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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":"6d0c2af87a15d96e21511ee948c5fcfaa017f9f2da3ce29f7015553f83c213d1","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-2779422d3a9a18c1","role":"sequence","name":"shift_x_plus_one of Triangle read by rows: T(n,k) = number of k-part order-consecutive partition of {1,2,...,n} (1 <= k <= n).","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:     3,      4;\n   3:    10,     26,      12;\n   4:    34,    130,     120,      32;\n   5:   116,    584,     807,     432,      80;\n   6:   396,   2472,    4545,    3656,    1360,     192;\n   7:  1352,  10072,   23106,   24804,   13880,    3936,     448;\n   8:  4616,  39960,  109842,  147460,  110240,   47040,   10752,   1024;\n   9: 15760, 155456,  497928,  802336,  750845,  428208,  147280,  28160,  2304;\n  10: 53808, 595712, 2178456, 4095776, 4604415, 3288492, 1511888, 434944, 71424, 5120;\n","recurrence_sentence":"The row polynomials satisfy -(32*n-32+(8*n^2+20*n-28)*x-(n^3-10*n^2+9*n)*x^2)*P(n,x) = -(32*n+32+(12*n^2+48*n+116)*x-(n^3-8*n^2-57*n-80)*x^2-(4*n^2-36*n)*x^3)*P(n-1,x) - (64+(32*n+176)*x-(4*n^2-84*n-152)*x^2-(6*n^2-70*n-40)*x^3-(2*n^2-18*n)*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(32*n + 32 + (12*n^2 + 48*n + 116)*x - (n^3 - 8*n^2 - 57*n - 80)*x^2 - (4*n^2 - 36*n)*x^3)*P[n - 1, x] - (64 + (32*n + 176)*x - (4*n^2 - 84*n - 152)*x^2 - (6*n^2 - 70*n - 40)*x^3 - (2*n^2 - 18*n)*x^4)*D[P[n - 1, x], x])/(-32*n + 32 - (8*n^2 + 20*n - 28)*x + (n^3 - 10*n^2 + 9*n)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(32*n + 32 + (12*n^2 + 48*n + 116)*x - (n^3 - 8*n^2 - 57*n - 80)*x^2 - (4*n^2 - 36*n)*x^3)*v[i - 1] - (64 + (32*n + 176)*x - (4*n^2 - 84*n - 152)*x^2 - (6*n^2 - 70*n - 40)*x^3 - (2*n^2 - 18*n)*x^4)*deriv(v[i - 1], x))/(-32*n + 32 - (8*n^2 + 20*n - 28)*x + (n^3 - 10*n^2 + 9*n)*x^2)); 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-2821c21537bbbcdd","steps":["x \u21a6 x+1","\u03b8+1"],"edges":["e-991bb8f582af0f63","e-8b4bc884d54417b2"],"text":"A056242 \u2192 x \u21a6 x+1 \u2192 \u03b8+1"},"headline":"A056242 \u2192 x \u21a6 x+1 \u2192 \u03b8+1","chain":null}