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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":"ea0685de219aa329692942fe5be7b498ec06a992126ed4f673721f0dfa9a532e","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-43836771acfe9d89","role":"sequence","name":"basis_map_A007318 of Triangle T(n, k) is coefficient of z^n*w^k in 1/(1 - 2*z - 2*w - 2*z*w) read by rows in order 00, 10, 01, 20, 11, 02, ...","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,       4;\n  2:     18,      36,       12;\n  3:     80,     240,      168,       32;\n  4:    356,    1424,     1524,      608,       80;\n  5:   1584,    7920,    11424,     7008,     1920,      192;\n  6:   7048,   42288,    76776,    63776,    26880,     5568,     448;\n  7:  31360,  219520,   480576,   504320,   288320,    91776,   15232,   1024;\n  8: 139536, 1116288,  2861280,  3631104,  2627280,  1134720,  288960,  39936,   2304;\n  9: 620864, 5587776, 16413696, 24444672, 21424320, 11719872, 4048128, 857088, 101376, 5120;\n","recurrence_sentence":"The row polynomials satisfy (12*n*x+(6*n^2+12*n)*x^2+(n^3+3*n^2+2*n)*x^3)*P(n,x) = -(48*n-48+(36*n^2+48*n-132)*x+(6*n^3+36*n^2-34*n-128)*x^2+(4*n^3-48*n-52)*x^3-(4*n^2+12*n+8)*x^4)*P(n-1,x) + (48+(48*n+168)*x+(12*n^2+132*n+224)*x^2+(24*n^2+128*n+140)*x^3+(14*n^2+50*n+40)*x^4+(2*n^2+6*n+4)*x^5)*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[(-(48*n - 48 + (36*n^2 + 48*n - 132)*x + (6*n^3 + 36*n^2 - 34*n - 128)*x^2 + (4*n^3 - 48*n - 52)*x^3 - (4*n^2 + 12*n + 8)*x^4)*P[n - 1, x] + (48 + (48*n + 168)*x + (12*n^2 + 132*n + 224)*x^2 + (24*n^2 + 128*n + 140)*x^3 + (14*n^2 + 50*n + 40)*x^4 + (2*n^2 + 6*n + 4)*x^5)*D[P[n - 1, x], x])/(12*n*x + (6*n^2 + 12*n)*x^2 + (n^3 + 3*n^2 + 2*n)*x^3)]];\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] = (-(48*n - 48 + (36*n^2 + 48*n - 132)*x + (6*n^3 + 36*n^2 - 34*n - 128)*x^2 + (4*n^3 - 48*n - 52)*x^3 - (4*n^2 + 12*n + 8)*x^4)*v[i - 1] + (48 + (48*n + 168)*x + (12*n^2 + 132*n + 224)*x^2 + (24*n^2 + 128*n + 140)*x^3 + (14*n^2 + 50*n + 40)*x^4 + (2*n^2 + 6*n + 4)*x^5)*deriv(v[i - 1], x))/(12*n*x + (6*n^2 + 12*n)*x^2 + (n^3 + 3*n^2 + 2*n)*x^3)); 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-1bffaa6198344f93","steps":["x \u21a6 x+1","\u03b8+1"],"edges":["e-c090df23694a462d","e-e12cdbe964099667"],"text":"A059473 \u2192 x \u21a6 x+1 \u2192 \u03b8+1"},"headline":"A059473 \u2192 x \u21a6 x+1 \u2192 \u03b8+1","chain":null}