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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":"57aec465f90a129b79410c83eb449065f281ec84da4680ac224499dab2eb8792","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-44b1dbf5d009bf35","role":"sequence","name":"A convolution triangle of numbers obtained from A036224.","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:            42,             2;\n  1:          1113,           126,             3;\n  2:         23184,          4662,           252,            4;\n  3:        412272,        130410,         11970,          420,           5;\n  4:       6531840,       3036096,        419580,        24360,         630,          6;\n  5:      94618368,      61825680,      12032307,      1025640,       43155,        882,         7;\n  6:    1274921856,    1134724032,     297906336,     34895700,     2120580,      69678,      1176,        8;\n  7:   16176101760,   19147482432,    6579379872,   1015163604,    83427435,    3913434,    105252,     1512,      9;\n  8:  195030291456,  301257605376,  132461671680,  26136527040,  2785427190,  174707442,   6650280,   151200,   1890,   10;\n  9: 2250184914432, 4465750423680, 2468708057088, 609388621920, 81792868365, 6579546246, 332066826, 10614240, 208845, 2310, 11;\n","recurrence_sentence":"The row polynomials satisfy (n^2+5*n+6)*P(n,x) = -(36*n^2+720*n+3276+(36*n+252)*x+6*x^2)*P(n-2,x) - ((216*n+1512)*x+(6*n+114)*x^2+2*x^3)*d/dx P(n-2,x) + (12*n^2+150*n+252+(6*n+12)*x)*P(n-1,x) + ((36*n+72)*x+(n+2)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 42 + 2*x, P(1,x) = 1113 + 126*x + 3*x^2.","mathematica":"P[0, x_] := 42 + 2*x;\nP[1, x_] := 1113 + 126*x + 3*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(36*n^2 + 720*n + 3276 + (36*n + 252)*x + 6*x^2)*P[n - 2, x] - ((216*n + 1512)*x + (6*n + 114)*x^2 + 2*x^3)*D[P[n - 2, x], x] + (12*n^2 + 150*n + 252 + (6*n + 12)*x)*P[n - 1, x] + ((36*n + 72)*x + (n + 2)*x^2)*D[P[n - 1, x], x])/(n^2 + 5*n + 6)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 42 + 2*x; v[2] = 1113 + 126*x + 3*x^2; for(n = 2, N, my(i = n + 1); v[i] = (-(36*n^2 + 720*n + 3276 + (36*n + 252)*x + 6*x^2)*v[i - 2] - ((216*n + 1512)*x + (6*n + 114)*x^2 + 2*x^3)*deriv(v[i - 2], x) + (12*n^2 + 150*n + 252 + (6*n + 12)*x)*v[i - 1] + ((36*n + 72)*x + (n + 2)*x^2)*deriv(v[i - 1], x))/(n^2 + 5*n + 6)); 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-44b1dbf5d009bf35","steps":["D"],"edges":["e-8b8fded416737aa9"],"text":"A132166 \u2192 D"},"headline":"A132166 \u2192 D","chain":null}