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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":"30b7a744df8e351fc73cec5e0cec3c24834eadd504da94daab7426b84ef0d596","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-29bd79a8a49e73a9","role":"sequence","name":"basis_map_A013609 of Triangle of coefficients of polynomials u(n,x) jointly generated with A207607; see the Formula section.","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:    13,      44,      24;\n  1:    34,     176,     204,       64;\n  2:    89,     620,    1116,      736,      160;\n  3:   233,    2040,    5004,     5120,     2320,      384;\n  4:   610,    6432,   20016,    27968,    19600,     6720,      896;\n  5:  1597,   19696,   74340,   132160,   126400,    66816,    18368,     2048;\n  6:  4181,   59044,  262092,   566752,   690000,   497088,   210112,    48128,     4608;\n  7: 10946,  174160,  889068,  2267008,  3360480,  3080448,  1771840,   622592,   122112,   10240;\n  8: 28657,  507220, 2927592,  8605120, 15058960, 16809408, 12293120,  5869568,  1762560,  302080,  22528;\n  9: 75025, 1462128, 9415368, 31355264, 63329200, 83489280, 74329472, 45084672, 18372096, 4812800, 732160, 49152;\n","recurrence_sentence":"The row polynomials satisfy -(40*n+(264*n+360)*x-(16*n^3-384*n-528)*x^2-(64*n^3+288*n^2+416*n+192)*x^3)*P(n,x) = -(60*n-40+(20*n^2+476*n+536)*x-(24*n^3-96*n^2-1464*n-2128)*x^2-(112*n^3+464*n^2-32*n-1024)*x^3-(64*n^3+384*n^2+704*n+384)*x^4)*P(n-1,x) - (25+290*x-(20*n^2+20*n-972)*x^2-(128*n^2+368*n-904)*x^3-(208*n^2+784*n+256)*x^4-(64*n^2+256*n+192)*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 13 + 44*x + 24*x^2.","mathematica":"P[0, x_] := 13 + 44*x + 24*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(60*n - 40 + (20*n^2 + 476*n + 536)*x - (24*n^3 - 96*n^2 - 1464*n - 2128)*x^2 - (112*n^3 + 464*n^2 - 32*n - 1024)*x^3 - (64*n^3 + 384*n^2 + 704*n + 384)*x^4)*P[n - 1, x] - (25 + 290*x - (20*n^2 + 20*n - 972)*x^2 - (128*n^2 + 368*n - 904)*x^3 - (208*n^2 + 784*n + 256)*x^4 - (64*n^2 + 256*n + 192)*x^5)*D[P[n - 1, x], x])/(-40*n - (264*n + 360)*x + (16*n^3 - 384*n - 528)*x^2 + (64*n^3 + 288*n^2 + 416*n + 192)*x^3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 13 + 44*x + 24*x^2; for(n = 1, N, my(i = n + 1); v[i] = (-(60*n - 40 + (20*n^2 + 476*n + 536)*x - (24*n^3 - 96*n^2 - 1464*n - 2128)*x^2 - (112*n^3 + 464*n^2 - 32*n - 1024)*x^3 - (64*n^3 + 384*n^2 + 704*n + 384)*x^4)*v[i - 1] - (25 + 290*x - (20*n^2 + 20*n - 972)*x^2 - (128*n^2 + 368*n - 904)*x^3 - (208*n^2 + 784*n + 256)*x^4 - (64*n^2 + 256*n + 192)*x^5)*deriv(v[i - 1], x))/(-40*n - (264*n + 360)*x + (16*n^3 - 384*n - 528)*x^2 + (64*n^3 + 288*n^2 + 416*n + 192)*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-dfab35176bd08e72","steps":["basis A013609","\u03b8+1"],"edges":["e-0adfae28f720d284","e-bdf971e649d9cc6a"],"text":"A207606 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A207606 \u2192 basis A013609 \u2192 \u03b8+1","chain":null}