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0<=k<=2n.","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: 19, 36, 16;\n 2: 361, 1368, 1904, 1152, 256;\n 3: 6859, 38988, 91200, 112320, 76800, 27648, 4096;\n 4: 130321, 987696, 3246112, 6041088, 6961920, 5087232, 2301952, 589824, 65536;\n 5: 2476099, 23457780, 99318320, 247443840, 401693440, 443943936, 338268160, 175472640, 59310080, 11796480, 1048576;\n 6: 47045881, 534837384, 2771145744, 8652216960, 18129246720, 26855000064, 28836171776, 22614736896, 12856197120, 5166858240, 1393557504, 226492416, 16777216;\n 7: 893871739, 11855562012, 72658649056, 272710767168, 700273830144, 1301333354496, 1804735213568, 1897462185984, 1519777021952, 922829193216, 418183643136, 137141157888, 30769414144, 4227858432, 268435456;\n 8: 16983563041, 257435060832, 1821616512320, 7986904934400, 24285328775680, 54298563895296, 92342351101952, 121843582894080, 126060165529600, 102605122437120, 65483772526592, 32425560244224, 12212638842880, 3382286745600, 649613803520, 77309411328, 4294967296;\n 9: 322687697779, 5502674425284, 44150112932688, 221448349170432, 778095688573440, 2033735028885504, 4097818231578624, 6508129736982528, 8258989746880512, 8437160611676160, 6954938734215168, 4615183414591488, 2447100667232256, 1022727410417664, 329507206594560, 78971563671552, 13258564042752, 1391569403904, 68719476736;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (19+36*x+16*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(19 + 36*x + 16*x^2)*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] = (19 + 36*x + 16*x^2)*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-b30e7a5b4198051a","steps":["basis A013609","basis A013609"],"edges":["e-89275777750acd05","e-842a0471538edcfa"],"text":"A272866 \u2192 basis A013609 \u2192 basis A013609"},"headline":"A272866 \u2192 basis A013609 \u2192 basis A013609","chain":null}