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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":"6e9b4eff0d66ff5bd118d343400a41bc91043480b6b860cc595f743192139e57","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-e2ca539e0aa62c2f","role":"sequence","name":"basis_map_A013611 of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","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:    1;\n   2:    5,     4;\n   3:   14,    44,     16;\n   4:   31,   188,    272,      64;\n   5:   65,   572,   1600,    1472,     256;\n   6:  128,  1496,   6256,   11072,    7424,     1024;\n   7:  245,  3560,  19904,   53184,   68096,    35840,     4096;\n   8:  457,  7956,  55792,  200640,  388864,   388096,   167936,    16384;\n   9:  838, 16996, 143456,  650304, 1702656,  2571264,  2097152,   770048,   65536;\n  10: 1515, 35092, 346480, 1897792, 6283776, 12840960, 15822848, 10895360, 3473408, 262144;\n","recurrence_sentence":"The row polynomials satisfy (64*n^3-240*n^2+228*n-52+(128*n^2-592*n+464)*x+(256*n^2-832*n+576)*x^2+(256*n-256)*x^3)*P(n,x) = -(128*n^3-1056*n^2+2200*n-980-(512*n^2-1184*n-896)*x+(512*n^2-1920*n+2944)*x^2-512*n*x^3-1024*x^4)*P(n-1,x) + (80*n^2-220*n+105+(384*n^2-896*n+244)*x+(256*n^2+384*n-992)*x^2+(2048*n-896)*x^3+(1024*n+1280)*x^4+1024*x^5)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(128*n^3 - 1056*n^2 + 2200*n - 980 - (512*n^2 - 1184*n - 896)*x + (512*n^2 - 1920*n + 2944)*x^2 - 512*n*x^3 - 1024*x^4)*P[n - 1, x] + (80*n^2 - 220*n + 105 + (384*n^2 - 896*n + 244)*x + (256*n^2 + 384*n - 992)*x^2 + (2048*n - 896)*x^3 + (1024*n + 1280)*x^4 + 1024*x^5)*D[P[n - 1, x], x])/(64*n^3 - 240*n^2 + 228*n - 52 + (128*n^2 - 592*n + 464)*x + (256*n^2 - 832*n + 576)*x^2 + (256*n - 256)*x^3)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(128*n^3 - 1056*n^2 + 2200*n - 980 - (512*n^2 - 1184*n - 896)*x + (512*n^2 - 1920*n + 2944)*x^2 - 512*n*x^3 - 1024*x^4)*v[i - 1] + (80*n^2 - 220*n + 105 + (384*n^2 - 896*n + 244)*x + (256*n^2 + 384*n - 992)*x^2 + (2048*n - 896)*x^3 + (1024*n + 1280)*x^4 + 1024*x^5)*deriv(v[i - 1], x))/(64*n^3 - 240*n^2 + 228*n - 52 + (128*n^2 - 592*n + 464)*x + (256*n^2 - 832*n + 576)*x^2 + (256*n - 256)*x^3)); 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-392f49b496f79fbf","steps":["basis A013611","D+1"],"edges":["e-4d9d32144b57b963","e-625bba83807eaecc"],"text":"A026729 \u2192 basis A013611 \u2192 D+1"},"headline":"A026729 \u2192 basis A013611 \u2192 D+1","chain":null}