{"id":"n-4fda2e5d70c9292d","revision":0,"first_index":0,"fingerprint":"f6-1a2f142e31a877af","name":"mobius_x_over_x_plus_one of basis_map_A013609 of Coefficients of raising factorial polynomials, T(n,k) = [x^k] p(x, n) where p(x, n) = (m*x + n - 1)*p(x, n - 1) with p[x, 0] = 1, p[x, -1] = 0, p[x, 1] = m*x and m = 2. 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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":"51e33a45c9263bbcaa733ee8bd84de86f3c98fb40d1a34d5f530efeba9a126e3","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-52d9fa43425cb9e1","role":"sequence","name":"basis_map_A013609 of Coefficients of raising factorial polynomials, T(n,k) = [x^k] p(x, n) where p(x, n) = (m*x + n - 1)*p(x, n - 1) with p[x, 0] = 1, p[x, -1] = 0, p[x, 1] = m*x and m = 2. Triangle read by rows, for n >= 0 and 0 <= k <= n.","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,       32,        42;\n  3:      24,      176,       424,        336;\n  4:     120,     1096,      3704,       5496,       3024;\n  5:     720,     7776,     33184,      70016,      73104,      30240;\n  6:    5040,    62352,    317824,     855136,    1281904,    1015824,      332640;\n  7:   40320,   559296,   3290816,   10654976,   20516864,   23509440,    14851008,    3991680;\n  8:  362880,  5557824,  36888192,  138675392,  323166464,  478304192,   439281792,  228988224,   51891840;\n  9: 3628800, 60658560, 446691456, 1903188608, 5173120128, 9307372416, 11089076608, 8439827328, 3724753536, 726485760;\n","recurrence_sentence":"The row polynomials satisfy (1+x)*P(n,x) = (n+1+(2*n+6)*x+(n+5)*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 + (2*n + 6)*x + (n + 5)*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 + (2*n + 6)*x + (n + 5)*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-60c7e36cbe887b7e","steps":["basis A013609","x \u21a6 x/(x+1)"],"edges":["e-7070cc8d62757598","e-0c1592b57a977ad5"],"text":"A137320 \u2192 basis A013609 \u2192 x \u21a6 x/(x+1)"},"headline":"A137320 \u2192 basis A013609 \u2192 x \u21a6 x/(x+1)","chain":null}