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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":"4b2bc38377406b72f0c5256ae7f5841ed95ce9ef920fd0c626757efada92007a","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-aa9a15392d2524ca","role":"sequence","name":"derivative of Triangle of Dyck paths counted by number of long interior inclines.","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:    0;\n  1:    2;\n  2:   12,     4;\n  3:   50,    42,      6;\n  4:  160,   264,     96,      8;\n  5:  420,  1190,    840,    180,     10;\n  6:  952,  4200,   5040,   2080,    300,    12;\n  7: 1932, 12348,  22932,  15960,   4410,   462,    14;\n  8: 3600, 31584,  84672,  90720,  42000,  8400,   672,  16;\n  9: 6270, 72468, 266112, 410256, 291060, 97020, 14784, 936, 18;\n","recurrence_sentence":"The row polynomials satisfy (n^4-n^3-3*n^2+21*n-18+(2*n^3-n^2-16*n+15)*x+(n^2+2*n-3)*x^2)*P(n,x) = (n^4+4*n^3+5*n^2+14*n+24+(3*n^4-4*n^3+11*n^2-4*n-42)*x+(6*n^3-9*n^2-7*n+22)*x^2+(3*n^2+n-4)*x^3)*P(n-1,x) + ((2*n^3-n^2+5*n+6)*x-(2*n^3-5*n^2+7*n+12)*x^2-(4*n^2-4*n-8)*x^3-(2*n+2)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[((n^4 + 4*n^3 + 5*n^2 + 14*n + 24 + (3*n^4 - 4*n^3 + 11*n^2 - 4*n - 42)*x + (6*n^3 - 9*n^2 - 7*n + 22)*x^2 + (3*n^2 + n - 4)*x^3)*P[n - 1, x] + ((2*n^3 - n^2 + 5*n + 6)*x - (2*n^3 - 5*n^2 + 7*n + 12)*x^2 - (4*n^2 - 4*n - 8)*x^3 - (2*n + 2)*x^4)*D[P[n - 1, x], x])/(n^4 - n^3 - 3*n^2 + 21*n - 18 + (2*n^3 - n^2 - 16*n + 15)*x + (n^2 + 2*n - 3)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((n^4 + 4*n^3 + 5*n^2 + 14*n + 24 + (3*n^4 - 4*n^3 + 11*n^2 - 4*n - 42)*x + (6*n^3 - 9*n^2 - 7*n + 22)*x^2 + (3*n^2 + n - 4)*x^3)*v[i - 1] + ((2*n^3 - n^2 + 5*n + 6)*x - (2*n^3 - 5*n^2 + 7*n + 12)*x^2 - (4*n^2 - 4*n - 8)*x^3 - (2*n + 2)*x^4)*deriv(v[i - 1], x))/(n^4 - n^3 - 3*n^2 + 21*n - 18 + (2*n^3 - n^2 - 16*n + 15)*x + (n^2 + 2*n - 3)*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-c963abfc0266d66c","steps":["D","D+1"],"edges":["e-0041de9aacba5ab3","e-f5e8cf014c9bb49c"],"text":"A108838 \u2192 D \u2192 D+1"},"headline":"A108838 \u2192 D \u2192 D+1","chain":null}