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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":"520ad68996911a52cd86e7f60f089d74ddf973bf279b668af522b7211683f5ef","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-ec08416ae48317c3","role":"sequence","name":"shift_x_plus_one of Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:    1;\n   2:    1,     2;\n   3:    3,     8,     3;\n   4:    7,    26,    21,     4;\n   5:   17,    80,    96,    40,     5;\n   6:   41,   234,   372,   240,    65,     6;\n   7:   99,   664,  1311,  1144,   485,    96,     7;\n   8:  239,  1842,  4341,  4772,  2765,   858,   133,    8;\n   9:  577,  5024, 13752, 18224, 13410,  5712,  1386,  176,   9;\n  10: 1393, 13522, 42144, 65344, 58330, 31692, 10570, 2096, 225, 10;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(1+2*x+x^2)*P(n-4,x) - (4+6*x+2*x^2)*P(n-3,x) - (2+2*x+x^2)*P(n-2,x) + (4+2*x)*P(n-1,x) for n >= 5, with P(1,x) = 1, P(2,x) = 1 + 2*x, P(3,x) = 3 + 8*x + 3*x^2, P(4,x) = 7 + 26*x + 21*x^2 + 4*x^3.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 1 + 2*x;\nP[3, x_] := 3 + 8*x + 3*x^2;\nP[4, x_] := 7 + 26*x + 21*x^2 + 4*x^3;\nP[n_, x_] := P[n, x] = Expand[-(1 + 2*x + x^2)*P[n - 4, x] - (4 + 6*x + 2*x^2)*P[n - 3, x] - (2 + 2*x + x^2)*P[n - 2, x] + (4 + 2*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 2*x; v[3] = 3 + 8*x + 3*x^2; v[4] = 7 + 26*x + 21*x^2 + 4*x^3; for(n = 5, N, my(i = n); v[i] = -(1 + 2*x + x^2)*v[i - 4] - (4 + 6*x + 2*x^2)*v[i - 3] - (2 + 2*x + x^2)*v[i - 2] + (4 + 2*x)*v[i - 1]); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-b0fc620949fb5990","steps":["x \u21a6 x+1","\u03b8+1"],"edges":["e-3a1dbaeb4ca6a6b2","e-f4f0c9802e8e1974"],"text":"A122542 \u2192 x \u21a6 x+1 \u2192 \u03b8+1"},"headline":"A122542 \u2192 x \u21a6 x+1 \u2192 \u03b8+1","chain":null}