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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":"c16d33e727bcf495e474a4128605f7ac43ae2f459e1f1d24a098849ec200a972","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-4246e71c9dd6d2b5","role":"sequence","name":"basis_map_A013610 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:     9,      6;\n  3:    31,     60,     18;\n  4:   101,    336,    288,      54;\n  5:   305,   1452,   2214,    1188,     162;\n  6:   885,   5502,  12240,   11718,    4536,     486;\n  7:  2495,  19176,  56412,   79488,   54756,   16524,   1458;\n  8:  6889,  63120, 231552,  434484,  441612,  236196,  58320,   4374;\n  9: 18721, 199272, 876996, 2063448, 2802600, 2211300, 963738, 201204, 13122;\n","recurrence_sentence":"The row polynomials satisfy -(72*n-168-(162*n^3-270*n^2-576*n+1026)*x-(648*n^2-2160*n+1566)*x^2-(243*n^2-486*n-162)*x^3-(243*n-486)*x^4)*P(n,x) = -(216*n^3-576*n^2+96*n+168+(486*n^3-1080*n^2-774*n+1026)*x+(1296*n^2-2916*n+1566)*x^2+(729*n^2-162*n-162)*x^3+(729*n-486)*x^4)*P(n-1,x) + (144*n^2+48*n-32+(864*n^2+864*n-168)*x+(1458*n^2+4158*n+252)*x^2+(486*n^2+7290*n+2943)*x^3+(4131*n+5751)*x^4+(729*n+3645)*x^5+729*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[(-(216*n^3 - 576*n^2 + 96*n + 168 + (486*n^3 - 1080*n^2 - 774*n + 1026)*x + (1296*n^2 - 2916*n + 1566)*x^2 + (729*n^2 - 162*n - 162)*x^3 + (729*n - 486)*x^4)*P[n - 1, x] + (144*n^2 + 48*n - 32 + (864*n^2 + 864*n - 168)*x + (1458*n^2 + 4158*n + 252)*x^2 + (486*n^2 + 7290*n + 2943)*x^3 + (4131*n + 5751)*x^4 + (729*n + 3645)*x^5 + 729*x^6)*D[P[n - 1, x], x])/(-72*n + 168 + (162*n^3 - 270*n^2 - 576*n + 1026)*x + (648*n^2 - 2160*n + 1566)*x^2 + (243*n^2 - 486*n - 162)*x^3 + (243*n - 486)*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] = (-(216*n^3 - 576*n^2 + 96*n + 168 + (486*n^3 - 1080*n^2 - 774*n + 1026)*x + (1296*n^2 - 2916*n + 1566)*x^2 + (729*n^2 - 162*n - 162)*x^3 + (729*n - 486)*x^4)*v[i - 1] + (144*n^2 + 48*n - 32 + (864*n^2 + 864*n - 168)*x + (1458*n^2 + 4158*n + 252)*x^2 + (486*n^2 + 7290*n + 2943)*x^3 + (4131*n + 5751)*x^4 + (729*n + 3645)*x^5 + 729*x^6)*deriv(v[i - 1], x))/(-72*n + 168 + (162*n^3 - 270*n^2 - 576*n + 1026)*x + (648*n^2 - 2160*n + 1566)*x^2 + (243*n^2 - 486*n - 162)*x^3 + (243*n - 486)*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-a1540a6d32c29f8f","steps":["basis A013610","D+1"],"edges":["e-251e56a692041c9f","e-5d63f6d45bd49dc1"],"text":"A266213 \u2192 basis A013610 \u2192 D+1"},"headline":"A266213 \u2192 basis A013610 \u2192 D+1","chain":null}