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the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"5bfe1cda339f583b96142f24c4d7a90632146df66dc74ec193999b5d9396edb1","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-f7cf3d554e6e9f71","role":"sequence","name":"basis_map_A013611 of Triangle of coefficients of Chebyshev's S(n,x+4) polynomials (exponents of x in increasing order).","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:         1;\n  1:         9,          4;\n  2:        80,         88,          16;\n  3:       711,       1460,         624,          64;\n  4:      6319,      21736,       16432,        3840,         256;\n  5:     56160,     306384,      365664,      146304,       21760,        1024;\n  6:    499121,    4185896,     7425392,     4531200,     1128960,      116736,       4096;\n  7:   4435929,   56106748,   142609584,   124710464,    46327040,     7900160,     602112,     16384;\n  8:  39424240,  742945544,  2641364528,  3184826880,  1655888896,   414296064,   51466240,   3014656,    65536;\n  9: 350382231, 9759882284, 47732369568, 77322112128, 54090202880, 18630241280, 3354968064, 317390848, 14745600, 262144;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -P(n-2,x) + (9+4*x)*P(n-1,x) + 4*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 9 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 9 + 4*x;\nP[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (9 + 4*x)*P[n - 1, x] + 4*x*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 9 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = -v[i - 2] + (9 + 4*x)*v[i - 1] + 4*x*deriv(v[i - 1], x)); 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-30cb3cad216ed189","steps":["basis A013611","basis A008277"],"edges":["e-21c167d0f859d82c","e-13b04982e458fdc3"],"text":"A207823 \u2192 basis A013611 \u2192 basis A008277"},"headline":"A207823 \u2192 basis A013611 \u2192 basis A008277","chain":null}