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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:            1080,              1638,               441;\n  1:           26244,             68040,             51597,               9261;\n  2:          594864,           2204496,           2857680,            1444716,             194481;\n  3:        12912048,          62460720,         115736040,          100018800,           37923795,            4084101;\n  4:       272097792,        1626918048,        3935025360,         4860913680,         3150592200,         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7155668939648229, 350277500542221;\n","recurrence_sentence":"The row polynomials satisfy (7*n+13+7*x)*P(n,x) = (126*n+360+(147*n+546)*x+147*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1080 + 1638*x + 441*x^2.","mathematica":"P[0, x_] := 1080 + 1638*x + 441*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((126*n + 360 + (147*n + 546)*x + 147*x^2)*P[n - 1, x])/(7*n + 13 + 7*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1080 + 1638*x + 441*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((126*n + 360 + (147*n + 546)*x + 147*x^2)*v[i - 1])/(7*n + 13 + 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-7dcf11eddf2b9a8a","steps":["basis A013610","D+1"],"edges":["e-632cc4d34c6e419b","e-656b23e0e3858abc"],"text":"A038321 \u2192 basis A013610 \u2192 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