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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":"148724d6ce992a3458d1c41dbd4b7fcd18ec83f386a75a81fae6438688f37e9d","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-2821c21537bbbcdd","role":"sequence","name":"Triangle read by rows: T(n,k) = number of k-part order-consecutive partition of {1,2,...,n} (1 <= k <= n).","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:  0;\n   2:  2;\n   3:  5,    8;\n   4:  9,   32,   24;\n   5: 14,   82,  132,    64;\n   6: 20,  170,  438,   448,   160;\n   7: 27,  310, 1131,  1824,  1360,   384;\n   8: 35,  518, 2499,  5632,  6560,  3840,   896;\n   9: 44,  812, 4956, 14596, 23560, 21408, 10304,  2048;\n  10: 54, 1212, 9072, 33444, 70010, 87648, 65184, 26624, 4608;\n","recurrence_sentence":"The row polynomials satisfy (n^3-2*n^2-n+2+(4*n^2-12*n+8)*x)*P(n,x) = (n^3-3*n+2+(10*n^2-18*n+16)*x+(16*n-16)*x^2)*P(n-1,x) + ((2*n^2-2*n+4)*x+(2*n^2+6*n-4)*x^2+(8*n-8)*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 2.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[((n^3 - 3*n + 2 + (10*n^2 - 18*n + 16)*x + (16*n - 16)*x^2)*P[n - 1, x] + ((2*n^2 - 2*n + 4)*x + (2*n^2 + 6*n - 4)*x^2 + (8*n - 8)*x^3)*D[P[n - 1, x], x])/(n^3 - 2*n^2 - n + 2 + (4*n^2 - 12*n + 8)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 2; for(n = 3, N, my(i = n); v[i] = ((n^3 - 3*n + 2 + (10*n^2 - 18*n + 16)*x + (16*n - 16)*x^2)*v[i - 1] + ((2*n^2 - 2*n + 4)*x + (2*n^2 + 6*n - 4)*x^2 + (8*n - 8)*x^3)*deriv(v[i - 1], x))/(n^3 - 2*n^2 - n + 2 + (4*n^2 - 12*n + 8)*x)); 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-1bf99ad40f056d33","steps":["basis A056242","D"],"edges":["e-3cb5f3d64c4bb317","e-1da1d1636f7486ed"],"text":"basis-map coefficients of A026729 \u2192 basis A056242 \u2192 D"},"headline":"basis-map coefficients of A026729 \u2192 basis A056242 \u2192 D","chain":null}