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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":"8b2ab841d2d3c16a569f02a3ede6fcee0ea2f698f6d19fd8d5353cbd8910367d","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-b58ba5723c3a3a3f","role":"sequence","name":"Triangle, read by rows, T(n,k) = t(n-k, k) where t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(x) = x + 3.","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:       33,         42,           9;\n   2:      168,        423,         222,           27;\n   3:      807,       3546,        3588,         1050,           81;\n   4:     3714,      26283,       45624,        25290,         4686,          243;\n   5:    16557,     178482,      491943,       462180,       160467,        20202,          729;\n   6:    72060,    1136391,     4704822,      6952605,      4056816,       952497,        85182,        2187;\n   7: 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+ 42*x + 9*x^2;\nP[2, x_] := 168 + 423*x + 222*x^2 + 27*x^3;\nP[n_, x_] := P[n, x] = Expand[Together[(-(12*n + 24 + (6*n^2 + 15*n + 30)*x + (n^3 + 4*n^2 + 19*n + 12)*x^2 + (3*n^2 + 8*n + 10)*x^3 + (2*n + 4)*x^4)*P[n - 2, x] - ((3*n + 6)*x + (n^2 + 2)*x^2 - (n^2 + 4)*x^3 - (3*n + 2)*x^4 - 2*x^5)*D[P[n - 2, x], x] + (7*n + 10 + (2*n^2 + 5*n + 17)*x + (5*n + 6)*x^2 + 2*x^3)*P[n - 1, x] + ((n + 1)*x - (n - 1)*x^2 - 2*x^3)*D[P[n - 1, x], x])/(n + 1 + 2*x)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 33 + 42*x + 9*x^2; v[2] = 168 + 423*x + 222*x^2 + 27*x^3; for(n = 3, N, my(i = n); v[i] = (-(12*n + 24 + (6*n^2 + 15*n + 30)*x + (n^3 + 4*n^2 + 19*n + 12)*x^2 + (3*n^2 + 8*n + 10)*x^3 + (2*n + 4)*x^4)*v[i - 2] - ((3*n + 6)*x + (n^2 + 2)*x^2 - (n^2 + 4)*x^3 - (3*n + 2)*x^4 - 2*x^5)*deriv(v[i - 2], x) + (7*n + 10 + (2*n^2 + 5*n + 17)*x + (5*n + 6)*x^2 + 2*x^3)*v[i - 1] + ((n + 1)*x - (n - 1)*x^2 - 2*x^3)*deriv(v[i - 1], x))/(n + 1 + 2*x)); 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-b58ba5723c3a3a3f","steps":["D+1"],"edges":["e-d243a962f790122f"],"text":"A257180 \u2192 D+1"},"headline":"A257180 \u2192 D+1","chain":null}