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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":"6f4fb0b9dbfbdaa619b18ee5f45d2eb5f6d9d944edabbe7c4a22a648d77c7b16","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-3af57aaf5c129e4b","role":"sequence","name":"Triangle T(n,k), 0 <= k <= n, read by rows given by [0, 1, 0, 0, 0, 0, ...] DELTA [1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.","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,  1;\n  2: 1,  3,   1;\n  3: 1,  7,   6,    1;\n  4: 1, 11,  23,   10,    1;\n  5: 1, 15,  52,   55,   15,    1;\n  6: 1, 19,  93,  168,  110,   21,    1;\n  7: 1, 23, 146,  381,  434,  196,   28,   1;\n  8: 1, 27, 211,  726, 1215,  966,  322,  36,  1;\n  9: 1, 31, 288, 1235, 2761, 3267, 1932, 498, 45, 1;\n","recurrence_sentence":"The row polynomials satisfy (4*n^2-10*n+6+(8*n^3-28*n^2+32*n-12)*x+(16*n^2-50*n+39)*x^2+(8*n-12)*x^3)*P(n,x) = (4*n^2-10*n+6+(8*n^3-4*n^2-34*n+30)*x+(24*n^3-44*n^2-16*n+41)*x^2+(48*n^2-102*n+55)*x^3+(24*n-28)*x^4)*P(n-1,x) - ((2*n-2)*x+(4*n^2+6*n-10)*x^2+(16*n^2-13)*x^3+(32*n-16)*x^4+16*x^5)*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[((4*n^2 - 10*n + 6 + (8*n^3 - 4*n^2 - 34*n + 30)*x + (24*n^3 - 44*n^2 - 16*n + 41)*x^2 + (48*n^2 - 102*n + 55)*x^3 + (24*n - 28)*x^4)*P[n - 1, x] - ((2*n - 2)*x + (4*n^2 + 6*n - 10)*x^2 + (16*n^2 - 13)*x^3 + (32*n - 16)*x^4 + 16*x^5)*D[P[n - 1, x], x])/(4*n^2 - 10*n + 6 + (8*n^3 - 28*n^2 + 32*n - 12)*x + (16*n^2 - 50*n + 39)*x^2 + (8*n - 12)*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] = ((4*n^2 - 10*n + 6 + (8*n^3 - 4*n^2 - 34*n + 30)*x + (24*n^3 - 44*n^2 - 16*n + 41)*x^2 + (48*n^2 - 102*n + 55)*x^3 + (24*n - 28)*x^4)*v[i - 1] - ((2*n - 2)*x + (4*n^2 + 6*n - 10)*x^2 + (16*n^2 - 13)*x^3 + (32*n - 16)*x^4 + 16*x^5)*deriv(v[i - 1], x))/(4*n^2 - 10*n + 6 + (8*n^3 - 28*n^2 + 32*n - 12)*x + (16*n^2 - 50*n + 39)*x^2 + (8*n - 12)*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-3af57aaf5c129e4b","steps":["D+1"],"edges":["e-7b05e26c3067ec2d"],"text":"A121314 \u2192 D+1"},"headline":"A121314 \u2192 D+1","chain":null}