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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":"3bed966337add258b1bb39434907cdd4d9e018ce0de239bb45c243c94cbbe6f3","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-b8883330fe0b24f8","role":"sequence","name":"basis_map_A007318 of Triangle read by rows: coefficients of 1; 1(X+2); 1(X+2)(X+3); 1(X+2)(X+3)(X+4); ....","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,         5;\n  2:       12,        41,         35;\n  3:       60,       313,        544,         315;\n  4:      360,      2538,       6707,        7874,        3465;\n  5:     2520,     22446,      79943,      142309,      126617,       45045;\n  6:    20160,    217368,     976234,     2337617,     3147571,     2259615,      675675;\n  7:   181440,   2299032,   12481362,    37634531,    68067628,    73845242,    44494530,    11486475;\n  8:  1814400,  26437680,  168495228,   613491188,  1395732369,  2031737352,  1848004898,   960260820,   218243025;\n  9: 19958400, 328916880, 2408638788, 10286802856, 28236371007, 51659490621, 62994538270, 49370971878, 22566150495, 4583103525;\n","recurrence_sentence":"The row polynomials satisfy (1+x)*P(n,x) = (n+2+(3*n+5)*x+(2*n+3)*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 + 2 + (3*n + 5)*x + (2*n + 3)*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 + 2 + (3*n + 5)*x + (2*n + 3)*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-2feb6dc01f424b0a","steps":["basis A007318","x \u21a6 x/(x+1)"],"edges":["e-5281dccfa5f22cf2","e-8414ec7f90512903"],"text":"A145324 \u2192 basis A007318 \u2192 x \u21a6 x/(x+1)"},"headline":"A145324 \u2192 basis A007318 \u2192 x \u21a6 x/(x+1)","chain":null}