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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":"6b43d7d7be514adb7bbf816321e328e6ebe56336af7a36f26815fcc4ca458d5a","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-57cc3445e053f3c8","role":"sequence","name":"basis_map_A013610 of Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).","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:    3,      6;\n   3:    8,     36,      27;\n   4:   21,    150,     243,     108;\n   5:   55,    540,    1377,    1296,     405;\n   6:  144,   1800,    6318,    9288,    6075,     1458;\n   7:  377,   5724,   25677,   51840,   52650,    26244,     5103;\n   8:  987,  17634,   96471,  248940,  346275,   266814,   107163,   17496;\n   9: 2584,  53100,  343089, 1080864, 1919700,  2020788,  1250235,  419904,   59049;\n  10: 6765, 157170, 1171746, 4366008, 9466065, 12706470, 10716300, 5528736, 1594323, 196830;\n","recurrence_sentence":"The row polynomials satisfy -(60*n-60+(54*n-54)*x-(54*n^3-162*n^2+162*n-54)*x^2)*P(n,x) = -(90*n+(45*n^2+171*n)*x-(81*n^3-162*n^2-81*n)*x^2-(81*n^3-81*n^2)*x^3)*P(n-1,x) - (25+135*x-(45*n^2-45*n-207)*x^2-(162*n^2-162*n-81)*x^3-(81*n^2-81*n)*x^4)*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[(-(90*n + (45*n^2 + 171*n)*x - (81*n^3 - 162*n^2 - 81*n)*x^2 - (81*n^3 - 81*n^2)*x^3)*P[n - 1, x] - (25 + 135*x - (45*n^2 - 45*n - 207)*x^2 - (162*n^2 - 162*n - 81)*x^3 - (81*n^2 - 81*n)*x^4)*D[P[n - 1, x], x])/(-60*n + 60 - (54*n - 54)*x + (54*n^3 - 162*n^2 + 162*n - 54)*x^2)]];\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] = (-(90*n + (45*n^2 + 171*n)*x - (81*n^3 - 162*n^2 - 81*n)*x^2 - (81*n^3 - 81*n^2)*x^3)*v[i - 1] - (25 + 135*x - (45*n^2 - 45*n - 207)*x^2 - (162*n^2 - 162*n - 81)*x^3 - (81*n^2 - 81*n)*x^4)*deriv(v[i - 1], x))/(-60*n + 60 - (54*n - 54)*x + (54*n^3 - 162*n^2 + 162*n - 54)*x^2)); 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-976cdb6e4a693ec8","steps":["basis A013610","\u03b8+1"],"edges":["e-1c0d9915253416a6","e-9c33bfcbb595181e"],"text":"A078812 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"A078812 \u2192 basis A013610 \u2192 \u03b8+1","chain":null}