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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":"92d9318d6c10270a111e87adcf05e001c87eb073eca013409d293f27b1066799","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-ca8f77e380f6014f","role":"sequence","name":"basis_map_A013610 of Riordan array (1-x,x/(1-x)).","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,    24,     81;\n  1:   2,    60,    324,     432;\n  2:   4,   144,   1053,    2592,     2025;\n  3:   8,   336,   3078,   10800,    16200,      8748;\n  4:  16,   768,   8424,   38016,    83025,     87480,     35721;\n  5:  32,  1728,  22032,  120960,   344250,    533628,    428652,    139968;\n  6:  64,  3840,  55728,  359424,  1255500,   2554416,   3036285,   1959552,    531441;\n  7: 128,  8448, 137376, 1016064,  4195800,  10532592,  16503102,  15816384,   8503056,   1968300;\n  8: 256, 18432, 331776, 2764800, 13154400,  39191040,  76014288,  96297984,  77058945,  35429400,   7144929;\n  9: 512, 39936, 787968, 7299072, 39268800, 135209088, 312058656, 490447872, 519749298, 356262300, 142898580, 25509168;\n","recurrence_sentence":"The row polynomials satisfy (4+(18*n+90)*x+(9*n^2+180*n+432)*x^2+(54*n^2+486*n+648)*x^3+(81*n^2+324*n+324)*x^4)*P(n,x) = (8+(36*n+228)*x+(18*n^2+450*n+1566)*x^2+(135*n^2+1782*n+4239)*x^3+(324*n^2+2754*n+5022)*x^4+(243*n^2+1458*n+2187)*x^5)*P(n-1,x) for n >= 1, with P(0,x) = 1 + 24*x + 81*x^2.","mathematica":"P[0, x_] := 1 + 24*x + 81*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((8 + (36*n + 228)*x + (18*n^2 + 450*n + 1566)*x^2 + (135*n^2 + 1782*n + 4239)*x^3 + (324*n^2 + 2754*n + 5022)*x^4 + (243*n^2 + 1458*n + 2187)*x^5)*P[n - 1, x])/(4 + (18*n + 90)*x + (9*n^2 + 180*n + 432)*x^2 + (54*n^2 + 486*n + 648)*x^3 + (81*n^2 + 324*n + 324)*x^4)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1 + 24*x + 81*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((8 + (36*n + 228)*x + (18*n^2 + 450*n + 1566)*x^2 + (135*n^2 + 1782*n + 4239)*x^3 + (324*n^2 + 2754*n + 5022)*x^4 + (243*n^2 + 1458*n + 2187)*x^5)*v[i - 1])/(4 + (18*n + 90)*x + (9*n^2 + 180*n + 432)*x^2 + (54*n^2 + 486*n + 648)*x^3 + (81*n^2 + 324*n + 324)*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-7717563017cb9e5f","steps":["basis A013610","(\u03b8+1)\u00b2"],"edges":["e-02e979b2d9fb24b7","e-a69e39ca081e2942"],"text":"A159854 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2"},"headline":"A159854 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2","chain":null}