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D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"bed18dd8c81095ca4dfcb5441a6bb8077c667910086a4888043833b53e9043ac","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-194cb2d73cc003c5","role":"sequence","name":"basis_map_A013610 of Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:       1;\n   2:      13,        9;\n   3:      76,      150,        45;\n   4:     400,     1344,      1071,       189;\n   5:    1984,     9504,     13608,      6156,       729;\n   6:    9472,    59136,    125136,    104544,     31185,      2673;\n   7:   44032,   339456,    958464,   1213056,    678132,    145314,      9477;\n   8:  200704,  1843200,   6531840,  11232000,   9681120,   3919104,    637875,     32805;\n   9:  901120,  9609216,  41048064,  89773056, 106634880,  67417920,  20820240,   2676888,   111537;\n  10: 3997696, 48562176, 243044352, 647184384, 988505856, 871098624, 424173024, 103716288, 10845333, 373977;\n","recurrence_sentence":"The row polynomials satisfy (12*n^3-52*n^2+71*n-30+(24*n^2-86*n+75)*x+(12*n-18)*x^2)*P(n,x) = (48*n^3-160*n^2+116*n+24+(108*n^3-300*n^2+25*n+273)*x+(216*n^2-618*n+441)*x^2+(108*n-162)*x^3)*P(n-1,x) - ((24*n^2-32*n-6)*x+(72*n^2-48*n-70)*x^2+(144*n-132)*x^3+72*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((48*n^3 - 160*n^2 + 116*n + 24 + (108*n^3 - 300*n^2 + 25*n + 273)*x + (216*n^2 - 618*n + 441)*x^2 + (108*n - 162)*x^3)*P[n - 1, x] - ((24*n^2 - 32*n - 6)*x + (72*n^2 - 48*n - 70)*x^2 + (144*n - 132)*x^3 + 72*x^4)*D[P[n - 1, x], x])/(12*n^3 - 52*n^2 + 71*n - 30 + (24*n^2 - 86*n + 75)*x + (12*n - 18)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((48*n^3 - 160*n^2 + 116*n + 24 + (108*n^3 - 300*n^2 + 25*n + 273)*x + (216*n^2 - 618*n + 441)*x^2 + (108*n - 162)*x^3)*v[i - 1] - ((24*n^2 - 32*n - 6)*x + (72*n^2 - 48*n - 70)*x^2 + (144*n - 132)*x^3 + 72*x^4)*deriv(v[i - 1], x))/(12*n^3 - 52*n^2 + 71*n - 30 + (24*n^2 - 86*n + 75)*x + (12*n - 18)*x^2)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-ba8de4e02acfb6ca","steps":["basis A013610","D+1"],"edges":["e-4aa0aeb00b1923c9","e-f4f875716a927cc1"],"text":"A091042 \u2192 basis A013610 \u2192 D+1"},"headline":"A091042 \u2192 basis A013610 \u2192 D+1","chain":null}