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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":"52304e1d1fc0bbdee98d239e30f6ad477b5fac8d7aec7fe5495d02c26e590488","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-0444a4f352052985","role":"sequence","name":"basis_map_A007318 of T(n,k) = binomial(n,k)*binomial(n+k,k), 0 <= k <= n, triangle 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:       3,        4;\n   3:      13,       36,       18;\n   4:      63,      264,      270,        80;\n   5:     321,     1800,     2790,      1680,       350;\n   6:    1683,    11820,    24570,     22400,      9450,      1512;\n   7:    8989,    75852,   197820,    241920,    154350,     49896,     6468;\n   8:   48639,   479248,  1503684,   2308320,   1975050,    964656,   252252,    27456;\n   9:  265729,  2994192, 10980900,  20291040,  21794850,  14270256,  5633628,  1235520,  115830;\n  10: 1462563, 18548820, 77855580, 168241920, 217567350, 178810632, 94594500, 31299840, 5907330, 486200;\n","recurrence_sentence":"The row polynomials satisfy -(4*n-4+(3*n-3)*x-(n^3-4*n^2+5*n-2)*x^2)*P(n,x) = -(12*n-12+(4*n^2+13*n-17)*x-(3*n^3-12*n^2+3*n+6)*x^2-(2*n^3-6*n^2+4*n)*x^3)*P(n-1,x) - (8+24*x-(4*n^2-8*n-22)*x^2-(6*n^2-12*n-6)*x^3-(2*n^2-4*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[(-(12*n - 12 + (4*n^2 + 13*n - 17)*x - (3*n^3 - 12*n^2 + 3*n + 6)*x^2 - (2*n^3 - 6*n^2 + 4*n)*x^3)*P[n - 1, x] - (8 + 24*x - (4*n^2 - 8*n - 22)*x^2 - (6*n^2 - 12*n - 6)*x^3 - (2*n^2 - 4*n)*x^4)*D[P[n - 1, x], x])/(-4*n + 4 - (3*n - 3)*x + (n^3 - 4*n^2 + 5*n - 2)*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] = (-(12*n - 12 + (4*n^2 + 13*n - 17)*x - (3*n^3 - 12*n^2 + 3*n + 6)*x^2 - (2*n^3 - 6*n^2 + 4*n)*x^3)*v[i - 1] - (8 + 24*x - (4*n^2 - 8*n - 22)*x^2 - (6*n^2 - 12*n - 6)*x^3 - (2*n^2 - 4*n)*x^4)*deriv(v[i - 1], x))/(-4*n + 4 - (3*n - 3)*x + (n^3 - 4*n^2 + 5*n - 2)*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-4689fb1386d440ad","steps":["x \u21a6 x+1","\u03b8+1"],"edges":["e-437a2dd956a9df17","e-2b24c1c2945253ce"],"text":"A063007 \u2192 x \u21a6 x+1 \u2192 \u03b8+1"},"headline":"A063007 \u2192 x \u21a6 x+1 \u2192 \u03b8+1","chain":null}