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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":"50ed4ade6088976781d6c2ab0d78672690fa8f9c53e98b9fe0caa75429b8a66f","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-32779d2065847504","role":"sequence","name":"Triangle read by rows: T(n,k) = (n+1)*(n+2)*(k+3)*binomial(n,k)/6, 0 <= k <= n.","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:   7,    4;\n  2:  22,   36,    10;\n  3:  50,  140,   110,    20;\n  4:  95,  380,   510,   260,     35;\n  5: 161,  840,  1610,  1400,    525,     56;\n  6: 252, 1624,  4060,  5040,   3220,    952,    84;\n  7: 372, 2856,  8820, 14280,  13020,   6552,  1596,   120;\n  8: 525, 4680, 17220, 34440,  40950,  29400, 12180,  2520,  165;\n  9: 715, 7260, 31020, 73920, 108570, 101640, 60060, 21120, 3795, 220;\n","recurrence_sentence":"The row polynomials satisfy (4*n^2-n+(n^3+2*n^2+3*n)*x+(n^2+2*n)*x^2)*P(n,x) = (4*n^2+11*n+6+(n^3+10*n^2+25*n+18)*x+(n^3+7*n^2+19*n+18)*x^2+(n^2+5*n+6)*x^3)*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[((4*n^2 + 11*n + 6 + (n^3 + 10*n^2 + 25*n + 18)*x + (n^3 + 7*n^2 + 19*n + 18)*x^2 + (n^2 + 5*n + 6)*x^3)*P[n - 1, x])/(4*n^2 - n + (n^3 + 2*n^2 + 3*n)*x + (n^2 + 2*n)*x^2)]];\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] = ((4*n^2 + 11*n + 6 + (n^3 + 10*n^2 + 25*n + 18)*x + (n^3 + 7*n^2 + 19*n + 18)*x^2 + (n^2 + 5*n + 6)*x^3)*v[i - 1])/(4*n^2 - n + (n^3 + 2*n^2 + 3*n)*x + (n^2 + 2*n)*x^2)); 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-c8c19eba9d68881f","steps":["(1/3+(1/2)n+(1/6)n\u00b2)\u03b8+1+(3/2)n+(1/2)n\u00b2","D+1"],"edges":["e-6ba022abf95075ed","e-da387006150e2232"],"text":"A007318 \u2192 (1/3+(1/2)n+(1/6)n\u00b2)\u03b8+1+(3/2)n+(1/2)n\u00b2 \u2192 D+1"},"headline":"A007318 \u2192 (1/3+(1/2)n+(1/6)n\u00b2)\u03b8+1+(3/2)n+(1/2)n\u00b2 \u2192 D+1","chain":null}