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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:               47,                81;\n  2:             2209,              7614,               6561;\n  3:           103823,            536787,             925101,             531441;\n  4:          4879681,          33638652,           86959494,           99910908,            43046721;\n  5:        229345007,        1976270805,         6811827030,        11739531690,         10115979435,          3486784401;\n  6:      10779215329,      111461673402,       480233805615,      1103515978860,       1426353100335,        983273201082,        282429536481;\n  7:     506623120463,     6111815091543,     31599384409467,     90764189261235,     156423390003405,     161748441577989,      92919317502249,      22876792454961;\n  8:   23811286661761,   328291782060024,   1980228089659932,   6825467032444872,   14703798660320070,   20272471344441288,   17468831690422812,    8601673963065336,   1853020188851841;\n  9: 1119130473102767, 17358427976423769, 119662354560878748, 481195425787363476, 1243941366663077922, 2143813844674666206, 2463105268349616492, 1819254043188318564, 783827539884328743, 150094635296999121;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (47+81*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(47 + 81*x)*P[n - 1, 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] = (47 + 81*x)*v[i - 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-4e096b5929fd3f2e","steps":["basis A013610","basis A013610"],"edges":["e-3bcf549fe20349b4","e-4137e56c730ea8ca"],"text":"A038323 \u2192 basis A013610 \u2192 basis A013610"},"headline":"A038323 \u2192 basis A013610 \u2192 basis A013610","chain":null}