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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":"50711f6b3771baedf1907d10718f7f19b32e5d1433d755816fb910673b443088","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-47b361d1e904fbeb","role":"sequence","name":"basis_map_A007318 of Triangle read by rows: T(n,k) number of tilings of a 2n X 3 grid by dominoes, 2k of which are in a vertical position (0<=k<=n).","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:      14,        8;\n  1:      76,       88,        24;\n  2:     372,      648,       360,        64;\n  3:    1718,     3992,      3360,      1216,       160;\n  4:    7642,    22192,     25056,     13760,      3680,       384;\n  5:   33112,   115344,    163440,    120640,     48960,     10368,      896;\n  6:  140712,   571632,    974736,    904640,    493760,    158592,    27776,     2048;\n  7:  589098,  2734032,   5450400,   6097216,   4187680,   1809024,   480256,    71680,    4608;\n  8: 2437222, 12720888,  29028960,  38018624,  31500000,  17127936,  6114304,  1382400,  179712,  10240;\n  9: 9986596, 57898408, 148828104, 223387840, 216858880, 142302720, 63942144, 19433472, 3824640, 440320, 22528;\n","recurrence_sentence":"The row polynomials satisfy (2*n^2+5*n+3)*P(n,x) = (4*n^2+19*n+21+(2*n^2+10*n+12)*x)*P(n-1,x) + (6*n+12+(8*n+16)*x+(2*n+4)*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 14 + 8*x.","mathematica":"P[0, x_] := 14 + 8*x;\nP[n_, x_] := P[n, x] = Expand[Together[((4*n^2 + 19*n + 21 + (2*n^2 + 10*n + 12)*x)*P[n - 1, x] + (6*n + 12 + (8*n + 16)*x + (2*n + 4)*x^2)*D[P[n - 1, x], x])/(2*n^2 + 5*n + 3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 14 + 8*x; for(n = 1, N, my(i = n + 1); v[i] = ((4*n^2 + 19*n + 21 + (2*n^2 + 10*n + 12)*x)*v[i - 1] + (6*n + 12 + (8*n + 16)*x + (2*n + 4)*x^2)*deriv(v[i - 1], x))/(2*n^2 + 5*n + 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-8eb9bb3a09f0b7fd","steps":["basis A007318","D"],"edges":["e-1acefd6886502727","e-73efa9af5929e39d"],"text":"A123519 \u2192 basis A007318 \u2192 D"},"headline":"A123519 \u2192 basis A007318 \u2192 D","chain":null}