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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":"c43840dc057f63150ad3c7bdafc2bfd3cc85545576135e1d1905694058e38e33","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-47b361d1e904fbeb","role":"sequence","name":"basis_map_A007318 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,        36;\n  1:      41,      304,       396,       128;\n  2:     153,     1488,      2916,      1920,        400;\n  3:     571,     6872,     17964,     17920,       7600,       1152;\n  4:    2131,    30568,     99864,    133632,      86000,      26496,       3136;\n  5:    7953,   132448,    519048,    871680,     754000,     352512,      84672,      8192;\n  6:   29681,   562848,   2572344,   5198592,    5654000,    3555072,    1295168,    253952,     20736;\n  7:  110771,  2356392,  12303144,  29068800,   38107600,   30151296,   14773696,   4390912,    725760,    51200;\n  8:  413403,  9748888,  57243996, 154821120,  237616400,  226800000,  139878144,  55902208,  13996800,  1996800,  123904;\n  9: 1542841, 39946384, 260542836, 793749888, 1396174000, 1561383936, 1162138880, 584613888, 196763904, 42496000, 5327872, 294912;\n","recurrence_sentence":"The row polynomials satisfy (72*n+108+(102*n+153)*x-(60*n^3+270*n^2+364*n+141)*x^2-(38*n^3+171*n^2+257*n+129)*x^3+(2*n^5+15*n^4+44*n^3+63*n^2+44*n+12)*x^4)*P(n,x) = (144*n+252+(126*n^2+870*n+1185)*x-(120*n^3+528*n^2+284*n-571)*x^2-(12*n^4+250*n^3+1135*n^2+1751*n+765)*x^3+(4*n^5+31*n^4+28*n^3-257*n^2-666*n-448)*x^4+(2*n^5+20*n^4+76*n^3+136*n^2+114*n+36)*x^5)*P(n-1,x) + (54+270*x-(126*n^2+504*n+36)*x^2-(282*n^2+1128*n+760)*x^3+(6*n^4+48*n^3-62*n^2-632*n-588)*x^4+(8*n^4+64*n^3+138*n^2+40*n-62)*x^5+(2*n^4+16*n^3+44*n^2+48*n+18)*x^6)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 11 + 56*x + 36*x^2.","mathematica":"P[0, x_] := 11 + 56*x + 36*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((144*n + 252 + (126*n^2 + 870*n + 1185)*x - (120*n^3 + 528*n^2 + 284*n - 571)*x^2 - (12*n^4 + 250*n^3 + 1135*n^2 + 1751*n + 765)*x^3 + (4*n^5 + 31*n^4 + 28*n^3 - 257*n^2 - 666*n - 448)*x^4 + (2*n^5 + 20*n^4 + 76*n^3 + 136*n^2 + 114*n + 36)*x^5)*P[n - 1, x] + (54 + 270*x - (126*n^2 + 504*n + 36)*x^2 - (282*n^2 + 1128*n + 760)*x^3 + (6*n^4 + 48*n^3 - 62*n^2 - 632*n - 588)*x^4 + (8*n^4 + 64*n^3 + 138*n^2 + 40*n - 62)*x^5 + (2*n^4 + 16*n^3 + 44*n^2 + 48*n + 18)*x^6)*D[P[n - 1, x], x])/(72*n + 108 + (102*n + 153)*x - (60*n^3 + 270*n^2 + 364*n + 141)*x^2 - (38*n^3 + 171*n^2 + 257*n + 129)*x^3 + (2*n^5 + 15*n^4 + 44*n^3 + 63*n^2 + 44*n + 12)*x^4)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 11 + 56*x + 36*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((144*n + 252 + (126*n^2 + 870*n + 1185)*x - (120*n^3 + 528*n^2 + 284*n - 571)*x^2 - (12*n^4 + 250*n^3 + 1135*n^2 + 1751*n + 765)*x^3 + (4*n^5 + 31*n^4 + 28*n^3 - 257*n^2 - 666*n - 448)*x^4 + (2*n^5 + 20*n^4 + 76*n^3 + 136*n^2 + 114*n + 36)*x^5)*v[i - 1] + (54 + 270*x - (126*n^2 + 504*n + 36)*x^2 - (282*n^2 + 1128*n + 760)*x^3 + (6*n^4 + 48*n^3 - 62*n^2 - 632*n - 588)*x^4 + (8*n^4 + 64*n^3 + 138*n^2 + 40*n - 62)*x^5 + (2*n^4 + 16*n^3 + 44*n^2 + 48*n + 18)*x^6)*deriv(v[i - 1], x))/(72*n + 108 + (102*n + 153)*x - (60*n^3 + 270*n^2 + 364*n + 141)*x^2 - (38*n^3 + 171*n^2 + 257*n + 129)*x^3 + (2*n^5 + 15*n^4 + 44*n^3 + 63*n^2 + 44*n + 12)*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-8eb9bb3a09f0b7fd","steps":["basis A007318","(\u03b8+1)\u00b2"],"edges":["e-1acefd6886502727","e-9b09199f50ce6232"],"text":"A123519 \u2192 basis A007318 \u2192 (\u03b8+1)\u00b2"},"headline":"A123519 \u2192 basis A007318 \u2192 (\u03b8+1)\u00b2","chain":null}