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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":"d30dff77aafa6d662fd6c73d2b6d817f12ca03e59c05c6ff9cac5b76c4b8928e","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-45d316ccbe3d57d1","role":"sequence","name":"basis_map_A013609 of Square array A(n,r), the number of neighbors at a sharp Manhattan distance r in a finite n-hypercube lattice, read by upwards antidiagonals; A(n,r) = Sum_{k=0..min(n,r)} binomial(r-1,k-1)*binomial(n,k)* 2^k.","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:    23,    32,      8;\n  4:    73,   168,    104,     16;\n  5:   217,   704,    744,    288,     32;\n  6:   623,  2620,   3976,   2640,    736,     64;\n  7:  1743,  9024,  17968,  17280,   8256,   1792,   128;\n  8:  4785, 29456,  72816,  92512,  64192,  23808,  4224,  256;\n  9: 12945, 92416, 273328, 433472, 398720, 214784, 64896, 9728, 512;\n","recurrence_sentence":"The row polynomials satisfy -(2*n-4-(2*n^3-2*n^2-12*n+15)*x-(8*n^2-24*n+11)*x^2-(2*n^2-n-8)*x^3-(2*n-4)*x^4)*P(n,x) = -(4*n^3-8*n^2-2*n+4+(6*n^3-4*n^2-20*n+15)*x+(20*n^2-26*n+11)*x^2+(6*n^2+9*n-8)*x^3+(6*n-4)*x^4)*P(n-1,x) + (4*n^2+4*n+(16*n^2+32*n+6)*x+(18*n^2+86*n+38)*x^2+(4*n^2+92*n+87)*x^3+(34*n+85)*x^4+(4*n+32)*x^5+4*x^6)*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[(-(4*n^3 - 8*n^2 - 2*n + 4 + (6*n^3 - 4*n^2 - 20*n + 15)*x + (20*n^2 - 26*n + 11)*x^2 + (6*n^2 + 9*n - 8)*x^3 + (6*n - 4)*x^4)*P[n - 1, x] + (4*n^2 + 4*n + (16*n^2 + 32*n + 6)*x + (18*n^2 + 86*n + 38)*x^2 + (4*n^2 + 92*n + 87)*x^3 + (34*n + 85)*x^4 + (4*n + 32)*x^5 + 4*x^6)*D[P[n - 1, x], x])/(-2*n + 4 + (2*n^3 - 2*n^2 - 12*n + 15)*x + (8*n^2 - 24*n + 11)*x^2 + (2*n^2 - n - 8)*x^3 + (2*n - 4)*x^4)]];\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] = (-(4*n^3 - 8*n^2 - 2*n + 4 + (6*n^3 - 4*n^2 - 20*n + 15)*x + (20*n^2 - 26*n + 11)*x^2 + (6*n^2 + 9*n - 8)*x^3 + (6*n - 4)*x^4)*v[i - 1] + (4*n^2 + 4*n + (16*n^2 + 32*n + 6)*x + (18*n^2 + 86*n + 38)*x^2 + (4*n^2 + 92*n + 87)*x^3 + (34*n + 85)*x^4 + (4*n + 32)*x^5 + 4*x^6)*deriv(v[i - 1], x))/(-2*n + 4 + (2*n^3 - 2*n^2 - 12*n + 15)*x + (8*n^2 - 24*n + 11)*x^2 + (2*n^2 - n - 8)*x^3 + (2*n - 4)*x^4)); 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-e073e5e0b021f36a","steps":["basis A013609","D+1"],"edges":["e-064ccd10ffc87509","e-4fa6925f00dbd5cf"],"text":"A035607 \u2192 basis A013609 \u2192 D+1"},"headline":"A035607 \u2192 basis A013609 \u2192 D+1","chain":null}