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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":"c05647f1bd03a412e8ea4502e1c5af9ea0aec1e79da030638c788e7de4b31ef9","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-9ac33c0e4aa39b7d","role":"sequence","name":"basis_map_A007318 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:    5,     2;\n  3:   15,    12,     2;\n  4:   45,    56,    20,     2;\n  5:  129,   220,   126,    28,     2;\n  6:  361,   786,   628,   226,    36,    2;\n  7:  991,  2632,  2716,  1376,   356,   44,   2;\n  8: 2681,  8416, 10680,  7036,  2572,  516,  52,  2;\n  9: 7169, 25992, 39220, 31944, 15240, 4324, 706, 60, 2;\n","recurrence_sentence":"The row polynomials satisfy -(8*n-8-(2*n^3+2*n^2-24*n+6)*x-(8*n^2-16*n-14)*x^2-(n^2+4*n-14)*x^3-(n-2)*x^4)*P(n,x) = -(8*n^3-16*n+8+(6*n^3+24*n^2-22*n+6)*x+(24*n^2+12*n-14)*x^2+(3*n^2+20*n-14)*x^3+(3*n-2)*x^4)*P(n-1,x) + (16*n^2+48*n+32+(32*n^2+160*n+152)*x+(18*n^2+190*n+276)*x^2+(2*n^2+94*n+239)*x^3+(17*n+99)*x^4+(n+17)*x^5+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[(-(8*n^3 - 16*n + 8 + (6*n^3 + 24*n^2 - 22*n + 6)*x + (24*n^2 + 12*n - 14)*x^2 + (3*n^2 + 20*n - 14)*x^3 + (3*n - 2)*x^4)*P[n - 1, x] + (16*n^2 + 48*n + 32 + (32*n^2 + 160*n + 152)*x + (18*n^2 + 190*n + 276)*x^2 + (2*n^2 + 94*n + 239)*x^3 + (17*n + 99)*x^4 + (n + 17)*x^5 + x^6)*D[P[n - 1, x], x])/(-8*n + 8 + (2*n^3 + 2*n^2 - 24*n + 6)*x + (8*n^2 - 16*n - 14)*x^2 + (n^2 + 4*n - 14)*x^3 + (n - 2)*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] = (-(8*n^3 - 16*n + 8 + (6*n^3 + 24*n^2 - 22*n + 6)*x + (24*n^2 + 12*n - 14)*x^2 + (3*n^2 + 20*n - 14)*x^3 + (3*n - 2)*x^4)*v[i - 1] + (16*n^2 + 48*n + 32 + (32*n^2 + 160*n + 152)*x + (18*n^2 + 190*n + 276)*x^2 + (2*n^2 + 94*n + 239)*x^3 + (17*n + 99)*x^4 + (n + 17)*x^5 + x^6)*deriv(v[i - 1], x))/(-8*n + 8 + (2*n^3 + 2*n^2 - 24*n + 6)*x + (8*n^2 - 16*n - 14)*x^2 + (n^2 + 4*n - 14)*x^3 + (n - 2)*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 A007318","D+1"],"edges":["e-017120e86183432c","e-67e19e80afb71d69"],"text":"A035607 \u2192 basis A007318 \u2192 D+1"},"headline":"A035607 \u2192 basis A007318 \u2192 D+1","chain":null}