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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":"79afc06fb716910118e833621cf612f400f3db3bf364c3a42ecf84a1029bfb26","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-3bc26d3108ee6bfd","role":"sequence","name":"Triangle read by rows: T(n,k) (1 <= k <= n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*k-m+1)*T(n-1,k), where m = 6.","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: 1,        2;\n  2: 1,       28,          3;\n  3: 1,      222,        333,            4;\n  4: 1,     1592,       8658,         3184,            5;\n  5: 1,    11194,     157926,       210568,        27985,            6;\n  6: 1,    78420,    2472813,      8001584,      4121355,       235260,            7;\n  7: 1,   549014,   35793099,    234457196,    293071495,     71586198,      1921549,           8;\n  8: 1,  3843184,  495780564,   5934134816,  14653574750,   8901202224,   1156821316,    15372736,         9;\n  9: 1, 26902386, 6693032700, 137207069328, 596287092870, 715544511444, 240112371324, 17848087200, 121060737, 10;\n","recurrence_sentence":"The row polynomials satisfy (6*n-5)*P(n,x) = (30*n+5+(360*n^2-600*n-110)*x-(216*n^3-540*n^2+234*n+55)*x^2)*P(n-2,x) + ((180*n+30)*x-(216*n^2+36*n)*x^2+(216*n^2-144*n-30)*x^3)*d/dx P(n-2,x) - (24*n+10-(72*n^2-120*n+50)*x)*P(n-1,x) + ((36*n-30)*x-(36*n-30)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[((30*n + 5 + (360*n^2 - 600*n - 110)*x - (216*n^3 - 540*n^2 + 234*n + 55)*x^2)*P[n - 2, x] + ((180*n + 30)*x - (216*n^2 + 36*n)*x^2 + (216*n^2 - 144*n - 30)*x^3)*D[P[n - 2, x], x] - (24*n + 10 - (72*n^2 - 120*n + 50)*x)*P[n - 1, x] + ((36*n - 30)*x - (36*n - 30)*x^2)*D[P[n - 1, x], x])/(6*n - 5)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((30*n + 5 + (360*n^2 - 600*n - 110)*x - (216*n^3 - 540*n^2 + 234*n + 55)*x^2)*v[i - 2] + ((180*n + 30)*x - (216*n^2 + 36*n)*x^2 + (216*n^2 - 144*n - 30)*x^3)*deriv(v[i - 2], x) - (24*n + 10 - (72*n^2 - 120*n + 50)*x)*v[i - 1] + ((36*n - 30)*x - (36*n - 30)*x^2)*deriv(v[i - 1], x))/(6*n - 5)); 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-3bc26d3108ee6bfd","steps":["\u03b8+1"],"edges":["e-a9644f36139719e7"],"text":"A142461 \u2192 \u03b8+1"},"headline":"A142461 \u2192 \u03b8+1","chain":null}