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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":"56027510eac28861339b2d6ef27aa46c98d982bc85ec06a7c6773ccc9e21c8b0","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-0af3ffe51a3d7181","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 = 7.","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,       32,           3;\n  3: 1,      286,         429,            4;\n  4: 1,     2332,       12870,         4664,             5;\n  5: 1,    18714,      270006,       360008,         46785,             6;\n  6: 1,   149784,     4864149,     15840352,       8106915,        449352,            7;\n  7: 1,  1198358,    81050571,    536568172,     670710215,     162101142,      4194253,           8;\n  8: 1,  9586964,  1293052656,  15694927192,   38901192470,   23542390788,   3017122864,    38347856,         9;\n  9: 1, 76695826, 20114812140, 419423417168, 1834406305270, 2201287566324, 733990980044, 53639499040, 345131217, 10;\n","recurrence_sentence":"The row polynomials satisfy (7*n-6)*P(n,x) = (42*n+6+(588*n^2-1008*n-156)*x-(343*n^3-882*n^2+413*n+78)*x^2)*P(n-2,x) + ((294*n+42)*x-(343*n^2+49*n)*x^2+(343*n^2-245*n-42)*x^3)*d/dx P(n-2,x) - (35*n+12-(98*n^2-168*n+72)*x)*P(n-1,x) + ((49*n-42)*x-(49*n-42)*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[((42*n + 6 + (588*n^2 - 1008*n - 156)*x - (343*n^3 - 882*n^2 + 413*n + 78)*x^2)*P[n - 2, x] + ((294*n + 42)*x - (343*n^2 + 49*n)*x^2 + (343*n^2 - 245*n - 42)*x^3)*D[P[n - 2, x], x] - (35*n + 12 - (98*n^2 - 168*n + 72)*x)*P[n - 1, x] + ((49*n - 42)*x - (49*n - 42)*x^2)*D[P[n - 1, x], x])/(7*n - 6)]];\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] = ((42*n + 6 + (588*n^2 - 1008*n - 156)*x - (343*n^3 - 882*n^2 + 413*n + 78)*x^2)*v[i - 2] + ((294*n + 42)*x - (343*n^2 + 49*n)*x^2 + (343*n^2 - 245*n - 42)*x^3)*deriv(v[i - 2], x) - (35*n + 12 - (98*n^2 - 168*n + 72)*x)*v[i - 1] + ((49*n - 42)*x - (49*n - 42)*x^2)*deriv(v[i - 1], x))/(7*n - 6)); 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-0af3ffe51a3d7181","steps":["\u03b8+1"],"edges":["e-0548eebc0cf61360"],"text":"A142462 \u2192 \u03b8+1"},"headline":"A142462 \u2192 \u03b8+1","chain":null}