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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":"0c397c788a3f2bddb2bdfbbcdf6737f08ad40263753b1650137c326341088a0e","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-210f23d82ddfabd8","role":"sequence","name":"basis_map_A013609 of Triangle of coefficients in expansion of (1+4x)^n.","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:       5,       16;\n  2:      25,      160,       192;\n  3:     125,     1200,      2880,       2048;\n  4:     625,     8000,     28800,      40960,      20480;\n  5:    3125,    50000,    240000,     512000,     512000,      196608;\n  6:   15625,   300000,   1800000,    5120000,    7680000,     5898240,     1835008;\n  7:   78125,  1750000,  12600000,   44800000,   89600000,   103219200,    64225280,    16777216;\n  8:  390625, 10000000,  84000000,  358400000,  896000000,  1376256000,  1284505600,   671088640,  150994944;\n  9: 1953125, 56250000, 540000000, 2688000000, 8064000000, 15482880000, 19267584000, 15099494400, 6794772480, 1342177280;\n","recurrence_sentence":"The row polynomials satisfy (5+8*n*x)*P(n,x) = (25+(40*n+80)*x+(64*n+64)*x^2)*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[((25 + (40*n + 80)*x + (64*n + 64)*x^2)*P[n - 1, x])/(5 + 8*n*x)]];\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] = ((25 + (40*n + 80)*x + (64*n + 64)*x^2)*v[i - 1])/(5 + 8*n*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-01cea33b6485d9cc","steps":["basis A013611","\u03b8+1"],"edges":["e-391376c704e767eb","e-2ffd82b322178144"],"text":"A038220 \u2192 basis A013611 \u2192 \u03b8+1"},"headline":"A038220 \u2192 basis A013611 \u2192 \u03b8+1","chain":null}