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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":"852ab9182d4d8a995600345873df6a7428d2c11e94041ff63a7a94c412398a94","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-44370b381098ad54","role":"sequence","name":"basis_map_A013610 of A Riordan array of coordination sequences.","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:     6,      3;\n  2:    25,     36,       9;\n  3:    86,    231,     162,      27;\n  4:   269,   1092,    1404,     648,       81;\n  5:   798,   4371,    8478,    7074,     2430,      243;\n  6:  2285,  15816,   41553,   52056,    31995,     8748,     729;\n  7:  6382,  53487,  178308,  302589,   277830,   134865,   30618,    2187;\n  8: 17497, 172344,  698256, 1505952,  1870290,  1350108,  540918,  104976,   6561;\n  9: 47286, 535539, 2558196, 6718788, 10606788, 10335762, 6133806, 2090772, 354294, 19683;\n","recurrence_sentence":"The row polynomials satisfy (216*n^2-432*n+48+(162*n^3-108*n^2-252*n-828)*x+(162*n^2-540*n-1188)*x^2+(243*n^2-81)*x^3+(243*n+243)*x^4)*P(n,x) = -(216*n^3-1440*n^2+1608*n+624+(486*n^3-4158*n^2+4086*n+6246)*x-(1134*n^2+810*n-14418)*x^2+(729*n^2-2592*n+5589)*x^3-(729*n+2673)*x^4-1458*x^5)*P(n-1,x) + (144*n^2-240*n-80+(864*n^2-1296*n-1032)*x+(1458*n^2-1350*n-3906)*x^2+(486*n^2+1944*n-4347)*x^3+(2673*n+1620)*x^4+(729*n+2916)*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 - 1440*n^2 + 1608*n + 624 + (486*n^3 - 4158*n^2 + 4086*n + 6246)*x - (1134*n^2 + 810*n - 14418)*x^2 + (729*n^2 - 2592*n + 5589)*x^3 - (729*n + 2673)*x^4 - 1458*x^5)*P[n - 1, x] + (144*n^2 - 240*n - 80 + (864*n^2 - 1296*n - 1032)*x + (1458*n^2 - 1350*n - 3906)*x^2 + (486*n^2 + 1944*n - 4347)*x^3 + (2673*n + 1620)*x^4 + (729*n + 2916)*x^5 + 729*x^6)*D[P[n - 1, x], x])/(216*n^2 - 432*n + 48 + (162*n^3 - 108*n^2 - 252*n - 828)*x + (162*n^2 - 540*n - 1188)*x^2 + (243*n^2 - 81)*x^3 + (243*n + 243)*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 - 1440*n^2 + 1608*n + 624 + (486*n^3 - 4158*n^2 + 4086*n + 6246)*x - (1134*n^2 + 810*n - 14418)*x^2 + (729*n^2 - 2592*n + 5589)*x^3 - (729*n + 2673)*x^4 - 1458*x^5)*v[i - 1] + (144*n^2 - 240*n - 80 + (864*n^2 - 1296*n - 1032)*x + (1458*n^2 - 1350*n - 3906)*x^2 + (486*n^2 + 1944*n - 4347)*x^3 + (2673*n + 1620)*x^4 + (729*n + 2916)*x^5 + 729*x^6)*deriv(v[i - 1], x))/(216*n^2 - 432*n + 48 + (162*n^3 - 108*n^2 - 252*n - 828)*x + (162*n^2 - 540*n - 1188)*x^2 + (243*n^2 - 81)*x^3 + (243*n + 243)*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-ec35d1145e9a1211","steps":["basis A013610","D+1"],"edges":["e-3d8a64e4b99d997d","e-ae84aeefcf164a5b"],"text":"A113413 \u2192 basis A013610 \u2192 D+1"},"headline":"A113413 \u2192 basis A013610 \u2192 D+1","chain":null}