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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":"0d9821dfb178e4a12042dc3db83e3651e33cb980776d4420333a3c03b31f683f","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-c16b107c87dd2b64","role":"sequence","name":"basis_map_A013610 of Riordan array ((1+x)/(1-2x),x(1+x)/(1-2x)).","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:     4,       6;\n  2:    13,      48,      27;\n  3:    43,     252,     324,      108;\n  4:   142,    1140,    2349,     1728,      405;\n  5:   469,    4782,   13635,    15984,     8100,     1458;\n  6:  1549,   19152,   69903,   113616,    91125,    34992,     5103;\n  7:  5116,   74346,  331047,   690984,   767475,   463644,   142884,   17496;\n  8: 16897,  282180, 1483785,  3790368,  5410800,  4516884,  2178981,  559872,   59049;\n  9: 55807, 1052964, 6386769, 19321416, 33804945, 36256086, 24111675, 9657792, 2125764, 196830;\n","recurrence_sentence":"The row polynomials satisfy -(78*n-234+(117*n^2-657*n-486)*x+(81*n^3-837*n^2-810*n-324)*x^2-(243*n^3+567*n^2-324*n)*x^3+(243*n^2+243*n)*x^4)*P(n,x) = -(624*n-1248+(1287*n^2-927*n-5400)*x+(648*n^3+1323*n^2-9045*n-6642)*x^2+(972*n^3-3807*n^2-10287*n-162)*x^3-(3159*n^2+1215*n-3402)*x^4+(1458*n+1458)*x^5)*P(n-1,x) + (169+(507*n+1092)*x+(351*n^2+3042*n+1971)*x^2+(1863*n^2+5373*n-378)*x^3+(2673*n^2+1863*n-4131)*x^4+(729*n^2-1944*n-3402)*x^5-(729*n+729)*x^6)*d/dx 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[(-(624*n - 1248 + (1287*n^2 - 927*n - 5400)*x + (648*n^3 + 1323*n^2 - 9045*n - 6642)*x^2 + (972*n^3 - 3807*n^2 - 10287*n - 162)*x^3 - (3159*n^2 + 1215*n - 3402)*x^4 + (1458*n + 1458)*x^5)*P[n - 1, x] + (169 + (507*n + 1092)*x + (351*n^2 + 3042*n + 1971)*x^2 + (1863*n^2 + 5373*n - 378)*x^3 + (2673*n^2 + 1863*n - 4131)*x^4 + (729*n^2 - 1944*n - 3402)*x^5 - (729*n + 729)*x^6)*D[P[n - 1, x], x])/(-78*n + 234 - (117*n^2 - 657*n - 486)*x - (81*n^3 - 837*n^2 - 810*n - 324)*x^2 + (243*n^3 + 567*n^2 - 324*n)*x^3 - (243*n^2 + 243*n)*x^4)]];\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] = (-(624*n - 1248 + (1287*n^2 - 927*n - 5400)*x + (648*n^3 + 1323*n^2 - 9045*n - 6642)*x^2 + (972*n^3 - 3807*n^2 - 10287*n - 162)*x^3 - (3159*n^2 + 1215*n - 3402)*x^4 + (1458*n + 1458)*x^5)*v[i - 1] + (169 + (507*n + 1092)*x + (351*n^2 + 3042*n + 1971)*x^2 + (1863*n^2 + 5373*n - 378)*x^3 + (2673*n^2 + 1863*n - 4131)*x^4 + (729*n^2 - 1944*n - 3402)*x^5 - (729*n + 729)*x^6)*deriv(v[i - 1], x))/(-78*n + 234 - (117*n^2 - 657*n - 486)*x - (81*n^3 - 837*n^2 - 810*n - 324)*x^2 + (243*n^3 + 567*n^2 - 324*n)*x^3 - (243*n^2 + 243*n)*x^4)); 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-d6b7687ba0ff59dd","steps":["basis A013610","\u03b8+1"],"edges":["e-dddeff0317c778fa","e-d78b6fff4e393247"],"text":"A116412 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"A116412 \u2192 basis A013610 \u2192 \u03b8+1","chain":null}