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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":"3a30e26947cebf2f72d9657a7223338ffa080358d6dc92498cd11c83b431cee4","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-63d702ebcae1deea","role":"sequence","name":"basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^j.","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:            324,             3024,              3969;\n  1:           5832,            81648,            214326,             148176;\n  2:         104976,          1959552,           7715736,           10668672,            4862025;\n  3:        1889568,         44089920,         231472080,          480090240,          437582250,           147027636;\n  4:       34012224,        952342272,        6249746160,        17283248640,        23629441500,         15878984688,          4202539929;\n  5:      612220032,      19999187712,      157493603232,       544422332160,       992436543000,       1000376035344,        529520031054,        115269666624;\n  6:    11019960576,     411411861504,     3779846477568,     15679363166208,     35727715548000,      48018049696512,      38125442235888,      16598831993856,       3063651608241;\n  7:   198359290368,    8331090195456,    87476447052288,    423342805487616,   1157577983755200,    1944731012708736,    2058773880737952,    1344505391502336,     496311560535042,      79428004658100;\n  8:  3570467226624,  166621803909120,  1968220058676480,  10885957855395840,  34727339512656000,   70010316457514496,   92644824633207840,   80670323490140160,   44668040448153780,   14297040838458000,   2018265598362321;\n  9: 64268410079232, 3299111717400576, 43300841290882560, 269427456921047040, 982287603357984000, 2310340443097978368, 3668735055475030464, 3993181012761937920, 2948090669578149480, 1415407043007342000, 399616588475739558, 50439960078079824;\n","recurrence_sentence":"The row polynomials satisfy (36+(126*n+210)*x+(49*n^2+196*n+196)*x^2)*P(n,x) = (648+(2268*n+6804)*x+(882*n^2+7938*n+14994)*x^2+(1029*n^2+6174*n+9261)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 324 + 3024*x + 3969*x^2.","mathematica":"P[0, x_] := 324 + 3024*x + 3969*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((648 + (2268*n + 6804)*x + (882*n^2 + 7938*n + 14994)*x^2 + (1029*n^2 + 6174*n + 9261)*x^3)*P[n - 1, x])/(36 + (126*n + 210)*x + (49*n^2 + 196*n + 196)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 324 + 3024*x + 3969*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((648 + (2268*n + 6804)*x + (882*n^2 + 7938*n + 14994)*x^2 + (1029*n^2 + 6174*n + 9261)*x^3)*v[i - 1])/(36 + (126*n + 210)*x + (49*n^2 + 196*n + 196)*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-7dcf11eddf2b9a8a","steps":["basis A013610","(\u03b8+1)\u00b2"],"edges":["e-632cc4d34c6e419b","e-f23c3c3b66bbb5f7"],"text":"A038321 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2"},"headline":"A038321 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2","chain":null}