2916*n^3 - 13122*n^2 + 19602*n - 9720)*x^3 + (486*n^5 - 3645*n^4 + 10692*n^3 - 15309*n^2 + 10692*n - 2916)*x^4)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-318700a428d7b97b","steps":["basis A013610","(\u03b8+1)\u00b2"],"edges":["e-258aa4ea5bc7ae92","e-e2726d90af41930f"],"text":"A111125 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2"},"headline":"A111125 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2","chain":null}