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a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"7a96e4e8dd4cf2a145a246f389532cfe448b2f0dfd69d7c3c93227938dd27b92","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-2f02b2035f8f4541","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*8^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:           19,            32;\n  2:          361,          1216,           1024;\n  3:         6859,         34656,          58368,           32768;\n  4:       130321,        877952,        2217984,         2490368,         1048576;\n  5:      2476099,      20851360,       70236160,       118292480,        99614720,        33554432;\n  6:     47045881,     475411008,     2001730560,      4495114240,      5678039040,      3825205248,      1073741824;\n  7:    893871739,   10538277344,    53246032896,    149462548480,    251726397440,    254376148992,    142807662592,     34359738368;\n  8:  16983563041,  228831165184,  1348899500032,   4543661473792,   9565603102720,  12888391548928,  10853382356992,   5222680231936,   1099511627776;\n  9: 322687697779, 4891266155808, 32951687786496, 129494352003072, 327143626113024, 550978738716672, 618642794348544, 446539159830528, 188016488349696, 35184372088832;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (19+32*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[(19 + 32*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] = (19 + 32*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-2f02b2035f8f4541","steps":["basis A013611"],"edges":["e-36980f01022d8136"],"text":"A038322 \u2192 basis A013611"},"headline":"A038322 \u2192 basis A013611","chain":null}