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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":"4037d85c9c08a5472eeed4e16843fe2a2a9b0100b5848719f9317cb68ac3c1a8","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-c72653d9ab0ee8be","role":"sequence","name":"Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.","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:   1;\n   2:   2,    1;\n   3:   5,    5,    1;\n   4:  10,   17,    8,    1;\n   5:  20,   47,   37,   11,    1;\n   6:  38,  116,  130,   65,   14,    1;\n   7:  71,  266,  389,  279,  101,   17,    1;\n   8: 130,  579, 1048,  987,  514,  145,   20,   1;\n   9: 235, 1213, 2621, 3069, 2106,  855,  197,  23,  1;\n  10: 420, 2467, 6202, 8701, 7446, 3990, 1322, 257, 26, 1;\n","recurrence_sentence":"The row polynomials satisfy (n^3-6*n+5+(5*n^2-10*n+5)*x+(n^2+2*n-3)*x^2+(n-1)*x^3)*P(n,x) = -(2*n^3-9*n^2+4*n+10+(7*n^2-28*n+17)*x+(2*n^2-3*n-17)*x^2+(n-9)*x^3-x^4)*P(n-1,x) + (5*n^2+5*n+(6*n^2+31*n+10)*x+(n^2+36*n+37)*x^2+(11*n+37)*x^3+(n+11)*x^4+x^5)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(2*n^3 - 9*n^2 + 4*n + 10 + (7*n^2 - 28*n + 17)*x + (2*n^2 - 3*n - 17)*x^2 + (n - 9)*x^3 - x^4)*P[n - 1, x] + (5*n^2 + 5*n + (6*n^2 + 31*n + 10)*x + (n^2 + 36*n + 37)*x^2 + (11*n + 37)*x^3 + (n + 11)*x^4 + x^5)*D[P[n - 1, x], x])/(n^3 - 6*n + 5 + (5*n^2 - 10*n + 5)*x + (n^2 + 2*n - 3)*x^2 + (n - 1)*x^3)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(2*n^3 - 9*n^2 + 4*n + 10 + (7*n^2 - 28*n + 17)*x + (2*n^2 - 3*n - 17)*x^2 + (n - 9)*x^3 - x^4)*v[i - 1] + (5*n^2 + 5*n + (6*n^2 + 31*n + 10)*x + (n^2 + 36*n + 37)*x^2 + (11*n + 37)*x^3 + (n + 11)*x^4 + x^5)*deriv(v[i - 1], x))/(n^3 - 6*n + 5 + (5*n^2 - 10*n + 5)*x + (n^2 + 2*n - 3)*x^2 + (n - 1)*x^3)); 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-c72653d9ab0ee8be","steps":["D+1"],"edges":["e-c9969a1cd93f190d"],"text":"A063967 \u2192 D+1"},"headline":"A063967 \u2192 D+1","chain":null}