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(f(k))_(k >= 0) is a multiplier sequence, so x^k -> f(k) x^k maps every real-rooted polynomial to a real-rooted polynomial or zero (Laguerre; see T. 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":"84111a2556d950608494f9df250c2c68b516c73f2b12475b0b08055bca6a949c","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-07af2e6e3c9f0c77","role":"sequence","name":"basis_map_A007318 of Triangle read by rows: T(n, k) is the number of diagonal dissections of a convex n-gon into k+1 regions.","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:      3,        8;\n  2:     11,       60,       45;\n  3:     45,      372,      567,       224;\n  4:    197,     2184,     5040,      4032,      1050;\n  5:    903,    12560,    38880,     47040,     24750,      4752;\n  6:   4279,    71604,   278235,    451440,    358875,    138996,     21021;\n  7:  20793,   406700,  1902285,   3874640,   4129125,   2414412,    735735,    91520;\n  8: 103049,  2306832, 12618144,  30942912,  41391350,  32432400,  14910896,  3734016,   393822;\n  9: 518859, 13081752, 81931824, 235015872, 378500850, 372206016, 229228272, 86561280, 18366426, 1679600;\n","recurrence_sentence":"The row polynomials satisfy (16*n+32-(12*n+24)*x-(20*n^3+60*n^2+32*n-16)*x^2-(3*n^3+9*n^2+6*n)*x^3+(n^5+4*n^4+5*n^3+2*n^2)*x^4)*P(n,x) = (48*n+96+(56*n^2+148*n+72)*x-(60*n^3+204*n^2+216*n+96)*x^2-(8*n^4+93*n^3+287*n^2+322*n+112)*x^3+(3*n^5+12*n^4+3*n^3-42*n^2-60*n-24)*x^4+(2*n^5+12*n^4+28*n^3+32*n^2+18*n+4)*x^5)*P(n-1,x) + (16-(56*n^2+112*n+76)*x^2-(96*n^2+192*n+96)*x^3+(4*n^4+16*n^3-22*n^2-76*n-38)*x^4+(6*n^4+24*n^3+30*n^2+12*n)*x^5+(2*n^4+8*n^3+12*n^2+8*n+2)*x^6)*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[((48*n + 96 + (56*n^2 + 148*n + 72)*x - (60*n^3 + 204*n^2 + 216*n + 96)*x^2 - (8*n^4 + 93*n^3 + 287*n^2 + 322*n + 112)*x^3 + (3*n^5 + 12*n^4 + 3*n^3 - 42*n^2 - 60*n - 24)*x^4 + (2*n^5 + 12*n^4 + 28*n^3 + 32*n^2 + 18*n + 4)*x^5)*P[n - 1, x] + (16 - (56*n^2 + 112*n + 76)*x^2 - (96*n^2 + 192*n + 96)*x^3 + (4*n^4 + 16*n^3 - 22*n^2 - 76*n - 38)*x^4 + (6*n^4 + 24*n^3 + 30*n^2 + 12*n)*x^5 + (2*n^4 + 8*n^3 + 12*n^2 + 8*n + 2)*x^6)*D[P[n - 1, x], x])/(16*n + 32 - (12*n + 24)*x - (20*n^3 + 60*n^2 + 32*n - 16)*x^2 - (3*n^3 + 9*n^2 + 6*n)*x^3 + (n^5 + 4*n^4 + 5*n^3 + 2*n^2)*x^4)]];\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] = ((48*n + 96 + (56*n^2 + 148*n + 72)*x - (60*n^3 + 204*n^2 + 216*n + 96)*x^2 - (8*n^4 + 93*n^3 + 287*n^2 + 322*n + 112)*x^3 + (3*n^5 + 12*n^4 + 3*n^3 - 42*n^2 - 60*n - 24)*x^4 + (2*n^5 + 12*n^4 + 28*n^3 + 32*n^2 + 18*n + 4)*x^5)*v[i - 1] + (16 - (56*n^2 + 112*n + 76)*x^2 - (96*n^2 + 192*n + 96)*x^3 + (4*n^4 + 16*n^3 - 22*n^2 - 76*n - 38)*x^4 + (6*n^4 + 24*n^3 + 30*n^2 + 12*n)*x^5 + (2*n^4 + 8*n^3 + 12*n^2 + 8*n + 2)*x^6)*deriv(v[i - 1], x))/(16*n + 32 - (12*n + 24)*x - (20*n^3 + 60*n^2 + 32*n - 16)*x^2 - (3*n^3 + 9*n^2 + 6*n)*x^3 + (n^5 + 4*n^4 + 5*n^3 + 2*n^2)*x^4)); 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-66e2c3448245b0b7","steps":["x \u21a6 x+1","(\u03b8+1)\u00b2"],"edges":["e-190c283dd8b47413","e-0ca5f87f702acc81"],"text":"A033282 \u2192 x \u21a6 x+1 \u2192 (\u03b8+1)\u00b2"},"headline":"A033282 \u2192 x \u21a6 x+1 \u2192 (\u03b8+1)\u00b2","chain":null}