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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":"f5e3ecb425db890401834b4b2b7476a1ed55b5cb47d8ed305f1aad19b48e4a3d","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-66d8297eef1ced69","role":"sequence","name":"basis_map_A007318 of Triangle read by rows whose (i,j)-th entry is binomial(i,j)*11^(i-j)*10^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:            1;\n  1:           21,            20;\n  2:          441,           840,            300;\n  3:         9261,         26460,          18900,           4000;\n  4:       194481,        740880,         793800,         336000,          50000;\n  5:      4084101,      19448100,       27783000,       17640000,        5250000,         600000;\n  6:     85766121,     490092120,      875164500,      740880000,      330750000,       75600000,       7000000;\n  7:   1801088541,   12007256940,    25729836300,    27227340000,    16206750000,     5556600000,    1029000000,      80000000;\n  8:  37822859361,  288174166560,   720435416400,   914838624000,   680683500000,   311169600000,   86436000000,   13440000000,    900000000;\n  9: 794280046581, 6808114684980, 19451756242800, 28817416656000, 25729836300000, 14702763600000, 5445468000000, 1270080000000, 170100000000, 10000000000;\n","recurrence_sentence":"The row polynomials satisfy (21+10*n*x)*P(n,x) = (441+(210*n+420)*x+(100*n+100)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((441 + (210*n + 420)*x + (100*n + 100)*x^2)*P[n - 1, x])/(21 + 10*n*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((441 + (210*n + 420)*x + (100*n + 100)*x^2)*v[i - 1])/(21 + 10*n*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-e1d1088f7d1ff894","steps":["basis A007318","\u03b8+1"],"edges":["e-3706f9e3ae504ed3","e-4c74b3784c701238"],"text":"A038324 \u2192 basis A007318 \u2192 \u03b8+1"},"headline":"A038324 \u2192 basis A007318 \u2192 \u03b8+1","chain":null}