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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":"01b967d803229d57cf882e5b7794e5d663280e14cd8276d4e089da8d8bf375ae","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-d5dd236ae5fa65ba","role":"sequence","name":"basis_map_A013609 of Triangle read by rows: T(n,k) number of tilings of a 2n X 3 grid by dominoes, 2k of which are in a vertical position (0<=k<=n).","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:      11,       56,        48;\n  1:      41,      304,       528,        256;\n  2:     153,     1488,      3888,       3840,       1280;\n  3:     571,     6872,     23952,      35840,      24320,       6144;\n  4:    2131,    30568,    133152,     267264,     275200,     141312,       28672;\n  5:    7953,   132448,    692064,    1743360,    2412800,    1880064,      774144,     131072;\n  6:   29681,   562848,   3429792,   10397184,   18092800,   18960384,    11841536,    4063232,     589824;\n  7:  110771,  2356392,  16404192,   58137600,  121944320,  160806912,   135073792,   70254592,   20643840,    2621440;\n  8:  413403,  9748888,  76325328,  309642240,  760372480, 1209600000,  1278885888,  894435328,  398131200,  102236160,  11534336;\n  9: 1542841, 39946384, 347390448, 1587499776, 4467756800, 8327380992, 10625269760, 9353822208, 5596839936, 2175795200, 495976448, 50331648;\n","recurrence_sentence":"The row polynomials satisfy -(12*n+18+(20*n+30)*x-(8*n^3+36*n^2+52*n+24)*x^2)*P(n,x) = -(24*n+42+(12*n^2+124*n+190)*x-(16*n^3+76*n^2+36*n-104)*x^2-(16*n^3+96*n^2+176*n+96)*x^3)*P(n-1,x) - (9+54*x-(12*n^2+48*n-56)*x^2-(32*n^2+128*n+56)*x^3-(16*n^2+64*n+48)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 11 + 56*x + 48*x^2.","mathematica":"P[0, x_] := 11 + 56*x + 48*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(24*n + 42 + (12*n^2 + 124*n + 190)*x - (16*n^3 + 76*n^2 + 36*n - 104)*x^2 - (16*n^3 + 96*n^2 + 176*n + 96)*x^3)*P[n - 1, x] - (9 + 54*x - (12*n^2 + 48*n - 56)*x^2 - (32*n^2 + 128*n + 56)*x^3 - (16*n^2 + 64*n + 48)*x^4)*D[P[n - 1, x], x])/(-12*n - 18 - (20*n + 30)*x + (8*n^3 + 36*n^2 + 52*n + 24)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 11 + 56*x + 48*x^2; for(n = 1, N, my(i = n + 1); v[i] = (-(24*n + 42 + (12*n^2 + 124*n + 190)*x - (16*n^3 + 76*n^2 + 36*n - 104)*x^2 - (16*n^3 + 96*n^2 + 176*n + 96)*x^3)*v[i - 1] - (9 + 54*x - (12*n^2 + 48*n - 56)*x^2 - (32*n^2 + 128*n + 56)*x^3 - (16*n^2 + 64*n + 48)*x^4)*deriv(v[i - 1], x))/(-12*n - 18 - (20*n + 30)*x + (8*n^3 + 36*n^2 + 52*n + 24)*x^2)); 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-8eb9bb3a09f0b7fd","steps":["basis A013609","\u03b8+1"],"edges":["e-125a67b606fd2347","e-96c14534a419d5b4"],"text":"A123519 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A123519 \u2192 basis A013609 \u2192 \u03b8+1","chain":null}