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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":"bf916a6d31362a440ec37db7df7f798ee528c5a8b24c7e306654da794afd04af","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-8e6b51ac707373f6","role":"sequence","name":"Triangle read by rows: T(n,k) is number of paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch the horizontal axis), consisting of steps u=(2,1),U=(1,2), or d=(1,-1) and have k peaks of the form Ud.","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:       1;\n  2:       5,       4;\n  3:      28,      42,       15;\n  4:     165,     360,      252,       56;\n  5:    1001,    2860,     2970,     1320,      210;\n  6:    6188,   21840,    30030,    20020,     6435,      792;\n  7:   38760,  162792,   278460,   247520,   120120,    30030,    3003;\n  8:  245157, 1193808,  2441880,  2713200,  1763580,   668304,  136136,  11440;\n  9: 1562275, 8652600, 20593188, 27457584, 22383900, 11395440, 3527160, 604656, 43758;\n","recurrence_sentence":"The row polynomials satisfy (4*n^3-6*n^2+2*n)*P(n,x) = (27*n^3-54*n^2+33*n-6+(24*n^3-52*n^2+36*n-8)*x+(n^3-4*n^2+5*n-2)*x^2)*P(n-1,x) - ((9*n^2-9*n+2)*x+(10*n^2-11*n+3)*x^2+(n^2-2*n+1)*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((27*n^3 - 54*n^2 + 33*n - 6 + (24*n^3 - 52*n^2 + 36*n - 8)*x + (n^3 - 4*n^2 + 5*n - 2)*x^2)*P[n - 1, x] - ((9*n^2 - 9*n + 2)*x + (10*n^2 - 11*n + 3)*x^2 + (n^2 - 2*n + 1)*x^3)*D[P[n - 1, x], x])/(4*n^3 - 6*n^2 + 2*n)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((27*n^3 - 54*n^2 + 33*n - 6 + (24*n^3 - 52*n^2 + 36*n - 8)*x + (n^3 - 4*n^2 + 5*n - 2)*x^2)*v[i - 1] - ((9*n^2 - 9*n + 2)*x + (10*n^2 - 11*n + 3)*x^2 + (n^2 - 2*n + 1)*x^3)*deriv(v[i - 1], x))/(4*n^3 - 6*n^2 + 2*n)); 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-8e6b51ac707373f6","steps":["D"],"edges":["e-5769aaed6c405931"],"text":"A108426 \u2192 D"},"headline":"A108426 \u2192 D","chain":null}