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for an input row that is a polynomial the degree condition holds automatically","lean_route":"fTransform","lean_declarations":["RealRooted.splits_fPolynomial"],"certificate_hash":"95a58b421d464fe98c93066838e7e54feb22a77dd80251af7974437e24b5ef1b","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-c20d7dca3713c0eb","role":"sequence","name":"basis_map_A007318 of Triangle whose (n,k)-th entry is 15^(n-k)*binomial(n,k).","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:          16,           17;\n  2:         256,          544,           289;\n  3:        4096,        13056,         13872,          4913;\n  4:       65536,       278528,        443904,        314432,          83521;\n  5:     1048576,      5570560,      11837440,      12577280,        6681680,        1419857;\n  6:    16777216,    106954752,     284098560,     402472960,      320720640,      136306272,      24137569;\n  7:   268435456,   1996488704,    6363807744,   11269242880,    11973570560,     7633151232,    2703407728,     410338673;\n  8:  4294967296,  36507222016,  135761231872,  288492617728,   383154257920,   325681119232,  173018094592,   52523350144,    6975757441;\n  9: 68719476736, 657129996288, 2792802484224, 6923822825472, 11034842628096, 11724520292352, 8304868540416, 3781681210368, 1004509071504, 118587876497;\n","recurrence_sentence":"The row polynomials satisfy (1+x)*P(n,x) = (16+33*x+17*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[Together[((16 + 33*x + 17*x^2)*P[n - 1, x])/(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] = ((16 + 33*x + 17*x^2)*v[i - 1])/(1 + 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-ff0214c333bfdcca","steps":["x \u21a6 x+1","x \u21a6 x/(x+1)"],"edges":["e-5b39e32205b8abe3","e-1e96fecde40a0389"],"text":"A027467 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x/(x+1)"},"headline":"A027467 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x/(x+1)","chain":null}