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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":"d0cbb5a7b65ffce9d14e856869e9b266906e9fd8966d6aeea80b25850d710cf6","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-e6d483569f3966cd","role":"sequence","name":"basis_map_A013610 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,       12;\n  2:     18,      108,       108;\n  3:     80,      720,      1512,       864;\n  4:    356,     4272,     13716,     16416,       6480;\n  5:   1584,    23760,    102816,    189216,     155520,      46656;\n  6:   7048,   126864,    690984,   1721952,    2177280,    1353024,     326592;\n  7:  31360,   658560,   4325184,  13616640,   23353920,   22301568,   11104128,    2239488;\n  8: 139536,  3348864,  25751520,  98039808,  212809680,  275736960,  210651840,   87340032,  15116544;\n  9: 620864, 16763328, 147723264, 660006144, 1735369920, 2847928896, 2951085312, 1874451456, 665127936, 100776960;\n","recurrence_sentence":"The row polynomials satisfy (36*n*x+(54*n^2+108*n)*x^2+(27*n^3+81*n^2+54*n)*x^3)*P(n,x) = -(48*n-48+(108*n^2+144*n-396)*x+(54*n^3+324*n^2-306*n-1152)*x^2+(108*n^3-1296*n-1404)*x^3-(324*n^2+972*n+648)*x^4)*P(n-1,x) + (16+(48*n+168)*x+(36*n^2+396*n+672)*x^2+(216*n^2+1152*n+1260)*x^3+(378*n^2+1350*n+1080)*x^4+(162*n^2+486*n+324)*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 + (108*n^2 + 144*n - 396)*x + (54*n^3 + 324*n^2 - 306*n - 1152)*x^2 + (108*n^3 - 1296*n - 1404)*x^3 - (324*n^2 + 972*n + 648)*x^4)*P[n - 1, x] + (16 + (48*n + 168)*x + (36*n^2 + 396*n + 672)*x^2 + (216*n^2 + 1152*n + 1260)*x^3 + (378*n^2 + 1350*n + 1080)*x^4 + (162*n^2 + 486*n + 324)*x^5)*D[P[n - 1, x], x])/(36*n*x + (54*n^2 + 108*n)*x^2 + (27*n^3 + 81*n^2 + 54*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 + (108*n^2 + 144*n - 396)*x + (54*n^3 + 324*n^2 - 306*n - 1152)*x^2 + (108*n^3 - 1296*n - 1404)*x^3 - (324*n^2 + 972*n + 648)*x^4)*v[i - 1] + (16 + (48*n + 168)*x + (36*n^2 + 396*n + 672)*x^2 + (216*n^2 + 1152*n + 1260)*x^3 + (378*n^2 + 1350*n + 1080)*x^4 + (162*n^2 + 486*n + 324)*x^5)*deriv(v[i - 1], x))/(36*n*x + (54*n^2 + 108*n)*x^2 + (27*n^3 + 81*n^2 + 54*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":["basis A013610","\u03b8+1"],"edges":["e-04dfb33ef5bb2a04","e-3fd57f575f666bf7"],"text":"A059473 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"A059473 \u2192 basis A013610 \u2192 \u03b8+1","chain":null}