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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":"12d7a821cb6eb6371c9456e43d138d99fae27aa350760918771a21d7c7acfb1a","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-f07088d208415183","role":"sequence","name":"basis_map_A013609 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,        8;\n  2:     18,       72,       48;\n  3:     80,      480,      672,       256;\n  4:    356,     2848,     6096,      4864,      1280;\n  5:   1584,    15840,    45696,     56064,     30720,      6144;\n  6:   7048,    84576,   307104,    510208,    430080,    178176,     28672;\n  7:  31360,   439040,  1922304,   4034560,   4613120,   2936832,    974848,    131072;\n  8: 139536,  2232576, 11445120,  29048832,  42036480,  36311040,  18493440,   5111808,   589824;\n  9: 620864, 11175552, 65654784, 195557376, 342789120, 375035904, 259080192, 109707264, 25952256, 2621440;\n","recurrence_sentence":"The row polynomials satisfy (3*n*x+(3*n^2+6*n)*x^2+(n^3+3*n^2+2*n)*x^3)*P(n,x) = -(6*n-6+(9*n^2+12*n-33)*x+(3*n^3+18*n^2-17*n-64)*x^2+(4*n^3-48*n-52)*x^3-(8*n^2+24*n+16)*x^4)*P(n-1,x) + (3+(6*n+21)*x+(3*n^2+33*n+56)*x^2+(12*n^2+64*n+70)*x^3+(14*n^2+50*n+40)*x^4+(4*n^2+12*n+8)*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[(-(6*n - 6 + (9*n^2 + 12*n - 33)*x + (3*n^3 + 18*n^2 - 17*n - 64)*x^2 + (4*n^3 - 48*n - 52)*x^3 - (8*n^2 + 24*n + 16)*x^4)*P[n - 1, x] + (3 + (6*n + 21)*x + (3*n^2 + 33*n + 56)*x^2 + (12*n^2 + 64*n + 70)*x^3 + (14*n^2 + 50*n + 40)*x^4 + (4*n^2 + 12*n + 8)*x^5)*D[P[n - 1, x], x])/(3*n*x + (3*n^2 + 6*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] = (-(6*n - 6 + (9*n^2 + 12*n - 33)*x + (3*n^3 + 18*n^2 - 17*n - 64)*x^2 + (4*n^3 - 48*n - 52)*x^3 - (8*n^2 + 24*n + 16)*x^4)*v[i - 1] + (3 + (6*n + 21)*x + (3*n^2 + 33*n + 56)*x^2 + (12*n^2 + 64*n + 70)*x^3 + (14*n^2 + 50*n + 40)*x^4 + (4*n^2 + 12*n + 8)*x^5)*deriv(v[i - 1], x))/(3*n*x + (3*n^2 + 6*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":["basis A013609","\u03b8+1"],"edges":["e-c90acfb06fc8929f","e-a5e95914ebeb92f2"],"text":"A059473 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A059473 \u2192 basis A013609 \u2192 \u03b8+1","chain":null}