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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":"a16801d7682fa468164d1c06a293bd0f5ac2f5ea030d5699953d26636d0853de","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-e0a0f9191d8a0512","role":"sequence","name":"basis_map_A007318 of Triangle read by rows: coefficients of the polynomials given by KummerU(-n, 1 - n - x, 3).","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:       4,        5;\n  2:      17,       43,        27;\n  3:      78,      299,       379,        159;\n  4:     393,     2024,      3875,       3272,       1029;\n  5:    2208,    14269,     36556,      46459,      29316,       7353;\n  6:   13977,   108351,    346954,     588001,     556714,     279387,      58095;\n  7:  100026,   900927,   3449965,    7287384,    9176931,    6893732,    2861811,     506691;\n  8:  806769,  8248104,  36630446,   92365344,  144721779,  144357088,   89559462,   31608216,   4860297;\n  9: 7280604, 83041749, 418346424, 1222462142, 2284570884, 2832828711, 2331526744, 1228594518, 376225740, 51023277;\n","recurrence_sentence":"The row polynomials satisfy (1+x)*P(n,x) = -(3*n-3+(9*n-9)*x+(9*n-9)*x^2+(3*n-3)*x^3)*P(n-2,x) + (n+3+(2*n+7)*x+(n+4)*x^2)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 5*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4 + 5*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(3*n - 3 + (9*n - 9)*x + (9*n - 9)*x^2 + (3*n - 3)*x^3)*P[n - 2, x] + (n + 3 + (2*n + 7)*x + (n + 4)*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; v[2] = 4 + 5*x; for(n = 2, N, my(i = n + 1); v[i] = (-(3*n - 3 + (9*n - 9)*x + (9*n - 9)*x^2 + (3*n - 3)*x^3)*v[i - 2] + (n + 3 + (2*n + 7)*x + (n + 4)*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-ff69dadb0811f5e6","steps":["x \u21a6 x+1","x \u21a6 x/(x+1)"],"edges":["e-29bcfb0ecbd3591f","e-8bc647f8e83f1183"],"text":"A327997 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x/(x+1)"},"headline":"A327997 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x/(x+1)","chain":null}