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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":"0185a4a886b26a75a4673f9e1ea4330de5cbf40cb61512b5ecd990d95139dae7","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-99e5bf113a0a46cd","role":"sequence","name":"Triangle read by rows, A007318 rows repeated three times.","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: 1;\n  3: 2, 1;\n  4: 2, 1;\n  5: 2, 1;\n  6: 3, 4, 1;\n  7: 3, 4, 1;\n  8: 3, 4, 1;\n  9: 4, 9, 6, 1;\n","recurrence_sentence":"The row polynomials satisfy (n^3-3*n^2-25*n+54+(9*n^2-45*n-21)*x+(27*n-108)*x^2+27*x^3)*P(n,x) = (n^3+6*n^2-97*n+168+(n^3+15*n^2-115*n+12)*x+(9*n^2+9*n-237)*x^2+(27*n-54)*x^3+27*x^4)*P(n-3,x) - (27*n-162+(54*n-297)*x+(27*n-108)*x^2+27*x^3)*d/dx P(n-3,x) + (27+81*x+81*x^2+27*x^3)*d^2/dx^2 P(n-3,x) for n >= 3, with P(0,x) = 1, P(1,x) = 1, P(2,x) = 1.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[2, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((n^3 + 6*n^2 - 97*n + 168 + (n^3 + 15*n^2 - 115*n + 12)*x + (9*n^2 + 9*n - 237)*x^2 + (27*n - 54)*x^3 + 27*x^4)*P[n - 3, x] - (27*n - 162 + (54*n - 297)*x + (27*n - 108)*x^2 + 27*x^3)*D[P[n - 3, x], x] + (27 + 81*x + 81*x^2 + 27*x^3)*D[P[n - 3, x], {x, 2}])/(n^3 - 3*n^2 - 25*n + 54 + (9*n^2 - 45*n - 21)*x + (27*n - 108)*x^2 + 27*x^3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; v[3] = 1; for(n = 3, N, my(i = n + 1); v[i] = ((n^3 + 6*n^2 - 97*n + 168 + (n^3 + 15*n^2 - 115*n + 12)*x + (9*n^2 + 9*n - 237)*x^2 + (27*n - 54)*x^3 + 27*x^4)*v[i - 3] - (27*n - 162 + (54*n - 297)*x + (27*n - 108)*x^2 + 27*x^3)*deriv(v[i - 3], x) + (27 + 81*x + 81*x^2 + 27*x^3)*deriv(deriv(v[i - 3], x), x))/(n^3 - 3*n^2 - 25*n + 54 + (9*n^2 - 45*n - 21)*x + (27*n - 108)*x^2 + 27*x^3)); 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-99e5bf113a0a46cd","steps":["D+1"],"edges":["e-809cfb1679ce14dc"],"text":"A152828 \u2192 D+1"},"headline":"A152828 \u2192 D+1","chain":null}