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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":"8d4da4f4ec358a9eff4af642b148321fea0b287ec421f814739ccede0a2290a6","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-656046abebe1865b","role":"sequence","name":"basis_map_A013609 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,        4;\n  2:       6,       26,        28;\n  3:      24,      164,       372,        280;\n  4:     120,     1132,      3992,       6236,        3640;\n  5:     720,     8712,     42064,     101288,      121616,       58240;\n  6:    5040,    74664,    459976,    1508232,     2775784,     2718384,     1106560;\n  7:   40320,   708192,   5322416,   22185328,    55387376,    82814320,    68656928,    24344320;\n  8:  362880,  7381728,  65606544,  332728352,  1053119584,  2130013280,  2688270352,  1935522080,   608608000;\n  9: 3628800, 83977920, 862753824, 5164266752, 19847589696, 50787481152, 86523075360, 94626790656, 60280698240, 17041024000;\n","recurrence_sentence":"The row polynomials satisfy (1+x)*P(n,x) = (n+1+(4*n+2)*x+(3*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 + (4*n + 2)*x + (3*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 + (4*n + 2)*x + (3*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 A013609","x \u21a6 x/(x+1)"],"edges":["e-23134a27b4e4bcc8","e-144520f9a112c23d"],"text":"A094638 \u2192 basis A013609 \u2192 x \u21a6 x/(x+1)"},"headline":"A094638 \u2192 basis A013609 \u2192 x \u21a6 x/(x+1)","chain":null}