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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":"cdd714a996bb68edaccb1d109ef9a6017ac7def6ecac768dd7e2ef1f8efe748c","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-f07088d208415183","role":"sequence","name":"basis_map_A013609 of Triangle T(n, k) is coefficient of z^n*w^k in 1/(1 - 2*z - 2*w - 2*z*w) read by rows in order 00, 10, 01, 20, 11, 02, ...","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:       8,        4;\n  2:      54,       68,        16;\n  3:     320,      688,       416,        64;\n  4:    1780,     5488,      5680,      2240,       256;\n  5:    9504,    38384,     57280,     38592,     11264,      1024;\n  6:   49336,   247024,    485024,    471616,    234496,     54272,      4096;\n  7:  250880,  1501056,   3666688,   4699136,   3369984,   1325056,    253952,    16384;\n  8: 1255824,  8746368,  25601664,  40891392,  38666496,  21903360,   7114752,  1163264,   65536;\n  9: 6208640, 49357632, 168552960, 323120640, 381087744, 284574720, 133005312, 36782080, 5242880, 262144;\n","recurrence_sentence":"The row polynomials satisfy (3*n^2+(3*n^3+3*n^2)*x+(8*n^2-6*n)*x^2+(4*n^2+2*n)*x^3+4*n*x^4)*P(n,x) = -(9*n^3-24*n^2-21*n+12+(12*n^3-42*n^2-78*n+48)*x+(8*n^2-86*n+38)*x^2+(16*n^2-28*n-44)*x^3-56*x^4-16*x^5)*P(n-1,x) + (9*n^2+9*n+(36*n^2+60*n+12)*x+(42*n^2+150*n+78)*x^2+(12*n^2+172*n+188)*x^3+(88*n+204)*x^4+(16*n+96)*x^5+16*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[(-(9*n^3 - 24*n^2 - 21*n + 12 + (12*n^3 - 42*n^2 - 78*n + 48)*x + (8*n^2 - 86*n + 38)*x^2 + (16*n^2 - 28*n - 44)*x^3 - 56*x^4 - 16*x^5)*P[n - 1, x] + (9*n^2 + 9*n + (36*n^2 + 60*n + 12)*x + (42*n^2 + 150*n + 78)*x^2 + (12*n^2 + 172*n + 188)*x^3 + (88*n + 204)*x^4 + (16*n + 96)*x^5 + 16*x^6)*D[P[n - 1, x], x])/(3*n^2 + (3*n^3 + 3*n^2)*x + (8*n^2 - 6*n)*x^2 + (4*n^2 + 2*n)*x^3 + 4*n*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] = (-(9*n^3 - 24*n^2 - 21*n + 12 + (12*n^3 - 42*n^2 - 78*n + 48)*x + (8*n^2 - 86*n + 38)*x^2 + (16*n^2 - 28*n - 44)*x^3 - 56*x^4 - 16*x^5)*v[i - 1] + (9*n^2 + 9*n + (36*n^2 + 60*n + 12)*x + (42*n^2 + 150*n + 78)*x^2 + (12*n^2 + 172*n + 188)*x^3 + (88*n + 204)*x^4 + (16*n + 96)*x^5 + 16*x^6)*deriv(v[i - 1], x))/(3*n^2 + (3*n^3 + 3*n^2)*x + (8*n^2 - 6*n)*x^2 + (4*n^2 + 2*n)*x^3 + 4*n*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-1bffaa6198344f93","steps":["basis A013609","D+1"],"edges":["e-c90acfb06fc8929f","e-6f41b455c7c6cce3"],"text":"A059473 \u2192 basis A013609 \u2192 D+1"},"headline":"A059473 \u2192 basis A013609 \u2192 D+1","chain":null}