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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":"b3b321112ed91983f768449d3da2a1964c5bf05b24ebbbbb29e19564be5308ec","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-f07088d208415183","role":"sequence","name":"basis_map_A013609 of Triangle T(n, k) is coefficient of z^n*w^k in 1/(1 - 2*z - 2*w - 2*z*w) read by rows in order 00, 10, 01, 20, 11, 02, ...","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:       0;\n  1:       4;\n  2:      36,       32;\n  3:     240,      448,       192;\n  4:    1424,     4064,      3648,      1024;\n  5:    7920,    30464,     42048,     24576,      5120;\n  6:   42288,   204736,    382656,    344064,    148480,     24576;\n  7:  219520,  1281536,   3025920,   3690496,   2447360,    835584,   114688;\n  8: 1116288,  7630080,  21786624,  33629184,  30259200,  15851520,  4472832,   524288;\n  9: 5587776, 43769856, 146668032, 274231296, 312529920, 222068736, 95993856, 23068672, 2359296;\n","recurrence_sentence":"The row polynomials satisfy (3*n-3+(3*n^2-3*n)*x-(4*n-4)*x^2)*P(n,x) = -(9*n^2-42*n+21+(12*n^2-102*n+78)*x-(52*n-92)*x^2+32*x^3)*P(n-1,x) + (9*n-9+(36*n-48)*x+(42*n-90)*x^2+(12*n-68)*x^3-16*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 4;\nP[n_, x_] := P[n, x] = Expand[Together[(-(9*n^2 - 42*n + 21 + (12*n^2 - 102*n + 78)*x - (52*n - 92)*x^2 + 32*x^3)*P[n - 1, x] + (9*n - 9 + (36*n - 48)*x + (42*n - 90)*x^2 + (12*n - 68)*x^3 - 16*x^4)*D[P[n - 1, x], x])/(3*n - 3 + (3*n^2 - 3*n)*x - (4*n - 4)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 4; for(n = 2, N, my(i = n + 1); v[i] = (-(9*n^2 - 42*n + 21 + (12*n^2 - 102*n + 78)*x - (52*n - 92)*x^2 + 32*x^3)*v[i - 1] + (9*n - 9 + (36*n - 48)*x + (42*n - 90)*x^2 + (12*n - 68)*x^3 - 16*x^4)*deriv(v[i - 1], x))/(3*n - 3 + (3*n^2 - 3*n)*x - (4*n - 4)*x^2)); 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-1bffaa6198344f93","steps":["basis A013609","D"],"edges":["e-c90acfb06fc8929f","e-508f45fa7ff72ea3"],"text":"A059473 \u2192 basis A013609 \u2192 D"},"headline":"A059473 \u2192 basis A013609 \u2192 D","chain":null}