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real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); 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":"dec0fd78f2c6c078f6027d538b64d1771dfc627f9bf0fe4a5501f4d34ea3871c","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-5ffbcfe26ecef156","role":"sequence","name":"basis_map_A013610 of Triangle of central factorial numbers |t(2n,2n-2k)| read by rows.","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:              3;\n  2:             39,              72;\n  3:            660,            2826,             2916;\n  4:          16020,          111402,           253044,            186624;\n  5:         544020,         5299452,         19111869,          30156624,          17496000;\n  6:       24902340,       313588044,       1565650377,        3867904224,        4718496240,        2267481600;\n  7:     1485520800,     23000690160,     147428682552,      500262685092,      946652213160,      945716816736,      388873094400;\n  8:   112258692000,   2065484847600,   16207063131960,    70258817264292,   181595438277480,   279580262999904,   237117318084864,    85329867571200;\n  9: 10496769894000, 223927481815200, 2081848403770920, 11012311470426984, 36231941682782055, 75878811367705296, 98704624643431632, 72847915763263488, 23327052547276800;\n","recurrence_sentence":"The row polynomials satisfy (n^2-2*n+1)*P(n,x) = -(n^6-4*n^5+8*n^4-8*n^3+4*n^2+(6*n^6-24*n^5+42*n^4-36*n^3+12*n^2)*x+(9*n^6-36*n^5+54*n^4-36*n^3+9*n^2)*x^2)*P(n-2,x) + (2*n^4-4*n^3+4*n^2-2*n+1+(6*n^4-12*n^3+6*n^2)*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 3;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^6 - 4*n^5 + 8*n^4 - 8*n^3 + 4*n^2 + (6*n^6 - 24*n^5 + 42*n^4 - 36*n^3 + 12*n^2)*x + (9*n^6 - 36*n^5 + 54*n^4 - 36*n^3 + 9*n^2)*x^2)*P[n - 2, x] + (2*n^4 - 4*n^3 + 4*n^2 - 2*n + 1 + (6*n^4 - 12*n^3 + 6*n^2)*x)*P[n - 1, x])/(n^2 - 2*n + 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 3; for(n = 2, N, my(i = n + 1); v[i] = (-(n^6 - 4*n^5 + 8*n^4 - 8*n^3 + 4*n^2 + (6*n^6 - 24*n^5 + 42*n^4 - 36*n^3 + 12*n^2)*x + (9*n^6 - 36*n^5 + 54*n^4 - 36*n^3 + 9*n^2)*x^2)*v[i - 2] + (2*n^4 - 4*n^3 + 4*n^2 - 2*n + 1 + (6*n^4 - 12*n^3 + 6*n^2)*x)*v[i - 1])/(n^2 - 2*n + 1)); 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-e734a631f8279507","steps":["basis A013610","D"],"edges":["e-5cbcb537f7289c2f","e-5d4a850c4b79c370"],"text":"A008955 \u2192 basis A013610 \u2192 D"},"headline":"A008955 \u2192 basis A013610 \u2192 D","chain":null}