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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":"10871d71fe3f566bef2a00b0cb75a69235d55862779b624c4b634c536259fa6f","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-141ab74a8d7e58b7","role":"sequence","name":"shift_x_plus_one of Triangle read by rows: T(n, k) = [z^k] P(n, z) where P(n, z) = Sum_{k=0..n} binomial(n, k) * Pochhammer(n - k + c, k) * z^k / k! and c = 4.","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:      5,       8;\n   3:     21,      60,      30;\n   4:     84,     336,     315,       80;\n   5:    330,    1680,    2268,     1120,      175;\n   6:   1287,    7920,   13860,    10080,     3150,     336;\n   7:   5005,   36036,   77220,    73920,    34650,    7560,     588;\n   8:  19448,  160160,  405405,   480480,   300300,   99792,   16170,    960;\n   9:  75582,  700128, 2042040,  2882880,  2252250, 1009008,  252252,  31680,  1485;\n  10: 293930, 3023280, 9976824, 16336320, 15315300, 8648640, 2942940, 576576, 57915, 2200;\n","recurrence_sentence":"The row polynomials satisfy (2*n^3+3*n^2-3*n-2+(n^4+2*n^3-3*n^2-4*n+4)*x)*P(n,x) = (8*n^3+8*n^2+2*n+(4*n^4+18*n^3+4*n^2+6*n+4)*x+(3*n^4+8*n^3+9*n-2)*x^2)*P(n-1,x) - ((4*n^2+4*n+1)*x+(2*n^3+10*n^2+4*n+2)*x^2+(2*n^3+6*n^2+1)*x^3)*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[((8*n^3 + 8*n^2 + 2*n + (4*n^4 + 18*n^3 + 4*n^2 + 6*n + 4)*x + (3*n^4 + 8*n^3 + 9*n - 2)*x^2)*P[n - 1, x] - ((4*n^2 + 4*n + 1)*x + (2*n^3 + 10*n^2 + 4*n + 2)*x^2 + (2*n^3 + 6*n^2 + 1)*x^3)*D[P[n - 1, x], x])/(2*n^3 + 3*n^2 - 3*n - 2 + (n^4 + 2*n^3 - 3*n^2 - 4*n + 4)*x)]];\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] = ((8*n^3 + 8*n^2 + 2*n + (4*n^4 + 18*n^3 + 4*n^2 + 6*n + 4)*x + (3*n^4 + 8*n^3 + 9*n - 2)*x^2)*v[i - 1] - ((4*n^2 + 4*n + 1)*x + (2*n^3 + 10*n^2 + 4*n + 2)*x^2 + (2*n^3 + 6*n^2 + 1)*x^3)*deriv(v[i - 1], x))/(2*n^3 + 3*n^2 - 3*n - 2 + (n^4 + 2*n^3 - 3*n^2 - 4*n + 4)*x)); 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-a2304b34e998691c","steps":["x \u21a6 x+1","\u03b8+1"],"edges":["e-1067be4ea4032d33","e-7fe116191fd89191"],"text":"A062145 \u2192 x \u21a6 x+1 \u2192 \u03b8+1"},"headline":"A062145 \u2192 x \u21a6 x+1 \u2192 \u03b8+1","chain":null}