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the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"56ad35cbeac57678a6018a1ed4c880ab563032ecd1c8e4eadd8c2835b435e038","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-7196be931c2ed996","role":"sequence","name":"Triangle T(n,k), 0 <= k <= n, read by rows, given by [1,0,-1,0,0,0,0,0,0,...] DELTA [3,-2,-1,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.","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:     4,       3;\n  2:     8,      13,        3;\n  3:    32,     115,       69,        9;\n  4:    64,     377,      391,      114,        9;\n  5:   256,    2831,     5041,     2655,      486,       27;\n  6:   512,    8749,    22995,    18316,     5571,      675,      27;\n  7:  2048,   62779,   256197,   307093,   144084,    29538,    2619,     81;\n  8:  4096,  190385,  1087567,  1791662,  1171597,   350850,   50490,   3348,    81;\n  9: 16384, 1344983, 11446825, 26554307, 24120538, 10180941, 2163330, 235170, 12312, 243;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (8+13*x+3*x^2)*P(n-2,x) + (13*x+6*x^2)*d/dx P(n-2,x) + 3*x^2*d^2/dx^2 P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 3*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4 + 3*x;\nP[n_, x_] := P[n, x] = Expand[(8 + 13*x + 3*x^2)*P[n - 2, x] + (13*x + 6*x^2)*D[P[n - 2, x], x] + 3*x^2*D[P[n - 2, x], {x, 2}]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (8 + 13*x + 3*x^2)*v[i - 2] + (13*x + 6*x^2)*deriv(v[i - 2], x) + 3*x^2*deriv(deriv(v[i - 2], x), 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-7196be931c2ed996","steps":["basis A008277"],"edges":["e-79898a409f2d103b"],"text":"A152842 \u2192 basis A008277"},"headline":"A152842 \u2192 basis A008277","chain":null}