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real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); 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":"4a02420f488dfcce7b60d6fc1e2503882abf37abd4d50c8d7a8a6cfcb6e2353d","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-d2d581dc25d5445a","role":"sequence","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*9^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:             1;\n  1:            29,             9;\n  2:           760,           522,             81;\n  3:         18800,         20520,           7047,            729;\n  4:        448000,        676800,         369360,          84564,          6561;\n  5:      10400000,      20160000,       15228000,        5540400,        951345,         59049;\n  6:     236800000,     561600000,      544320000,      274104000,      74795400,      10274526,       531441;\n  7:    5312000000,   14918400000,    17690400000,    11430720000,    4317138000,     942422040,    107882523,      4782969;\n  8:  117760000000,  382464000000,   537062400000,   424569600000,  205752960000,   62166787200,  11309064480,   1109648808,    43046721;\n  9: 2585600000000, 9538560000000, 15489792000000, 14500684800000, 8597534400000, 3333197952000, 839251627200, 130862031840, 11235194181, 387420489;\n","recurrence_sentence":"The row polynomials satisfy (9*n+11+9*x)*P(n,x) = (180*n+400+(81*n+360)*x+81*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((180*n + 400 + (81*n + 360)*x + 81*x^2)*P[n - 1, x])/(9*n + 11 + 9*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((180*n + 400 + (81*n + 360)*x + 81*x^2)*v[i - 1])/(9*n + 11 + 9*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-4e096b5929fd3f2e","steps":["x \u21a6 x+1","D+1"],"edges":["e-e91f5a6c0f5b0271","e-aacacccab6312a94"],"text":"A038323 \u2192 x \u21a6 x+1 \u2192 D+1"},"headline":"A038323 \u2192 x \u21a6 x+1 \u2192 D+1","chain":null}