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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":"9576eb2593d98ccc215e763a1d743eb23019d439c2efbc4d6197b366d9abba5e","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-4d9510ff3052f23c","role":"sequence","name":"basis_map_A013610 of Triangle of coefficients of (x+1)*(x+3)*(x+6)*...*(x+n(n+1)/2).","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:           2,            12;\n  2:           8,           120,            243;\n  3:          56,          1416,           6561,            7776;\n  4:         616,         22296,         167751,          435456,           364500;\n  5:        9856,        467616,        4941486,        20387376,         36450000,         23619600;\n  6:      216832,      12771264,      174997260,      1001968704,       2808782325,       3826375200,        2025380700;\n  7:     6288128,     443222208,     7488689436,     55190016576,     212963079825,     450715190832,      496218271500,      222213196800;\n  8:   232660736,   19115692992,   384784505676,   3479858985024,   17192949250725,   49796480235168,    84615209097804,    78219045273600,    30373766337600;\n  9: 10702393856, 1004958675072, 23506479007416, 252421794673344, 1524908420186850, 5632947425158668, 13042402861711104, 18517982339831040, 14761650440073600, 5062294389600000;\n","recurrence_sentence":"The row polynomials satisfy (4*n-4)*P(n,x) = -(n^5-n^4+3*n^3+n^2+4+(12*n^5-12*n^4+12*n^3+12*n^2-24*n)*x-(81*n^5-81*n^4-81*n^3+81*n^2)*x^2)*P(n-2,x) + ((12*n^5-12*n^4+12*n^3+12*n^2-24*n)*x^2+(36*n^5-36*n^4-36*n^3+36*n^2)*x^3)*d/dx P(n-2,x) + (4*n^3+4*n+(24*n^3-24*n)*x)*P(n-1,x) - (24*n^3-24*n)*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 12*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 12*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^5 - n^4 + 3*n^3 + n^2 + 4 + (12*n^5 - 12*n^4 + 12*n^3 + 12*n^2 - 24*n)*x - (81*n^5 - 81*n^4 - 81*n^3 + 81*n^2)*x^2)*P[n - 2, x] + ((12*n^5 - 12*n^4 + 12*n^3 + 12*n^2 - 24*n)*x^2 + (36*n^5 - 36*n^4 - 36*n^3 + 36*n^2)*x^3)*D[P[n - 2, x], x] + (4*n^3 + 4*n + (24*n^3 - 24*n)*x)*P[n - 1, x] - (24*n^3 - 24*n)*x^2*D[P[n - 1, x], x])/(4*n - 4)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 12*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^5 - n^4 + 3*n^3 + n^2 + 4 + (12*n^5 - 12*n^4 + 12*n^3 + 12*n^2 - 24*n)*x - (81*n^5 - 81*n^4 - 81*n^3 + 81*n^2)*x^2)*v[i - 2] + ((12*n^5 - 12*n^4 + 12*n^3 + 12*n^2 - 24*n)*x^2 + (36*n^5 - 36*n^4 - 36*n^3 + 36*n^2)*x^3)*deriv(v[i - 2], x) + (4*n^3 + 4*n + (24*n^3 - 24*n)*x)*v[i - 1] - (24*n^3 - 24*n)*x^2*deriv(v[i - 1], x))/(4*n - 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-ecd77b030b8fab94","steps":["basis A013610","(\u03b8+1)\u00b2"],"edges":["e-201bedfe8f6594c1","e-0715b42ac0535ab1"],"text":"A128813 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2"},"headline":"A128813 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2","chain":null}