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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":"44c7e494e421b56e365888dbb0ed987e0d1ada5f3ae477e436f39c2a31d381b9","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-fdb186db0412847e","role":"sequence","name":"basis_map_A013610 of Triangle of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/2).","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:         1;\n   2:         3,          24;\n   3:        15,         192,          324;\n   4:       105,        1704,         4860,         3456;\n   5:       945,       17856,        66744,        82944,        32400;\n   6:     10395,      219096,       975240,      1624320,      1134000,       279936;\n   7:    135135,     3097728,     15635916,     31518720,     29970000,     13436928,      2286144;\n   8:   2027025,    49709160,    276358068,    639563904,    745038000,    460494720,    144027072,    17915904;\n   9:  34459425,   893704320,   5369160816,  13820405760,  18661557600,  14265538560,   6209167104,  1433272320,   136048896;\n  10: 654729075, 17807408280, 114079063824, 319858758144, 484135898400, 432281090304, 234476073216, 75891769344, 13468840704, 1007769600;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(4*n^2-12*n+9+(48*n-72)*x-324*x^2)*P(n-2,x) + ((48*n-72)*x^2+144*x^3)*d/dx P(n-2,x) + (4*n-4+24*x)*P(n-1,x) - 24*x^2*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 3 + 24*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 3 + 24*x;\nP[n_, x_] := P[n, x] = Expand[-(4*n^2 - 12*n + 9 + (48*n - 72)*x - 324*x^2)*P[n - 2, x] + ((48*n - 72)*x^2 + 144*x^3)*D[P[n - 2, x], x] + (4*n - 4 + 24*x)*P[n - 1, x] - 24*x^2*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 3 + 24*x; for(n = 3, N, my(i = n); v[i] = -(4*n^2 - 12*n + 9 + (48*n - 72)*x - 324*x^2)*v[i - 2] + ((48*n - 72)*x^2 + 144*x^3)*deriv(v[i - 2], x) + (4*n - 4 + 24*x)*v[i - 1] - 24*x^2*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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-7c9c430277778c10","steps":["basis A013610","(\u03b8+1)\u00b2"],"edges":["e-1b98e7733a40a4e4","e-efc12e35b0336d2c"],"text":"A161198 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2"},"headline":"A161198 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2","chain":null}