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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":"686afa9141d73244e1b9113f224a59062c6a26012a913b5b9713c3950c1a2156","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-f00f6de2d4bbc6f5","role":"sequence","name":"Triangle read by rows: the reversed x = 1+q Narayana triangle at m=2.","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:      12,       32,        9;\n   4:      55,      220,      135,       16;\n   5:     273,     1456,     1404,      384,       25;\n   6:    1428,     9520,    12600,     5600,      875,       36;\n   7:    7752,    62016,   104652,    65280,    17000,     1728,      49;\n   8:   43263,   403788,   829521,   670320,   249375,    43092,    3087,     64;\n   9:  246675,  2631200,  6375600,  6347264,  3099250,   776160,   96040,   5120,   81;\n  10: 1430715, 17168580, 47953620, 56833920, 34534500, 11476080, 2082696, 194304, 8019, 100;\n","recurrence_sentence":"The row polynomials satisfy (36*n^4-6*n^3-8*n^2+2*n+(36*n^5-50*n^4-6*n^3+10*n^2-2*n)*x+(4*n^6-6*n^5+2*n^3)*x^2)*P(n,x) = (243*n^4-405*n^3+243*n^2-63*n+6+(243*n^5-270*n^4-162*n^3+258*n^2-91*n+10)*x+(27*n^6+99*n^5-273*n^4+87*n^3+134*n^2-86*n+12)*x^2+(12*n^6-8*n^5-24*n^4+24*n^3+6*n^2-14*n+4)*x^3)*P(n-1,x) - ((81*n^3-81*n^2+27*n-3)*x+(81*n^4-72*n^2+38*n-5)*x^2+(9*n^5+75*n^4-55*n^3-31*n^2+32*n-6)*x^3+(8*n^5+6*n^4-12*n^3+6*n-2)*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[((243*n^4 - 405*n^3 + 243*n^2 - 63*n + 6 + (243*n^5 - 270*n^4 - 162*n^3 + 258*n^2 - 91*n + 10)*x + (27*n^6 + 99*n^5 - 273*n^4 + 87*n^3 + 134*n^2 - 86*n + 12)*x^2 + (12*n^6 - 8*n^5 - 24*n^4 + 24*n^3 + 6*n^2 - 14*n + 4)*x^3)*P[n - 1, x] - ((81*n^3 - 81*n^2 + 27*n - 3)*x + (81*n^4 - 72*n^2 + 38*n - 5)*x^2 + (9*n^5 + 75*n^4 - 55*n^3 - 31*n^2 + 32*n - 6)*x^3 + (8*n^5 + 6*n^4 - 12*n^3 + 6*n - 2)*x^4)*D[P[n - 1, x], x])/(36*n^4 - 6*n^3 - 8*n^2 + 2*n + (36*n^5 - 50*n^4 - 6*n^3 + 10*n^2 - 2*n)*x + (4*n^6 - 6*n^5 + 2*n^3)*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] = ((243*n^4 - 405*n^3 + 243*n^2 - 63*n + 6 + (243*n^5 - 270*n^4 - 162*n^3 + 258*n^2 - 91*n + 10)*x + (27*n^6 + 99*n^5 - 273*n^4 + 87*n^3 + 134*n^2 - 86*n + 12)*x^2 + (12*n^6 - 8*n^5 - 24*n^4 + 24*n^3 + 6*n^2 - 14*n + 4)*x^3)*v[i - 1] - ((81*n^3 - 81*n^2 + 27*n - 3)*x + (81*n^4 - 72*n^2 + 38*n - 5)*x^2 + (9*n^5 + 75*n^4 - 55*n^3 - 31*n^2 + 32*n - 6)*x^3 + (8*n^5 + 6*n^4 - 12*n^3 + 6*n - 2)*x^4)*deriv(v[i - 1], x))/(36*n^4 - 6*n^3 - 8*n^2 + 2*n + (36*n^5 - 50*n^4 - 6*n^3 + 10*n^2 - 2*n)*x + (4*n^6 - 6*n^5 + 2*n^3)*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-f00f6de2d4bbc6f5","steps":["(\u03b8+1)\u00b2"],"edges":["e-0c136f29e2dce1bd"],"text":"A243662 \u2192 (\u03b8+1)\u00b2"},"headline":"A243662 \u2192 (\u03b8+1)\u00b2","chain":null}