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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":"fd015786dbc95d2ef72f19397e366e76c0b6325d26c0340e9b0cab5f26460cf5","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-f712c326ca8e5415","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of rooted trees with k nodes which are disjoint sets of labels with union {1..n}. If a node has an empty set of labels then it must have at least two children.","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: 1,    8,      9;\n  2: 1,   24,    108,      160,        75;\n  3: 1,   56,    549,     1984,      3275,       2520,         735;\n  4: 1,  120,   2160,    14240,     45750,      80136,       78400,        40320,          8505;\n  5: 1,  248,   7569,    80960,    424750,    1267776,     2306038,      2610944,       1800225,        693000,        114345;\n  6: 1,  504,  24948,   406560,   3198825,   14451696,    41011824,     76578048,      95757795,      79571800,      42231420,      12972960,       1756755;\n  7: 1, 1016,  79389,  1896384,  21284025,  136432296,   554475621,   1516706304,    2887091019,    3880894600,    3677593535,    2407204800,    1037071035,     264864600,      30405375;\n  8: 1, 2040, 247320,  8440480, 130786500, 1142151192,  6318306960,  23727911040,   63113196870,  121925353000,  173186513016,  181057425120,  137786068420,   74297463240,   26918892000,    5881075200,    585810225;\n  9: 1, 4088, 760329, 36416320, 761499500, 8812197072, 64164992556, 317540990208, 1121620808070, 2918367904000, 5704167054806, 8462751979968, 9551777639404, 8152215671600, 5176675003500, 2371510901760, 741049934625, 141421480200, 12439852425;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(6*x^2+12*x^3+6*x^4)*P(n-2,x) - (2*x^2+6*x^3+6*x^4+2*x^5)*d/dx P(n-2,x) + (1+8*x+7*x^2)*P(n-1,x) + (x+2*x^2+x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 8*x + 9*x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 8*x + 9*x^2;\nP[n_, x_] := P[n, x] = Expand[-(6*x^2 + 12*x^3 + 6*x^4)*P[n - 2, x] - (2*x^2 + 6*x^3 + 6*x^4 + 2*x^5)*D[P[n - 2, x], x] + (1 + 8*x + 7*x^2)*P[n - 1, x] + (x + 2*x^2 + x^3)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 8*x + 9*x^2; for(n = 2, N, my(i = n + 1); v[i] = -(6*x^2 + 12*x^3 + 6*x^4)*v[i - 2] - (2*x^2 + 6*x^3 + 6*x^4 + 2*x^5)*deriv(v[i - 2], x) + (1 + 8*x + 7*x^2)*v[i - 1] + (x + 2*x^2 + x^3)*deriv(v[i - 1], 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-f712c326ca8e5415","steps":["(\u03b8+1)\u00b2"],"edges":["e-a473098a8e206989"],"text":"A094262 \u2192 (\u03b8+1)\u00b2"},"headline":"A094262 \u2192 (\u03b8+1)\u00b2","chain":null}