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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":"5ad07fe515c503089033b281a98cbab79e3f0f9226a6e1071790e7d1b44c382d","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-9116f92e53f934a9","role":"sequence","name":"basis_map_A013609 of Triangle of central factorial numbers |t(2n,2n-2k)| read by rows.","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:            2,              8;\n  2:           10,            104,             144;\n  3:          100,           1760,            5652,             4608;\n  4:         1700,          42720,          222804,           399872,            230400;\n  5:        44200,        1450720,        10598904,         30201472,          37230400,          16588800;\n  6:      1635400,       66406240,       627176088,       2474114176,        4775190400,        4473833472,        1625702400;\n  7:     81770000,     3961388800,     46001380320,     232973720576,      617608253200,      897566542848,      678044795904,      208089907200;\n  8:   5315050000,   299356512000,   4130969695200,   25611161492480,    86739280573200,   172179378514944,   200448949421056,   126883863724032,    33710564966400;\n  9: 435834100000, 27991386384000, 447854963630400, 3289834514600960, 13595446259786400, 34353248410341504, 54402366815071744, 52817838204911616, 28779423511412736, 6742112993280000;\n","recurrence_sentence":"The row polynomials satisfy (n^2-2*n+1)*P(n,x) = -(n^6-4*n^5+8*n^4-8*n^3+4*n^2+(8*n^6-32*n^5+56*n^4-48*n^3+16*n^2)*x-(36*n^6-144*n^5+216*n^4-144*n^3+36*n^2)*x^2)*P(n-2,x) + ((8*n^6-32*n^5+56*n^4-48*n^3+16*n^2)*x^2+(16*n^6-64*n^5+96*n^4-64*n^3+16*n^2)*x^3)*d/dx P(n-2,x) + (2*n^4-4*n^3+4*n^2-2*n+1+(8*n^4-16*n^3+8*n^2)*x)*P(n-1,x) - (8*n^4-16*n^3+8*n^2)*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 8*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 8*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^6 - 4*n^5 + 8*n^4 - 8*n^3 + 4*n^2 + (8*n^6 - 32*n^5 + 56*n^4 - 48*n^3 + 16*n^2)*x - (36*n^6 - 144*n^5 + 216*n^4 - 144*n^3 + 36*n^2)*x^2)*P[n - 2, x] + ((8*n^6 - 32*n^5 + 56*n^4 - 48*n^3 + 16*n^2)*x^2 + (16*n^6 - 64*n^5 + 96*n^4 - 64*n^3 + 16*n^2)*x^3)*D[P[n - 2, x], x] + (2*n^4 - 4*n^3 + 4*n^2 - 2*n + 1 + (8*n^4 - 16*n^3 + 8*n^2)*x)*P[n - 1, x] - (8*n^4 - 16*n^3 + 8*n^2)*x^2*D[P[n - 1, x], x])/(n^2 - 2*n + 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^6 - 4*n^5 + 8*n^4 - 8*n^3 + 4*n^2 + (8*n^6 - 32*n^5 + 56*n^4 - 48*n^3 + 16*n^2)*x - (36*n^6 - 144*n^5 + 216*n^4 - 144*n^3 + 36*n^2)*x^2)*v[i - 2] + ((8*n^6 - 32*n^5 + 56*n^4 - 48*n^3 + 16*n^2)*x^2 + (16*n^6 - 64*n^5 + 96*n^4 - 64*n^3 + 16*n^2)*x^3)*deriv(v[i - 2], x) + (2*n^4 - 4*n^3 + 4*n^2 - 2*n + 1 + (8*n^4 - 16*n^3 + 8*n^2)*x)*v[i - 1] - (8*n^4 - 16*n^3 + 8*n^2)*x^2*deriv(v[i - 1], x))/(n^2 - 2*n + 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-e734a631f8279507","steps":["basis A013609","(\u03b8+1)\u00b2"],"edges":["e-031a48225080e193","e-efd0e783498ca8e1"],"text":"A008955 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2"},"headline":"A008955 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2","chain":null}