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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":"5ed1c8bc7121162b660d29aa70389da60303927543dd1cf7dcadd792b9075c28","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"-1"}}}],"premises":[{"position":0,"node_id":"n-60ec3928030a20ba","role":"sequence","name":"veronese_section_2_0 of Triangle read by rows: T(n,k) = number of k-part order-consecutive partition of {1,2,...,n} (1 <= k <= n).","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:     1;\n   2:     1;\n   3:     5,     4;\n   4:    17,    16;\n   5:    58,    73,    16;\n   6:   198,   309,   112;\n   7:   676,  1259,   648,    64;\n   8:  2308,  4995,  3328,   640;\n   9:  7880, 19432, 15889,  4592,  256;\n  10: 26904, 74464, 72153, 27920, 3328;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = x*P(n-4,x) - 4*x*P(n-3,x) - (2-4*x)*P(n-2,x) + 4*P(n-1,x) for n >= 5, with P(1,x) = 1, P(2,x) = 1, P(3,x) = 5 + 4*x, P(4,x) = 17 + 16*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 1;\nP[3, x_] := 5 + 4*x;\nP[4, x_] := 17 + 16*x;\nP[n_, x_] := P[n, x] = Expand[x*P[n - 4, x] - 4*x*P[n - 3, x] - (2 - 4*x)*P[n - 2, x] + 4*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1; v[3] = 5 + 4*x; v[4] = 17 + 16*x; for(n = 5, N, my(i = n); v[i] = x*v[i - 4] - 4*x*v[i - 3] - (2 - 4*x)*v[i - 2] + 4*v[i - 1]); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-2821c21537bbbcdd","steps":["Veronese 2","x \u21a6 x+1"],"edges":["e-37a36b1b0e8be851","e-8cb9b616162dec46"],"text":"A056242 \u2192 Veronese 2 \u2192 x \u21a6 x+1"},"headline":"A056242 \u2192 Veronese 2 \u2192 x \u21a6 x+1","chain":null}