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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":"05eff8e9c413b7419d2956816b3553a9e87a0ad76f021c0e6c67f6d59199b391","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-194cb2d73cc003c5","role":"sequence","name":"basis_map_A013610 of Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:      1;\n   2:      4,      18;\n   3:     16,     120,      135;\n   4:     64,     672,     1512,       756;\n   5:    256,    3456,    11664,     12960,      3645;\n   6:   1024,   16896,    76032,    133056,     89100,      16038;\n   7:   4096,   79872,   449280,   1078272,   1179360,     530712,     66339;\n   8:  16384,  368640,  2488320,   7603200,  11664000,    8817984,   2857680,    262440;\n   9:  65536, 1671168, 13160448,  48881664,  96940800,  104696064,  58296672,  14276736,  1003833;\n  10: 262144, 7471104, 67239936, 294174720, 717050880, 1014114816, 819092736, 351039744, 67315860, 3739770;\n","recurrence_sentence":"The row polynomials satisfy (8*n^2-20*n+12+(12*n^3-36*n^2+33*n-9)*x)*P(n,x) = (32*n^2-80*n+48+(48*n^3-228*n+180)*x+(108*n^3-180*n^2-45*n+108)*x^2)*P(n-1,x) - ((16*n-16)*x+(24*n^2+48*n-66)*x^2+(72*n^2-54)*x^3)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((32*n^2 - 80*n + 48 + (48*n^3 - 228*n + 180)*x + (108*n^3 - 180*n^2 - 45*n + 108)*x^2)*P[n - 1, x] - ((16*n - 16)*x + (24*n^2 + 48*n - 66)*x^2 + (72*n^2 - 54)*x^3)*D[P[n - 1, x], x])/(8*n^2 - 20*n + 12 + (12*n^3 - 36*n^2 + 33*n - 9)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((32*n^2 - 80*n + 48 + (48*n^3 - 228*n + 180)*x + (108*n^3 - 180*n^2 - 45*n + 108)*x^2)*v[i - 1] - ((16*n - 16)*x + (24*n^2 + 48*n - 66)*x^2 + (72*n^2 - 54)*x^3)*deriv(v[i - 1], x))/(8*n^2 - 20*n + 12 + (12*n^3 - 36*n^2 + 33*n - 9)*x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-ba8de4e02acfb6ca","steps":["basis A013610","\u03b8+1"],"edges":["e-4aa0aeb00b1923c9","e-26d65a67760d727d"],"text":"A091042 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"A091042 \u2192 basis A013610 \u2192 \u03b8+1","chain":null}