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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":"f716230db2bc21c4756bfc7ab42e227c308a5b4f18d0fbaf485ed7ef248c3a23","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-b8883330fe0b24f8","role":"sequence","name":"basis_map_A007318 of Triangle read by rows: coefficients of 1; 1(X+2); 1(X+2)(X+3); 1(X+2)(X+3)(X+4); ....","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:        3,         8;\n  2:       12,        68,         54;\n  3:       60,       532,        882,         384;\n  4:      360,      4392,      11277,       10144,        3000;\n  5:     2520,     39384,     138231,      191296,      116100,       25920;\n  6:    20160,    385632,    1726146,     3250576,     3021100,     1377648,      246960;\n  7:   181440,   4115808,   22476690,    53804816,    67822100,    47201904,    17223696,     2580480;\n  8:  1814400,  47689920,  308112012,   897675200,  1434851225,  1350993456,   750460284,   228271104,   29393280;\n  9: 19958400, 597165120, 4462255332, 15351974080, 29809538475, 35522785656, 26643585164, 12312912384, 3212382240, 362880000;\n","recurrence_sentence":"The row polynomials satisfy n*P(n,x) = -(n^3+3*n^2+3*n+1+(4*n^3+8*n^2+4*n)*x-(9*n^3+9*n^2)*x^2)*P(n-2,x) + ((4*n^3+8*n^2+4*n)*x^2+(4*n^3+4*n^2)*x^3)*d/dx P(n-2,x) + (2*n^2+4*n+1+(4*n^2+4*n)*x)*P(n-1,x) - (4*n^2+4*n)*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 8*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + 8*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + 3*n^2 + 3*n + 1 + (4*n^3 + 8*n^2 + 4*n)*x - (9*n^3 + 9*n^2)*x^2)*P[n - 2, x] + ((4*n^3 + 8*n^2 + 4*n)*x^2 + (4*n^3 + 4*n^2)*x^3)*D[P[n - 2, x], x] + (2*n^2 + 4*n + 1 + (4*n^2 + 4*n)*x)*P[n - 1, x] - (4*n^2 + 4*n)*x^2*D[P[n - 1, x], x])/(n)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 + 3*n^2 + 3*n + 1 + (4*n^3 + 8*n^2 + 4*n)*x - (9*n^3 + 9*n^2)*x^2)*v[i - 2] + ((4*n^3 + 8*n^2 + 4*n)*x^2 + (4*n^3 + 4*n^2)*x^3)*deriv(v[i - 2], x) + (2*n^2 + 4*n + 1 + (4*n^2 + 4*n)*x)*v[i - 1] - (4*n^2 + 4*n)*x^2*deriv(v[i - 1], x))/(n)); 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-2feb6dc01f424b0a","steps":["basis A007318","(\u03b8+1)\u00b2"],"edges":["e-5281dccfa5f22cf2","e-a7763b102cb5d3a7"],"text":"A145324 \u2192 basis A007318 \u2192 (\u03b8+1)\u00b2"},"headline":"A145324 \u2192 basis A007318 \u2192 (\u03b8+1)\u00b2","chain":null}