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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":"e15a221795d367d2f7f6766b8625e723a21c5c42ec6f03c0fb54f4c2ede65edb","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-546dca52670dddd1","role":"sequence","name":"basis_map_A013609 of A Jacobsthal triangle.","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:    3,     2;\n  2:   11,    16,      4;\n  3:   27,    78,     52,      8;\n  4:   71,   284,    352,    144,     16;\n  5:  167,   898,   1672,   1264,    368,     32;\n  6:  395,  2616,   6508,   7488,   3984,    896,     64;\n  7:  899,  7158,  22444,  35032,  28432,  11552,   2112,   128;\n  8: 2031, 18804,  71112, 140944, 156416,  96704,  31616,  4864,   256;\n  9: 4511, 47754, 211872, 510464, 723936, 613952, 304128, 82944, 11008, 512;\n","recurrence_sentence":"The row polynomials satisfy (16*n^3+12*n^2-34*n+(64*n^2-52*n)*x+(16*n^2+40*n)*x^2+16*n*x^3)*P(n,x) = -(64*n^3-192*n^2-28*n+174+(160*n^2-456*n+272)*x+(64*n^2-16*n-240)*x^2+(32*n-192)*x^3-32*x^4)*P(n-1,x) + (72*n^2+54*n-9+(160*n^2+408*n+106)*x+(32*n^2+736*n+600)*x^2+(288*n+848)*x^3+(32*n+304)*x^4+32*x^5)*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[(-(64*n^3 - 192*n^2 - 28*n + 174 + (160*n^2 - 456*n + 272)*x + (64*n^2 - 16*n - 240)*x^2 + (32*n - 192)*x^3 - 32*x^4)*P[n - 1, x] + (72*n^2 + 54*n - 9 + (160*n^2 + 408*n + 106)*x + (32*n^2 + 736*n + 600)*x^2 + (288*n + 848)*x^3 + (32*n + 304)*x^4 + 32*x^5)*D[P[n - 1, x], x])/(16*n^3 + 12*n^2 - 34*n + (64*n^2 - 52*n)*x + (16*n^2 + 40*n)*x^2 + 16*n*x^3)]];\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] = (-(64*n^3 - 192*n^2 - 28*n + 174 + (160*n^2 - 456*n + 272)*x + (64*n^2 - 16*n - 240)*x^2 + (32*n - 192)*x^3 - 32*x^4)*v[i - 1] + (72*n^2 + 54*n - 9 + (160*n^2 + 408*n + 106)*x + (32*n^2 + 736*n + 600)*x^2 + (288*n + 848)*x^3 + (32*n + 304)*x^4 + 32*x^5)*deriv(v[i - 1], x))/(16*n^3 + 12*n^2 - 34*n + (64*n^2 - 52*n)*x + (16*n^2 + 40*n)*x^2 + 16*n*x^3)); 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-ba196dcd01800741","steps":["basis A013609","D+1"],"edges":["e-34478b6712498a37","e-141b17adc7cba724"],"text":"A113953 \u2192 basis A013609 \u2192 D+1"},"headline":"A113953 \u2192 basis A013609 \u2192 D+1","chain":null}