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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":"f1e8035098e13fa15adcc37c201e67e00c86e6b193dd9e238a5cc601be314544","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-43a1e62341e55229","role":"sequence","name":"basis_map_A013609 of Triangle of coefficients in expansion of (1+11x)^n.","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:          144,           1056,            1452;\n  1:         1728,          19008,           52272,            42592;\n  2:        20736,         304128,         1254528,          2044416,           1171280;\n  3:       248832,        4561920,        25090560,         61332480,          70276800,          30921792;\n  4:      2985984,       65691648,       451630080,       1471979520,        2529964800,        2226369024,         793659328;\n  5:     35831808,      919683072,      7587385344,      30911569920,       70839014400,       93507499008,       66667383552,        19954863104;\n  6:    429981696,    12612796416,    121398165504,     593502142464,     1700136345600,     2992239968256,     3200034410496,      1915666857984,       493882861824;\n  7:   5159780352,   170272751616,   1873000267776,   10683038564352,    36722945064960,    80790479142912,   115201238777856,    103446010331136,     53339349076992,    12072692177920;\n  8:  61917364224,  2270303354880,  28095004016640,  183137803960320,   734458901299200,  1938971499429888,  3456037163335680,   4137840413245440,   3200360944619520,  1448723061350400,   292159150705664;\n  9: 743008370688, 29968004284416, 412060058910720, 3021773765345280, 13849796424499200, 42657372987457536, 91239381112061952, 136548733637099520, 140815881563258880, 95615722049126400, 38565007893147648, 7011819616935936;\n","recurrence_sentence":"The row polynomials satisfy (6+(11*n+22)*x)*P(n,x) = (72+(132*n+528)*x+(242*n+726)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 144 + 1056*x + 1452*x^2.","mathematica":"P[0, x_] := 144 + 1056*x + 1452*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((72 + (132*n + 528)*x + (242*n + 726)*x^2)*P[n - 1, x])/(6 + (11*n + 22)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 144 + 1056*x + 1452*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((72 + (132*n + 528)*x + (242*n + 726)*x^2)*v[i - 1])/(6 + (11*n + 22)*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-ec89fb3fc20684fc","steps":["x \u21a6 x/(x+1)","\u03b8+1"],"edges":["e-492aa71fabef2dc6","e-2fe802f2fe257174"],"text":"A038336 \u2192 x \u21a6 x/(x+1) \u2192 \u03b8+1"},"headline":"A038336 \u2192 x \u21a6 x/(x+1) \u2192 \u03b8+1","chain":null}