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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":"6a2db11f60998bf552f8e90764bd996e44811e6ee3abb0cb5c8d19cb5f9630dc","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-6186cc445ed1db3e","role":"sequence","name":"shift_x_plus_one of Triangle read by rows: A008277 * A007318.","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:       3,        4;\n  2:      11,       29,        19;\n  3:      47,      184,       239,       103;\n  4:     227,     1175,      2272,      1945,       622;\n  5:    1215,     7806,     19994,     25521,     16234,      4117;\n  6:    7107,    54461,    173381,    293524,    278701,    140724,     29521;\n  7:   44959,   399820,   1519825,   3201171,   4034939,   3043566,   1272123,    227290;\n  8:  305091,  3086291,  13626502,  34297087,  53824051,  53931947,  33695516,  12001715,   1865881;\n  9: 2206399, 25004986, 125670692, 367626953, 689838155, 861096517, 715033150, 380874505, 118094087, 16239523;\n","recurrence_sentence":"The row polynomials satisfy (1+x)*P(n,x) = -(2*n-5+(7*n-14)*x+(8*n-12)*x^2+(3*n-3)*x^3)*P(n-1,x) + (2+9*x+15*x^2+11*x^3+3*x^4)*d/dx 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 - 5 + (7*n - 14)*x + (8*n - 12)*x^2 + (3*n - 3)*x^3)*P[n - 1, x] + (2 + 9*x + 15*x^2 + 11*x^3 + 3*x^4)*D[P[n - 1, x], 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 - 5 + (7*n - 14)*x + (8*n - 12)*x^2 + (3*n - 3)*x^3)*v[i - 1] + (2 + 9*x + 15*x^2 + 11*x^3 + 3*x^4)*deriv(v[i - 1], x))/(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-bdae5aa3d1cac0ec","steps":["x \u21a6 x+1","x \u21a6 x/(x+1)"],"edges":["e-73e4f5467288c90f","e-d469a036b93143f5"],"text":"A137597 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x/(x+1)"},"headline":"A137597 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x/(x+1)","chain":null}