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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":"52a3f312c3a064fe3067ee6905acdc5b0191d23606716418c7ea8a91a025fdf7","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-054881b8a1092a7d","role":"sequence","name":"basis_map_A013611 of Triangle T(n, k) read by rows: T(n, k) = Sum_{j=0..n} binomial(j, n-k) * |Stirling1(n, n-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:        1,          5;\n  2:        3,         26,          55;\n  3:       15,        181,         717,          935;\n  4:      105,       1612,        9182,        23036,        21505;\n  5:      945,      17553,      129386,       473602,       861589,        623645;\n  6:    10395,     226158,     2037601,      9738132,     26053549,      37015710,      21827575;\n  7:   135135,    3366249,    35761291,    210137357,    737959549,    1549399739,    1801402585,     894930575;\n  8:  2027025,   56845080,   694633068,   4832841032,  20945849014,   57925094888,   99842826508,   98089880120,   42061737025;\n  9: 34459425, 1073798685, 14821551396, 118973850148, 612220007934, 2094856610838, 4767358079700, 6959197766964, 5913813175785, 2229272062325;\n","recurrence_sentence":"The row polynomials satisfy (1+x)*P(n,x) = (2*n-1+(8*n-2)*x+(6*n-1)*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[((2*n - 1 + (8*n - 2)*x + (6*n - 1)*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] = ((2*n - 1 + (8*n - 2)*x + (6*n - 1)*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-849de7332df28657","steps":["basis A013611","x \u21a6 x/(x+1)"],"edges":["e-3d3e7193ca830199","e-8bb923dd09499c74"],"text":"A088996 \u2192 basis A013611 \u2192 x \u21a6 x/(x+1)"},"headline":"A088996 \u2192 basis A013611 \u2192 x \u21a6 x/(x+1)","chain":null}