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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":"6308d44c699e5e2872e0ab3dffce076cac07218800b4cca3301f13454ea412e2","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-3afaca41e5482e8c","role":"sequence","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*12^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:            23,             48;\n  2:           529,           2208,            1728;\n  3:         12167,          76176,          119232,           55296;\n  4:        279841,        2336064,         5484672,         5087232,          1658880;\n  5:       6436343,       67161840,       210245760,       292515840,        190771200,         47775744;\n  6:     148035889,     1853666784,      7253478720,     13455728640,      13163212800,       6593052672,       1337720832;\n  7:    3404825447,    49740058704,    233562014784,    541593077760,     706425753600,     530740740096,     215373053952,     36691771392;\n  8:   78310985281,  1307452971648,   7162568453376,  19930625261568,   32495584665600,   32552098725888,   19814320963584,   6751285936128,    990677827584;\n  9: 1801152661463, 33830345641392, 211807381406976, 687606571524096, 1345317205155840, 1684571109064704, 1367188146487296, 698758094389248, 205070310309888, 26418075402240;\n","recurrence_sentence":"The row polynomials satisfy (23+24*n*x)*P(n,x) = (529+(552*n+1104)*x+(576*n+576)*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[((529 + (552*n + 1104)*x + (576*n + 576)*x^2)*P[n - 1, x])/(23 + 24*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] = ((529 + (552*n + 1104)*x + (576*n + 576)*x^2)*v[i - 1])/(23 + 24*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-9e472291b944d56e","steps":["basis A013609","\u03b8+1"],"edges":["e-15af34f262d68ea6","e-6984949df7a730d3"],"text":"A038326 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A038326 \u2192 basis A013609 \u2192 \u03b8+1","chain":null}