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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":"a756d0a05a90a6e953b383cea5a44b7b475a362e42a209c92281d4b15b760d16","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-ecc015942a4103af","role":"sequence","name":"basis_map_A013610 of T(n,k) = binomial(n,k)*binomial(n+k,k), 0 <= k <= n, triangle read by rows.","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,       12;\n   3:      13,      108,       162;\n   4:      63,      792,      2430,       2160;\n   5:     321,     5400,     25110,      45360,       28350;\n   6:    1683,    35460,    221130,     604800,      765450,      367416;\n   7:    8989,   227556,   1780380,    6531840,    12502350,    12124728,     4715172;\n   8:   48639,  1437744,  13533156,   62324640,   159979050,   234411408,   183891708,    60046272;\n   9:  265729,  8982576,  98828100,  547858080,  1765382850,  3467672208,  4106914812,  2702082240,   759960630;\n  10: 1462563, 55646460, 700700220, 4542531840, 17622955350, 43450983576, 68959390500, 68452750080, 38757992130, 9569874600;\n","recurrence_sentence":"The row polynomials satisfy -(12*n-12+(27*n-27)*x-(27*n^3-108*n^2+135*n-54)*x^2)*P(n,x) = -(36*n-36+(36*n^2+117*n-153)*x-(81*n^3-324*n^2+81*n+162)*x^2-(162*n^3-486*n^2+324*n)*x^3)*P(n-1,x) - (8+72*x-(36*n^2-72*n-198)*x^2-(162*n^2-324*n-162)*x^3-(162*n^2-324*n)*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[(-(36*n - 36 + (36*n^2 + 117*n - 153)*x - (81*n^3 - 324*n^2 + 81*n + 162)*x^2 - (162*n^3 - 486*n^2 + 324*n)*x^3)*P[n - 1, x] - (8 + 72*x - (36*n^2 - 72*n - 198)*x^2 - (162*n^2 - 324*n - 162)*x^3 - (162*n^2 - 324*n)*x^4)*D[P[n - 1, x], x])/(-12*n + 12 - (27*n - 27)*x + (27*n^3 - 108*n^2 + 135*n - 54)*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] = (-(36*n - 36 + (36*n^2 + 117*n - 153)*x - (81*n^3 - 324*n^2 + 81*n + 162)*x^2 - (162*n^3 - 486*n^2 + 324*n)*x^3)*v[i - 1] - (8 + 72*x - (36*n^2 - 72*n - 198)*x^2 - (162*n^2 - 324*n - 162)*x^3 - (162*n^2 - 324*n)*x^4)*deriv(v[i - 1], x))/(-12*n + 12 - (27*n - 27)*x + (27*n^3 - 108*n^2 + 135*n - 54)*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-4689fb1386d440ad","steps":["basis A013610","\u03b8+1"],"edges":["e-50c3f3c57fe6951a","e-082aa89876e21fe4"],"text":"A063007 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"A063007 \u2192 basis A013610 \u2192 \u03b8+1","chain":null}