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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":"2a294cf616e3318394747708df18d9d4aa77a188a304e5d1c364af76ba48b440","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-eb3f8d5fd252d0c9","role":"sequence","name":"basis_map_A013610 of basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A008288) regarded as a triangle. shifted by x -> -x + 0","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: 1,  24,    81;\n  2: 1,  48,   486,   1728,    2025;\n  3: 1,  72,  1215,   8640,   30375,    52488,     35721;\n  4: 1,  96,  2268,  24192,  141750,   489888,   1000188,    1119744,      531441;\n  5: 1, 120,  3645,  51840,  425250,  2204496,   7501410,   16796160,    23914845,    19683000,      7144929;\n  6: 1, 144,  5346,  95040, 1002375,  6928416,  33006204,  110854656,   263063295,   433026000,    471565314,    306110016,      89813529;\n  7: 1, 168,  7371, 157248, 2027025, 17513496, 107270163,  480370176,  1595917323,  3940536600,   7152073929,   9285337152,    8173031139,    4374822312,    1076168025;\n  8: 1, 192,  9720, 241920, 3685500, 38211264, 286053768, 1601233920,  6839645670, 22517352000,  57216591432, 111424045824,  163460622780,  174992892480,  129140163000,   58773123072,   12440502369;\n  9: 1, 216, 12393, 352512, 6196500, 74952864, 663124644, 4454341632, 23254795278, 95698746000, 312647803182, 811803762432, 1667298352356, 2677391254944, 3293074156500, 2997429276672, 1903396862457, 753145430616, 139858796529;\n","recurrence_sentence":"The row polynomials satisfy (1+(18*n-12)*x+(36*n^2-36*n+9)*x^2)*P(n,x) = (1+(18*n+12)*x+(36*n^2+144*n+54)*x^2+(216*n^2+378*n+108)*x^3+(324*n^2+324*n+81)*x^4)*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[((1 + (18*n + 12)*x + (36*n^2 + 144*n + 54)*x^2 + (216*n^2 + 378*n + 108)*x^3 + (324*n^2 + 324*n + 81)*x^4)*P[n - 1, x])/(1 + (18*n - 12)*x + (36*n^2 - 36*n + 9)*x^2)]];\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] = ((1 + (18*n + 12)*x + (36*n^2 + 144*n + 54)*x^2 + (216*n^2 + 378*n + 108)*x^3 + (324*n^2 + 324*n + 81)*x^4)*v[i - 1])/(1 + (18*n - 12)*x + (36*n^2 - 36*n + 9)*x^2)); 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-cdb0db4ad473ee28","steps":["basis A013610","(\u03b8+1)\u00b2"],"edges":["e-4aebfd11077ddb65","e-851c7e2755c4f1ec"],"text":"basis-map coefficients of A297191, shifted \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2"},"headline":"basis-map coefficients of A297191, shifted \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2","chain":null}