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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":"50db5d8b28fb1038d7f9f18d971d4449bcb94d9fdf83b6e00f6ab2d29d0209db","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-83675781d9687218","role":"sequence","name":"basis_map_A013609 of Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^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:        6,        56,          72;\n  2:       36,       672,        2628,         3584,          1600;\n  3:      216,      6048,       39528,       108416,        146400,         96768,          25088;\n  4:     1296,     48384,      443232,      1827840,       4128400,       5483520,        4290048,        1835008,          331776;\n  5:     7776,    362880,     4276800,     23546880,      74604000,     149030784,      194965120,      167444480,        91238400,        28672000,         3964928;\n  6:    46656,   2612736,    37651392,    259338240,    1057752000,    2822045184,     5179445824,     6689292288,      6092651520,      3842048000,      1599848448,       396361728,        44302336;\n  7:   279936,  18289152,   311568768,   2576738304,   12856233600,   42924639744,   101437865088,   174981373952,    223577335296,    211973529600,    147494330368,     73293889536,     24654249984,      5035261952,      471859200;\n  8:  1679616, 125411328,  2465676288,  23800283136,  140427302400,  563110797312,  1628165790720,  3515235827712,   5783379243264,   7323407974400,   7147684741120,   5339124596736,   3000220647424,   1228603916288,   346344652800,    60129542144,    4848615424;\n  9: 10077696, 846526464, 18865446912, 208182804480, 1417987814400, 6638084296704, 22701283928064, 58872276320256, 118517522167296, 187840536934400, 236059591477248, 235489105281024, 185591146610688, 114222399750144, 53778500812800, 18715319992320, 4538304036864, 684913065984, 48452599808;\n","recurrence_sentence":"The row polynomials satisfy (9+(63*n-21)*x+(49*n^2+96*n-72)*x^2+(112*n^2-28*n-28)*x^3+(64*n^2-64*n+16)*x^4)*P(n,x) = (54+(378*n+378)*x+(294*n^2+2046*n+1098)*x^2+(1358*n^2+4396*n+1694)*x^3+(2344*n^2+4680*n+1464)*x^4+(1792*n^2+2464*n+672)*x^5+(512*n^2+512*n+128)*x^6)*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[((54 + (378*n + 378)*x + (294*n^2 + 2046*n + 1098)*x^2 + (1358*n^2 + 4396*n + 1694)*x^3 + (2344*n^2 + 4680*n + 1464)*x^4 + (1792*n^2 + 2464*n + 672)*x^5 + (512*n^2 + 512*n + 128)*x^6)*P[n - 1, x])/(9 + (63*n - 21)*x + (49*n^2 + 96*n - 72)*x^2 + (112*n^2 - 28*n - 28)*x^3 + (64*n^2 - 64*n + 16)*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] = ((54 + (378*n + 378)*x + (294*n^2 + 2046*n + 1098)*x^2 + (1358*n^2 + 4396*n + 1694)*x^3 + (2344*n^2 + 4680*n + 1464)*x^4 + (1792*n^2 + 2464*n + 672)*x^5 + (512*n^2 + 512*n + 128)*x^6)*v[i - 1])/(9 + (63*n - 21)*x + (49*n^2 + 96*n - 72)*x^2 + (112*n^2 - 28*n - 28)*x^3 + (64*n^2 - 64*n + 16)*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-eed4f744ce063213","steps":["basis A013609","(\u03b8+1)\u00b2"],"edges":["e-185113cbc4b6c77b","e-0af28c4c2c9dde5a"],"text":"A200536 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2"},"headline":"A200536 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2","chain":null}