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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":"11d6d9129cbab871627ccacb9d901978b66a424c177cecdae5bb25a785155de6","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-5aa52e3acf29a627","role":"sequence","name":"Triangle read by rows, related to clusters of type D.","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:      4,       6;\n  2:     15,      40,      18;\n  3:     56,     210,     180,      40;\n  4:    210,    1008,    1260,     560,      75;\n  5:    792,    4620,    7560,    5040,    1400,     126;\n  6:   3003,   20592,   41580,   36960,   15750,    3024,     196;\n  7:  11440,   90090,  216216,  240240,  138600,   41580,    5880,    288;\n  8:  43758,  388960, 1081080, 1441440, 1051050,  432432,   97020,  10560,   405;\n  9: 167960, 1662804, 5250960, 8168160, 7207200, 3783780, 1177176, 205920, 17820, 550;\n","recurrence_sentence":"The row polynomials satisfy (2*n^3+6*n^2+4*n+(n^4+4*n^3+4*n^2)*x)*P(n,x) = (8*n^3+20*n^2+16*n+4+(4*n^4+26*n^3+42*n^2+28*n+8)*x+(3*n^4+14*n^3+19*n^2+14*n+4)*x^2)*P(n-1,x) - ((4*n^2+8*n+4)*x+(2*n^3+13*n^2+18*n+8)*x^2+(2*n^3+9*n^2+10*n+4)*x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((8*n^3 + 20*n^2 + 16*n + 4 + (4*n^4 + 26*n^3 + 42*n^2 + 28*n + 8)*x + (3*n^4 + 14*n^3 + 19*n^2 + 14*n + 4)*x^2)*P[n - 1, x] - ((4*n^2 + 8*n + 4)*x + (2*n^3 + 13*n^2 + 18*n + 8)*x^2 + (2*n^3 + 9*n^2 + 10*n + 4)*x^3)*D[P[n - 1, x], x])/(2*n^3 + 6*n^2 + 4*n + (n^4 + 4*n^3 + 4*n^2)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((8*n^3 + 20*n^2 + 16*n + 4 + (4*n^4 + 26*n^3 + 42*n^2 + 28*n + 8)*x + (3*n^4 + 14*n^3 + 19*n^2 + 14*n + 4)*x^2)*v[i - 1] - ((4*n^2 + 8*n + 4)*x + (2*n^3 + 13*n^2 + 18*n + 8)*x^2 + (2*n^3 + 9*n^2 + 10*n + 4)*x^3)*deriv(v[i - 1], x))/(2*n^3 + 6*n^2 + 4*n + (n^4 + 4*n^3 + 4*n^2)*x)); 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-5aa52e3acf29a627","steps":["\u03b8+1"],"edges":["e-8b6e26a04033d5bd"],"text":"A345013 \u2192 \u03b8+1"},"headline":"A345013 \u2192 \u03b8+1","chain":null}