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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":"727e341c6dc968f1171349e7f66cad599d2fcd6bcb388fce48185355dd332059","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-9e53622e0c440ce5","role":"sequence","name":"basis_map_A013609 of Triangle read by rows: T(n,k) is the number of ways to tile a 3 X n rectangle with k pieces of 2 X 2 tiles and 3n-4k pieces of 1 X 1 tiles (0 <= k <= floor(n/2)).","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:    1;\n  2:    7,     4;\n  3:   13,     8;\n  4:   39,    60,    16;\n  5:   85,   160,    48;\n  6:  207,   548,   384,   64;\n  7:  461,  1464,  1312,  256;\n  8: 1047,  4044,  5040, 2176,  256;\n  9: 2309, 10320, 15696, 8960, 1280;\n","recurrence_sentence":"The row polynomials satisfy (16*n^3-12*n^2-34*n+(64*n^2-92*n)*x+64*n*x^2)*P(n,x) = (80*n^3-108*n^2-122*n+120+(128*n^3+96*n^2-604*n+352)*x+(512*n^2-416*n-32)*x^2+(512*n-512)*x^3)*P(n-1,x) - (72*n^2+90*n+9+(272*n^2+628*n+196)*x+(256*n^2+1408*n+932)*x^2+(1024*n+1664)*x^3+1024*x^4)*d/dx 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[((80*n^3 - 108*n^2 - 122*n + 120 + (128*n^3 + 96*n^2 - 604*n + 352)*x + (512*n^2 - 416*n - 32)*x^2 + (512*n - 512)*x^3)*P[n - 1, x] - (72*n^2 + 90*n + 9 + (272*n^2 + 628*n + 196)*x + (256*n^2 + 1408*n + 932)*x^2 + (1024*n + 1664)*x^3 + 1024*x^4)*D[P[n - 1, x], x])/(16*n^3 - 12*n^2 - 34*n + (64*n^2 - 92*n)*x + 64*n*x^2)]];\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] = ((80*n^3 - 108*n^2 - 122*n + 120 + (128*n^3 + 96*n^2 - 604*n + 352)*x + (512*n^2 - 416*n - 32)*x^2 + (512*n - 512)*x^3)*v[i - 1] - (72*n^2 + 90*n + 9 + (272*n^2 + 628*n + 196)*x + (256*n^2 + 1408*n + 932)*x^2 + (1024*n + 1664)*x^3 + 1024*x^4)*deriv(v[i - 1], x))/(16*n^3 - 12*n^2 - 34*n + (64*n^2 - 92*n)*x + 64*n*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-e3d901814302c059","steps":["basis A013609","D+1"],"edges":["e-2ee1895757bd17cf","e-05ae637ad4f893cf"],"text":"A128099 \u2192 basis A013609 \u2192 D+1"},"headline":"A128099 \u2192 basis A013609 \u2192 D+1","chain":null}