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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":"a2a1bb8c1f9e742f117299bfabefda647d049eae999012ed0e548e3939b72648","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-57cc3445e053f3c8","role":"sequence","name":"basis_map_A013610 of Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:     1;\n   2:     6,      3;\n   3:    26,     36,       9;\n   4:    96,    237,     162,      27;\n   5:   325,   1188,    1431,     648,       81;\n   6:  1044,   5112,    9072,    7182,     2430,      243;\n   7:  3239,  19980,   47439,   55080,    32400,     8748,     729;\n   8:  9804,  73131,  218862,  339255,   291600,   136323,   30618,    2187;\n   9: 29134, 255276,  925011, 1805976,  2067930,  1408428,  546021,  104976,   6561;\n  10: 85350, 859749, 3665088, 8664354, 12481938, 11303145, 6368544, 2108268, 354294, 19683;\n","recurrence_sentence":"The row polynomials satisfy (54*n^3-162*n^2+240*n-132+(54*n-54)*x-(54*n-54)*x^2)*P(n,x) = (81*n^3+72*n+(81*n^3+36*n)*x-81*n*x^2-81*n*x^3)*P(n-1,x) + (45*n^2-45*n+20+(162*n^2-162*n+27)*x+(81*n^2-81*n-171)*x^2-243*x^3-81*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((81*n^3 + 72*n + (81*n^3 + 36*n)*x - 81*n*x^2 - 81*n*x^3)*P[n - 1, x] + (45*n^2 - 45*n + 20 + (162*n^2 - 162*n + 27)*x + (81*n^2 - 81*n - 171)*x^2 - 243*x^3 - 81*x^4)*D[P[n - 1, x], x])/(54*n^3 - 162*n^2 + 240*n - 132 + (54*n - 54)*x - (54*n - 54)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((81*n^3 + 72*n + (81*n^3 + 36*n)*x - 81*n*x^2 - 81*n*x^3)*v[i - 1] + (45*n^2 - 45*n + 20 + (162*n^2 - 162*n + 27)*x + (81*n^2 - 81*n - 171)*x^2 - 243*x^3 - 81*x^4)*deriv(v[i - 1], x))/(54*n^3 - 162*n^2 + 240*n - 132 + (54*n - 54)*x - (54*n - 54)*x^2)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-976cdb6e4a693ec8","steps":["basis A013610","D+1"],"edges":["e-1c0d9915253416a6","e-059500cb756d573e"],"text":"A078812 \u2192 basis A013610 \u2192 D+1"},"headline":"A078812 \u2192 basis A013610 \u2192 D+1","chain":null}