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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":"28643a5c59a368a1fe016914c48be0f8e518c6c7bdaff0ebfac36753edf18dfc","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-cab14672969f6507","role":"sequence","name":"Galton triangle T(n, k) = T(n-1, k-1) + (3k-2)*T(n-1, k) read by rows.","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: 1,      4;\n   3: 1,     20,        9;\n   4: 1,     84,      108,       16;\n   5: 1,    340,      945,      352,       25;\n   6: 1,   1364,     7380,     5200,      875,       36;\n   7: 1,   5460,    54729,    65120,    19500,     1836,      49;\n   8: 1,  21844,   395388,   748496,   355250,    57456,    3430,     64;\n   9: 1,  87380,  2816865,  8187872,  5787775,  1430856,  143374,   5888,   81;\n  10: 1, 349524, 19914660, 86886480, 88034625, 31228092, 4671660, 316800, 9477, 100;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2-4*x+9*x^2)*P(n-2,x) - (6*x-4*x^2-4*x^3)*d/dx P(n-2,x) - (9*x^2-12*x^3)*d^2/dx^2 P(n-2,x) - (1-4*x)*P(n-1,x) + (6*x-4*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 4*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 1 + 4*x;\nP[n_, x_] := P[n, x] = Expand[(2 - 4*x + 9*x^2)*P[n - 2, x] - (6*x - 4*x^2 - 4*x^3)*D[P[n - 2, x], x] - (9*x^2 - 12*x^3)*D[P[n - 2, x], {x, 2}] - (1 - 4*x)*P[n - 1, x] + (6*x - 4*x^2)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 4*x; for(n = 3, N, my(i = n); v[i] = (2 - 4*x + 9*x^2)*v[i - 2] - (6*x - 4*x^2 - 4*x^3)*deriv(v[i - 2], x) - (9*x^2 - 12*x^3)*deriv(deriv(v[i - 2], x), x) - (1 - 4*x)*v[i - 1] + (6*x - 4*x^2)*deriv(v[i - 1], 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-cab14672969f6507","steps":["(\u03b8+1)\u00b2"],"edges":["e-51ffa458e8b206be"],"text":"A111577 \u2192 (\u03b8+1)\u00b2"},"headline":"A111577 \u2192 (\u03b8+1)\u00b2","chain":null}