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binomial(i,j)*11^(i-j)*7^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:               1521,                4914,                 3969;\n  1:              59319,              287469,               464373,                250047;\n  2:            2313441,            14948388,             36221094,              39007332,              15752961;\n  3:           90224199,           728733915,           2354371110,            3803214870,            3071827395,              992436543;\n  4:         3518743761,         34104747222,         137730709935,          296650759860,          359403805215,           232230151062,            62523502209;\n  5:       137231006679,       1551765998601,        7520096762451,        20246414360445,        32705746274565,         31699415619963,         17068916103057,          3938980639167;\n  6:      5352009260481,      69164427366216,      391045031647452,      1263376256091768,      2551048209416070,       3296739224476152,       2662750912076892,       1228961959420104,        248155780267521;\n  7:    208728361158759,    3034589250692727,    19608115158322236,     73907510981368428,    179083584301008114,     289288866947782338,     311541856712996364,     215682823878228252,      87102678873899871,      15633814156853823;\n  8:   8140406085191601,  131498867530018170,   955895613968209005,   4117704183247669560,  11640432979565527410,   22564531621927022364,   30375331029517145490,   28038767104169672760,   16985022380410474845,    6097187521172990970,    984930291881790849;\n  9: 317475837322472439, 5641301417037779493, 45564357599151295905, 220811886826656280155, 713392249747658751270, 1613364010967782099026, 2606203402332571083042, 3007157771922197403510, 2428858200398697902835, 1307846723291606563065, 422535095217288274221, 62050608388552823487;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (39+63*x)*P(n-1,x) for n >= 1, with P(0,x) = 1521 + 4914*x + 3969*x^2.","mathematica":"P[0, x_] := 1521 + 4914*x + 3969*x^2;\nP[n_, x_] := P[n, x] = Expand[(39 + 63*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] = 1521 + 4914*x + 3969*x^2; for(n = 1, N, my(i = n + 1); v[i] = (39 + 63*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-7dcf11eddf2b9a8a","steps":["basis A013610","basis A013610"],"edges":["e-632cc4d34c6e419b","e-b94f8d1462b15b40"],"text":"A038321 \u2192 basis A013610 \u2192 basis A013610"},"headline":"A038321 \u2192 basis A013610 \u2192 basis A013610","chain":null}