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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":"3bbb859531c50b0198be8e361c455bb78018c65cfbf6f505c0c6977361688c16","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-63d702ebcae1deea","role":"sequence","name":"basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^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:            324,             1512,              1323;\n  1:           5832,            40824,             71442,             37044;\n  2:         104976,           979776,           2571912,           2667168,             972405;\n  3:        1889568,         22044960,          77157360,         120022560,           87516450,           24504606;\n  4:       34012224,        476171136,        2083248720,        4320812160,         4725888300,         2646497448,          600362847;\n  5:      612220032,       9999593856,       52497867744,      136105583040,       198487308600,       166729339224,        75645718722,        14408708328;\n  6:    11019960576,     205705930752,     1259948825856,     3919840791552,      7145543109600,      8003008282752,      5446491747984,      2074853999232,       340405734249;\n  7:   198359290368,    4165545097728,    29158815684096,   105835701371904,    231515596751040,    324121835451456,    294110554391136,    168063173937792,     55145728948338,      7942800465810;\n  8:  3570467226624,   83310901954560,   656073352892160,  2721489463848960,   6945467902531200,  11668386076252416,  13234974947601120,  10083790436267520,   4963115605350420,   1429704083845800,   183478690760211;\n  9: 64268410079232, 1649555858700288, 14433613763627520, 67356864230261760, 196457520671596800, 385056740516329728, 524105007925004352, 499147626595242240, 327565629953127720, 141540704300734200, 36328780770521778, 4203330006506652;\n","recurrence_sentence":"The row polynomials satisfy (6+(7*n+14)*x)*P(n,x) = (108+(126*n+504)*x+(147*n+441)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 324 + 1512*x + 1323*x^2.","mathematica":"P[0, x_] := 324 + 1512*x + 1323*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((108 + (126*n + 504)*x + (147*n + 441)*x^2)*P[n - 1, x])/(6 + (7*n + 14)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 324 + 1512*x + 1323*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((108 + (126*n + 504)*x + (147*n + 441)*x^2)*v[i - 1])/(6 + (7*n + 14)*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-7dcf11eddf2b9a8a","steps":["basis A013610","\u03b8+1"],"edges":["e-632cc4d34c6e419b","e-ad3cdf45f379d68f"],"text":"A038321 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"A038321 \u2192 basis A013610 \u2192 \u03b8+1","chain":null}