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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":"b2d3255503c878f137fd2b65e4e40a42bfa04e34418f3dc6bf74774d697346a6","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-90af30f849fc554d","role":"sequence","name":"Triangle read by rows: T(n,k) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*k-m+1)*T(n-1,k), where m = 8.","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: 1,         2;\n  2: 1,        36,           3;\n  3: 1,       358,         537,             4;\n  4: 1,      3272,       18258,          6544,             5;\n  5: 1,     29514,      433086,        577448,         73785,             6;\n  6: 1,    265708,     8823405,      28872400,      14705675,        797124,             7;\n  7: 1,   2391470,   166338927,    1110039820,    1387549775,     332677854,       8370145,            8;\n  8: 1,  21523344,  3003534804,   36844190176,   91578285150,   55266285264,    7008247876,     86093376,         9;\n  9: 1, 193710226, 52900337580, 1117335694928, 4910348156470, 5892417787764, 1955337466124, 141067566880, 871696017, 10;\n","recurrence_sentence":"The row polynomials satisfy (8*n-7)*P(n,x) = (56*n+7+(896*n^2-1568*n-210)*x-(512*n^3-1344*n^2+664*n+105)*x^2)*P(n-2,x) + ((448*n+56)*x-(512*n^2+64*n)*x^2+(512*n^2-384*n-56)*x^3)*d/dx P(n-2,x) - (48*n+14-(128*n^2-224*n+98)*x)*P(n-1,x) + ((64*n-56)*x-(64*n-56)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[((56*n + 7 + (896*n^2 - 1568*n - 210)*x - (512*n^3 - 1344*n^2 + 664*n + 105)*x^2)*P[n - 2, x] + ((448*n + 56)*x - (512*n^2 + 64*n)*x^2 + (512*n^2 - 384*n - 56)*x^3)*D[P[n - 2, x], x] - (48*n + 14 - (128*n^2 - 224*n + 98)*x)*P[n - 1, x] + ((64*n - 56)*x - (64*n - 56)*x^2)*D[P[n - 1, x], x])/(8*n - 7)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((56*n + 7 + (896*n^2 - 1568*n - 210)*x - (512*n^3 - 1344*n^2 + 664*n + 105)*x^2)*v[i - 2] + ((448*n + 56)*x - (512*n^2 + 64*n)*x^2 + (512*n^2 - 384*n - 56)*x^3)*deriv(v[i - 2], x) - (48*n + 14 - (128*n^2 - 224*n + 98)*x)*v[i - 1] + ((64*n - 56)*x - (64*n - 56)*x^2)*deriv(v[i - 1], x))/(8*n - 7)); 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-90af30f849fc554d","steps":["\u03b8+1"],"edges":["e-40e0af3027024d70"],"text":"A167884 \u2192 \u03b8+1"},"headline":"A167884 \u2192 \u03b8+1","chain":null}