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binomial(i,j)*12^(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:             627,              364,               49;\n  1:           14440,            13167,             3822,              343;\n  2:          322373,           404320,           184338,            35672,             2401;\n  3:         7037334,         11283055,          7075600,          2150610,           312130,            16807;\n  4:       151042039,        295568028,        236944155,         99058400,         22581405,          2621892,          117649;\n  5:      3199119908,       7401059911,       7241416686,       3870087865,       1213465400,        221297769,        21412118,         823543;\n  6:     67040380425,     179150714848,     207229677508,     135173111472,      54181230110,      13590812480,      2065445844,      171296944,        5764801;\n  7:   1392652169362,    4223543966775,    5643247517712,    4351823227668,    2128976505684,     682683499386,    142703531040,    18589012596,     1348963434,      40353607;\n  8:  28719205102331,   97485651855340,  147824038837125,  131675775413280,   76156906484190,   29805671079576,   7964640826170,  1427035310400,   162653860215,   10491937820,   282475249;\n  9: 588582360748896, 2211378792879487, 3753197596430590, 3794150330152875, 2534758676705640, 1172816359856526, 382506112187892, 87611049087870, 13735214862600, 1391594137395, 80787921214, 1977326743;\n","recurrence_sentence":"The row polynomials satisfy (7*n+26+7*x)*P(n,x) = (133*n+627+(49*n+364)*x+49*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 627 + 364*x + 49*x^2.","mathematica":"P[0, x_] := 627 + 364*x + 49*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((133*n + 627 + (49*n + 364)*x + 49*x^2)*P[n - 1, x])/(7*n + 26 + 7*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 627 + 364*x + 49*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((133*n + 627 + (49*n + 364)*x + 49*x^2)*v[i - 1])/(7*n + 26 + 7*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-5cf6acfdac607d6d","steps":["x \u21a6 x+1","D+1"],"edges":["e-737551651b8dc504","e-b0748bf555fe4a84"],"text":"A038333 \u2192 x \u21a6 x+1 \u2192 D+1"},"headline":"A038333 \u2192 x \u21a6 x+1 \u2192 D+1","chain":null}