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for an input row that is a polynomial the degree condition holds automatically","lean_route":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"3781c571f6114d39bca711b22330dad7ed987115cbc8a491afd69e42891a64d9","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}}],"premises":[{"position":0,"node_id":"n-eb51f16fedb3c293","role":"sequence","name":"Triangle read by rows, T(n, k) = Sum_{i=0..n} L'(n, n-i) * binomial(i, k), for k = 0..n-1.","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:          5,           2;\n  2:         37,          30,           6;\n  3:        361,         444,         180,          24;\n  4:       4361,        7220,        4440,        1200,         120;\n  5:      62701,      130830,      108300,       44400,        9000,         720;\n  6:    1044205,     2633442,     2747430,     1516200,      466200,       75600,       5040;\n  7:   19748177,    58475480,    73736376,    51285360,    21226800,     5221440,     705600,     40320;\n  8:  417787921,  1421868744,  2105117280,  1769673024,   923136480,   305665920,   62657280,   7257600,   362880;\n  9: 9770678101, 37600912890, 63984093480, 63153518400, 39817643040, 16616456640, 4584988800, 805593600, 81648000, 3628800;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (4*n+1+2*n*x)*P(n-1,x) - (4+4*x+x^2)*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[(4*n + 1 + 2*n*x)*P[n - 1, x] - (4 + 4*x + x^2)*D[P[n - 1, x], 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] = (4*n + 1 + 2*n*x)*v[i - 1] - (4 + 4*x + x^2)*deriv(v[i - 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-d55979c30b97975a","steps":["x \u21a6 x+1","x \u21a6 x+1"],"edges":["e-be016f00370af8fd","e-155ab131999d2c00"],"text":"A089231 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x+1"},"headline":"A089231 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x+1","chain":null}