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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":"8155f6cdaa01a253ded49fe87ea461104e58ecbad8b191be54c645d2ef423241","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-a893b3f9716e8c07","role":"sequence","name":"shift_x_plus_one of Coefficient triangle of certain polynomials N(4; m,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;\n  1:      6,      10;\n  2:     28,      84,       45;\n  3:    120,     504,      504,      140;\n  4:    495,    2640,     3780,     2016,      350;\n  5:   2002,   12870,    23760,    18480,     6300,      756;\n  6:   8008,   60060,   135135,   137280,    69300,    16632,    1470;\n  7:  31824,  272272,   720720,   900900,   600600,   216216,   38808,    2640;\n  8: 125970, 1209312,  3675672,  5445440,  4504500,  2162160,  588588,   82368,   4455;\n  9: 497420, 5290740, 18139680, 31039008, 30630600, 18378360, 6726720, 1441440, 162162, 7150;\n","recurrence_sentence":"The row polynomials satisfy (2*n^3+12*n^2+16*n+(n^4+8*n^3+16*n^2)*x)*P(n,x) = (8*n^3+44*n^2+80*n+48+(4*n^4+42*n^3+132*n^2+176*n+96)*x+(3*n^4+26*n^3+73*n^2+100*n+48)*x^2)*P(n-1,x) - ((4*n^2+16*n+16)*x+(2*n^3+19*n^2+44*n+32)*x^2+(2*n^3+15*n^2+28*n+16)*x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((8*n^3 + 44*n^2 + 80*n + 48 + (4*n^4 + 42*n^3 + 132*n^2 + 176*n + 96)*x + (3*n^4 + 26*n^3 + 73*n^2 + 100*n + 48)*x^2)*P[n - 1, x] - ((4*n^2 + 16*n + 16)*x + (2*n^3 + 19*n^2 + 44*n + 32)*x^2 + (2*n^3 + 15*n^2 + 28*n + 16)*x^3)*D[P[n - 1, x], x])/(2*n^3 + 12*n^2 + 16*n + (n^4 + 8*n^3 + 16*n^2)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((8*n^3 + 44*n^2 + 80*n + 48 + (4*n^4 + 42*n^3 + 132*n^2 + 176*n + 96)*x + (3*n^4 + 26*n^3 + 73*n^2 + 100*n + 48)*x^2)*v[i - 1] - ((4*n^2 + 16*n + 16)*x + (2*n^3 + 19*n^2 + 44*n + 32)*x^2 + (2*n^3 + 15*n^2 + 28*n + 16)*x^3)*deriv(v[i - 1], x))/(2*n^3 + 12*n^2 + 16*n + (n^4 + 8*n^3 + 16*n^2)*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-9516b04042fe64c0","steps":["x \u21a6 x+1","\u03b8+1"],"edges":["e-06bdf5b2aa1ff3d5","e-8b27a131453c5c7f"],"text":"A062264 \u2192 x \u21a6 x+1 \u2192 \u03b8+1"},"headline":"A062264 \u2192 x \u21a6 x+1 \u2192 \u03b8+1","chain":null}