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the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"5f5598ebed581c295ac9679403ec6e09a9e5e1d32c29584d0e66afe2a2c3fbe3","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-4e096b5929fd3f2e","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*9^j.","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:           20,              9;\n  2:          400,            441,             81;\n  3:         8000,          16389,           7047,             729;\n  4:       160000,         547281,         415287,           97686,           6561;\n  5:      3200000,       17311149,       20706435,         8328825,        1246590,          59049;\n  6:     64000000,      530823321,      942644871,       577812690,      144768465,       15057495,        531441;\n  7:   1280000000,    15969876309,    40597914987,     35641000269,    13307348250,     2281653360,     174844089,      4782969;\n  8:  25600000000,   474646412961,  1686449656287,   2040508247526,  1065980504421,   268073602650,   33473342826,   1970583228,    43046721;\n  9: 512000000000, 13995145975869, 68356904655555, 111081934540305, 78059482475310, 27018608712039, 4889689792974, 464818493358, 21695547384, 387420489;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (20+9*x)*P(n-1,x) + 9*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(20 + 9*x)*P[n - 1, x] + 9*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; for(n = 1, N, my(i = n + 1); v[i] = (20 + 9*x)*v[i - 1] + 9*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-4e096b5929fd3f2e","steps":["basis A008277"],"edges":["e-177cf07282fec367"],"text":"A038323 \u2192 basis A008277"},"headline":"A038323 \u2192 basis A008277","chain":null}