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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":"f95c5d23b46cd96e3847b719fe4961bc58307f757f2a339a2c31ba34f67c9327","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-60c7e36cbe887b7e","role":"sequence","name":"Coefficients of raising factorial polynomials, T(n,k) = [x^k] p(x, n) where p(x, n) = (m*x + n - 1)*p(x, n - 1) with p[x, 0] = 1, p[x, -1] = 0, p[x, 1] = m*x and m = 2. Triangle read by rows, for n >= 0 and 0 <= k <= n.","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: 1;\n  1: 0,      4;\n  2: 0,      4,      12;\n  3: 0,      8,      36,      32;\n  4: 0,     24,     132,     192,      80;\n  5: 0,     96,     600,    1120,     800,     192;\n  6: 0,    480,    3288,    7200,    6800,    2880,     448;\n  7: 0,   2880,   21168,   51968,   58800,   33600,    9408,   1024;\n  8: 0,  20160,  156816,  420224,  541520,  376320,  144256,  28672,  2304;\n  9: 0, 161280, 1315008, 3779968, 5382720, 4310208, 2032128, 559104, 82944, 5120;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^2-4*n+4+(4*n-8)*x+4*x^2)*P(n-2,x) + (2*n-3+4*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4*x;\nP[n_, x_] := P[n, x] = Expand[-(n^2 - 4*n + 4 + (4*n - 8)*x + 4*x^2)*P[n - 2, x] + (2*n - 3 + 4*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 - 4*n + 4 + (4*n - 8)*x + 4*x^2)*v[i - 2] + (2*n - 3 + 4*x)*v[i - 1]); 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-60c7e36cbe887b7e","steps":["\u03b8+1"],"edges":["e-0692d8447221de18"],"text":"A137320 \u2192 \u03b8+1"},"headline":"A137320 \u2192 \u03b8+1","chain":null}