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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":"acc49abbdfca7a914b0eac190e430c56992763990a578a7fd97cebdcd9bcb03d","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-aff30617856e629d","role":"sequence","name":"shift_x_plus_one of Triangle read by rows: T(n, k) = (n-k+1)*(k+1)*binomial(n, k).","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,      2;\n  2:    28,     20,      3;\n  3:   110,    126,     42,      4;\n  4:   384,    616,    336,     72,      5;\n  5:  1232,   2560,   1980,    700,    110,      6;\n  6:  3712,   9504,   9600,   4840,   1260,    156,     7;\n  7: 10656,  32480,  40656,  26880,  10010,   2058,   210,    8;\n  8: 29440, 104192, 155904, 128128,  62720,  18480,  3136,  272,   9;\n  9: 78848, 317952, 554112, 545664, 332640, 129024, 31416, 4536, 342, 10;\n","recurrence_sentence":"The row polynomials satisfy (n^3+2*n^2+3*n+2+(n^3+2*n^2+7*n+2)*x+(2*n^2+2*n+2)*x^2+n*x^3)*P(n,x) = (2*n^3+10*n^2+20*n+16+(3*n^3+15*n^2+38*n+32)*x+(n^3+9*n^2+26*n+24)*x^2+(2*n^2+8*n+8)*x^3+(n+1)*x^4)*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[((2*n^3 + 10*n^2 + 20*n + 16 + (3*n^3 + 15*n^2 + 38*n + 32)*x + (n^3 + 9*n^2 + 26*n + 24)*x^2 + (2*n^2 + 8*n + 8)*x^3 + (n + 1)*x^4)*P[n - 1, x])/(n^3 + 2*n^2 + 3*n + 2 + (n^3 + 2*n^2 + 7*n + 2)*x + (2*n^2 + 2*n + 2)*x^2 + n*x^3)]];\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] = ((2*n^3 + 10*n^2 + 20*n + 16 + (3*n^3 + 15*n^2 + 38*n + 32)*x + (n^3 + 9*n^2 + 26*n + 24)*x^2 + (2*n^2 + 8*n + 8)*x^3 + (n + 1)*x^4)*v[i - 1])/(n^3 + 2*n^2 + 3*n + 2 + (n^3 + 2*n^2 + 7*n + 2)*x + (2*n^2 + 2*n + 2)*x^2 + n*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-c8c19eba9d68881f","steps":["\u2212D\u00b2+(\u22121+n)D+1+n","x \u21a6 x+1","D+1"],"edges":["e-168270346f6f7c93","e-2c04e0d67d228006","e-82464008625092be"],"text":"A007318 \u2192 \u2212D\u00b2+(\u22121+n)D+1+n \u2192 x \u21a6 x+1 \u2192 D+1"},"headline":"A007318 \u2192 \u2212D\u00b2+(\u22121+n)D+1+n \u2192 x \u21a6 x+1 \u2192 D+1","chain":null}