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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":"455e6bad76bc1b8c3ba328d02defe029ea3e5ec52bd916e6262097019f7e37aa","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-5b7da9f56b3380cb","role":"sequence","name":"Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 7*x + 2.","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:   1;\n  1:   2,         8;\n  2:   4,       144,           36;\n  3:   8,      1552,         3492,           128;\n  4:  16,     14704,       111744,         58816,            400;\n  5:  32,    134256,      2548836,       4531264,         839100,           1152;\n  6:  64,   1213040,     49843656,     208438144,      138454600,       10917360,           3136;\n  7: 128,  10928624,    898484076,    7452405632,    11644383800,     3593936304,      133875644,          8192;\n  8: 256,  98383728,  15457678992,  230505615424,   698663028000,   518637634704,    84158474512,    1574139648,       20736;\n  9: 512, 885512944, 258612396660, 6510763155904, 34285996731700, 49371835293648, 19939212164956, 1839021487360, 17931637116, 51200;\n","recurrence_sentence":"The row polynomials satisfy (49*n^2-70*n+25)*P(n,x) = (490*n^2-210*n-100-(2744*n^3-5880*n^2-5404*n+12720)*x+(21609*n^4-66542*n^3+59633*n^2-14840*n-1200)*x^2)*P(n-2,x) - ((1372*n^2-588*n-2485)*x-(2744*n^3+6468*n^2-1435*n-11850)*x^2-(9604*n^4-61054*n^3+70217*n^2-20860*n-1175)*x^3)*d/dx P(n-2,x) - ((2401*n^2-1029*n-2205)*x^2-(9604*n^3-10976*n^2+3381*n-2030)*x^3+(9604*n^3-13377*n^2+4410*n+175)*x^4)*d^2/dx^2 P(n-2,x) - (147*n^2+35*n-100-(1372*n^3-2940*n^2+2100*n-500)*x)*P(n-1,x) + ((686*n^2-637*n-140)*x-(1372*n^3-1568*n^2+140*n+200)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 8*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 8*x;\nP[n_, x_] := P[n, x] = Expand[Together[((490*n^2 - 210*n - 100 - (2744*n^3 - 5880*n^2 - 5404*n + 12720)*x + (21609*n^4 - 66542*n^3 + 59633*n^2 - 14840*n - 1200)*x^2)*P[n - 2, x] - ((1372*n^2 - 588*n - 2485)*x - (2744*n^3 + 6468*n^2 - 1435*n - 11850)*x^2 - (9604*n^4 - 61054*n^3 + 70217*n^2 - 20860*n - 1175)*x^3)*D[P[n - 2, x], x] - ((2401*n^2 - 1029*n - 2205)*x^2 - (9604*n^3 - 10976*n^2 + 3381*n - 2030)*x^3 + (9604*n^3 - 13377*n^2 + 4410*n + 175)*x^4)*D[P[n - 2, x], {x, 2}] - (147*n^2 + 35*n - 100 - (1372*n^3 - 2940*n^2 + 2100*n - 500)*x)*P[n - 1, x] + ((686*n^2 - 637*n - 140)*x - (1372*n^3 - 1568*n^2 + 140*n + 200)*x^2)*D[P[n - 1, x], x])/(49*n^2 - 70*n + 25)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = ((490*n^2 - 210*n - 100 - (2744*n^3 - 5880*n^2 - 5404*n + 12720)*x + (21609*n^4 - 66542*n^3 + 59633*n^2 - 14840*n - 1200)*x^2)*v[i - 2] - ((1372*n^2 - 588*n - 2485)*x - (2744*n^3 + 6468*n^2 - 1435*n - 11850)*x^2 - (9604*n^4 - 61054*n^3 + 70217*n^2 - 20860*n - 1175)*x^3)*deriv(v[i - 2], x) - ((2401*n^2 - 1029*n - 2205)*x^2 - (9604*n^3 - 10976*n^2 + 3381*n - 2030)*x^3 + (9604*n^3 - 13377*n^2 + 4410*n + 175)*x^4)*deriv(deriv(v[i - 2], x), x) - (147*n^2 + 35*n - 100 - (1372*n^3 - 2940*n^2 + 2100*n - 500)*x)*v[i - 1] + ((686*n^2 - 637*n - 140)*x - (1372*n^3 - 1568*n^2 + 140*n + 200)*x^2)*deriv(v[i - 1], x))/(49*n^2 - 70*n + 25)); 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-5b7da9f56b3380cb","steps":["(\u03b8+1)\u00b2"],"edges":["e-d5a8609283eb0145"],"text":"A257617 \u2192 (\u03b8+1)\u00b2"},"headline":"A257617 \u2192 (\u03b8+1)\u00b2","chain":null}