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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":"df55591caeace73405d2f7be14a87de79854f3fb6f1e1829e3f43638e615501a","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-e63fe40b997d2bc1","role":"sequence","name":"basis_map_A013611 of Triangle read by rows: T(n,k) = |s(n,n+1-k)|, where s(n,k) are the signed Stirling numbers of the first kind A008276 (1 <= k <= n; in other words, the unsigned Stirling numbers of the first kind in reverse order).","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:       2,         6;\n  2:       6,        40,         66;\n  3:      24,       256,        904,        1056;\n  4:     120,      1784,       9896,       24264,        22176;\n  5:     720,     13824,     105760,      402880,       763920,       576576;\n  6:    5040,    119088,    1168864,     6098720,     17836720,     27717552,      17873856;\n  7:   40320,   1134144,   13638080,    90868864,    362247680,    863862336,    1140822720,     643458816;\n  8:  362880,  11860416,  169242624,  1376981056,   6985852544,  22626915904,   45685760256,   52564860864,   26381811456;\n  9: 3628800, 135296640, 2238005376, 21554971264, 133199654016, 547618376064, 1497695734144, 2627193580416, 2681801714304, 1213563326976;\n","recurrence_sentence":"The row polynomials satisfy (1+x)*P(n,x) = (n+1+(6*n+2)*x+(5*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[((n + 1 + (6*n + 2)*x + (5*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] = ((n + 1 + (6*n + 2)*x + (5*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-4d60c3dd8627c01d","steps":["basis A013611","x \u21a6 x/(x+1)"],"edges":["e-60ae4909e1eb1035","e-69ea00ce236005d6"],"text":"A094638 \u2192 basis A013611 \u2192 x \u21a6 x/(x+1)"},"headline":"A094638 \u2192 basis A013611 \u2192 x \u21a6 x/(x+1)","chain":null}