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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":"cd571c9734d5e2374ef1a595bd39cb21b7b381a0d5af6bf2f0ef3e4fba20c982","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-29bd79a8a49e73a9","role":"sequence","name":"basis_map_A013609 of Triangle of coefficients of polynomials u(n,x) jointly generated with A207607; see the Formula section.","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:     22,      16;\n  1:     88,     136,       48;\n  2:    310,     744,      552,      128;\n  3:   1020,    3336,     3840,     1856,      320;\n  4:   3216,   13344,    20976,    15680,     5600,      768;\n  5:   9848,   49560,    99120,   101120,    55680,    15744,     1792;\n  6:  29522,  174728,   425064,   552000,   414240,   180096,    42112,     4096;\n  7:  87080,  592712,  1700256,  2688384,  2567040,  1518720,   544768,   108544,    9216;\n  8: 253610, 1951728,  6453840, 12047168, 14007840, 10536960,  5135872,  1566720,  271872,  20480;\n  9: 731064, 6276912, 23516448, 50663360, 69574400, 63710976, 39449088, 16330752, 4331520, 665600, 45056;\n","recurrence_sentence":"The row polynomials satisfy (4*n^2-8*n-12+(16*n^2+40*n+24)*x)*P(n,x) = (6*n^2-4*n-66+(28*n^2+112*n+84)*x+(16*n^2+80*n+96)*x^2)*P(n-1,x) + (5*n-5+(32*n+28)*x+(52*n+92)*x^2+(16*n+32)*x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 22 + 16*x.","mathematica":"P[0, x_] := 22 + 16*x;\nP[n_, x_] := P[n, x] = Expand[Together[((6*n^2 - 4*n - 66 + (28*n^2 + 112*n + 84)*x + (16*n^2 + 80*n + 96)*x^2)*P[n - 1, x] + (5*n - 5 + (32*n + 28)*x + (52*n + 92)*x^2 + (16*n + 32)*x^3)*D[P[n - 1, x], x])/(4*n^2 - 8*n - 12 + (16*n^2 + 40*n + 24)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 22 + 16*x; for(n = 1, N, my(i = n + 1); v[i] = ((6*n^2 - 4*n - 66 + (28*n^2 + 112*n + 84)*x + (16*n^2 + 80*n + 96)*x^2)*v[i - 1] + (5*n - 5 + (32*n + 28)*x + (52*n + 92)*x^2 + (16*n + 32)*x^3)*deriv(v[i - 1], x))/(4*n^2 - 8*n - 12 + (16*n^2 + 40*n + 24)*x)); 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-dfab35176bd08e72","steps":["basis A013609","D"],"edges":["e-0adfae28f720d284","e-0db326cbf042e4b2"],"text":"A207606 \u2192 basis A013609 \u2192 D"},"headline":"A207606 \u2192 basis A013609 \u2192 D","chain":null}