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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":"e24112f59e696d2a892e8c4795ad6ac666a409b5bc0716b760d3ec1596d81a2e","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-c5818ec04781ce60","role":"sequence","name":"basis_map_A007318 of Triangle read by rows, Stirling cycle numbers of order 2, T(n, n) = 1, T(n, k) = 0 if k < 0 or k > n, otherwise T(n, k) = T(n-1, k-1) + (n-1)^2*T(n-1, k), for 0 <= k <= 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:          1,           4;\n  2:          2,          12,           9;\n  3:         10,          68,          72,          16;\n  4:        100,         720,         873,         288,         25;\n  5:       1700,       12640,       16461,        6448,        875,        36;\n  6:      44200,      335440,      456426,      196912,      32825,      2196,       49;\n  7:    1635400,    12588080,    17642502,     8097168,    1522200,    128520,     4802,      64;\n  8:   81770000,   635945600,   910448280,   436222848,   88761825,   8617968,   415030,    9472,    81;\n  9: 5315050000, 41663544000, 60610015800, 29973059840, 6451116825, 687984948, 38706962, 1157760, 17253, 100;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^4-8*n^3+26*n^2-40*n+25+(4*n^2-16*n+20)*x-9*x^2)*P(n-2,x) + ((4*n^2-16*n+20)*x^2+4*x^3)*d/dx P(n-2,x) + (2*n^2-6*n+7+4*x)*P(n-1,x) - 4*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 4*x;\nP[n_, x_] := P[n, x] = Expand[-(n^4 - 8*n^3 + 26*n^2 - 40*n + 25 + (4*n^2 - 16*n + 20)*x - 9*x^2)*P[n - 2, x] + ((4*n^2 - 16*n + 20)*x^2 + 4*x^3)*D[P[n - 2, x], x] + (2*n^2 - 6*n + 7 + 4*x)*P[n - 1, x] - 4*x^2*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^4 - 8*n^3 + 26*n^2 - 40*n + 25 + (4*n^2 - 16*n + 20)*x - 9*x^2)*v[i - 2] + ((4*n^2 - 16*n + 20)*x^2 + 4*x^3)*deriv(v[i - 2], x) + (2*n^2 - 6*n + 7 + 4*x)*v[i - 1] - 4*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 - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-54284d24a094e2d3","steps":["x \u21a6 x+1","(\u03b8+1)\u00b2"],"edges":["e-0cfc062841fe3ed0","e-2d747162443755ab"],"text":"A269944 \u2192 x \u21a6 x+1 \u2192 (\u03b8+1)\u00b2"},"headline":"A269944 \u2192 x \u21a6 x+1 \u2192 (\u03b8+1)\u00b2","chain":null}