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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":"1ba52b4f33fe4d2d252b693644d48c449bcddf73fc3c406c293dc97b282f567a","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-b69b5201d4a2fd8f","role":"sequence","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*6^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:            289,             1632,             1296;\n  1:           4913,            41616,            66096,             27648;\n  2:          83521,           943296,          2247264,           1880064,            518400;\n  3:        1419857,         20045040,         63672480,          79902720,          44064000,           8957952;\n  4:       24137569,        408918816,       1623648240,        2716692480,        2247264000,         913711104,         146313216;\n  5:      410338673,       8110223184,      38642828112,       80821601280,       89141472000,       54365810688,       17411272704,        2293235712;\n  6:     6975757441,     157570050432,     875904103872,     2198347554816,     3030810048000,     2464583417856,     1183966543872,      311880056832,       34828517376;\n  7:   118587876497,    3013527214512,   19144761127488,    56057862647808,    92742787468800,    94270315732992,    60382293737472,    23858824347648,     5328763158528,     515978035200;\n  8:  2015993900449,   56922180718560,  406826173959120,  1361405235732480,  2627712311616000,  3205190734921728,  2566247483842560,  1352000046366720,   452944868474880,   87716265984000,    7492001071104;\n  9: 34271896307633, 1064444779437072, 8452943836706160, 31822847385246720, 70197457467456000, 99895111238393856, 95977655895711744, 63206002167644160, 28233563468267520, 8201470869504000, 1401004200296448, 106993205379072;\n","recurrence_sentence":"The row polynomials satisfy (289+(612*n+1020)*x+(144*n^2+576*n+576)*x^2)*P(n,x) = (4913+(10404*n+31212)*x+(2448*n^2+22032*n+41616)*x^2+(1728*n^2+10368*n+15552)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 289 + 1632*x + 1296*x^2.","mathematica":"P[0, x_] := 289 + 1632*x + 1296*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((4913 + (10404*n + 31212)*x + (2448*n^2 + 22032*n + 41616)*x^2 + (1728*n^2 + 10368*n + 15552)*x^3)*P[n - 1, x])/(289 + (612*n + 1020)*x + (144*n^2 + 576*n + 576)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 289 + 1632*x + 1296*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((4913 + (10404*n + 31212)*x + (2448*n^2 + 22032*n + 41616)*x^2 + (1728*n^2 + 10368*n + 15552)*x^3)*v[i - 1])/(289 + (612*n + 1020)*x + (144*n^2 + 576*n + 576)*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-1f2dec04776afb2b","steps":["basis A013609","(\u03b8+1)\u00b2"],"edges":["e-7304be3f66cc3caf","e-1f9e3530f563b651"],"text":"A038320 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2"},"headline":"A038320 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2","chain":null}