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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":"15644b121e2d928684d66f5a071859e6fbe32c8f3d9fdd33e9e4360b77604c6f","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-71831045520c8a2d","role":"sequence","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*11^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:             529,              2024,               1452;\n  1:           12167,             69828,             100188,              42592;\n  2:          279841,           2141392,            4608648,            3918464,             1171280;\n  3:         6436343,          61565020,          176664840,          225311680,           134697200,            30921792;\n  4:       148035889,        1699194552,         6094936980,        10364337280,          9294106800,          4267207296,           793659328;\n  5:      3404825447,       45595053812,       196256970756,       417164575520,        498783731600,        343510187328,        127779151808,         19954863104;\n  6:     78310985281,     1198498557344,      6018547103184,     15351656379136,      22944051653600,      21068624822784,      11755681966336,       3671694811136,       493882861824;\n  7:   1801152661463,    31011150171276,    177977035765584,    529632145080192,     949883738459040,    1090301334579072,     811142055677184,     380020412952576,    102233752397568,     12072692177920;\n  8:  41426511213649,   792507171043720,   5116839778260540,  17402199052634880,   36412209974263200,   50153861390637312,   46640668201438080,   29134898326364160,  11756881525720320,   2776719200921600,   292159150705664;\n  9: 952809757913927, 20050431427406116, 143840051544435180, 550344545039578080, 1316041303355512800, 2114821155305206656, 2360017810992766848, 1842782319142533120, 991497008669080320, 351254978916582400, 73916265128532992, 7011819616935936;\n","recurrence_sentence":"The row polynomials satisfy (23+(22*n+44)*x)*P(n,x) = (529+(506*n+2024)*x+(484*n+1452)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 529 + 2024*x + 1452*x^2.","mathematica":"P[0, x_] := 529 + 2024*x + 1452*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((529 + (506*n + 2024)*x + (484*n + 1452)*x^2)*P[n - 1, x])/(23 + (22*n + 44)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 529 + 2024*x + 1452*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((529 + (506*n + 2024)*x + (484*n + 1452)*x^2)*v[i - 1])/(23 + (22*n + 44)*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-344e4bfa9933088a","steps":["basis A013609","\u03b8+1"],"edges":["e-9947ebaa7f991114","e-ae11628b645caf8d"],"text":"A038337 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A038337 \u2192 basis A013609 \u2192 \u03b8+1","chain":null}