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the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"c214b65bdb1d92d5581ba2f06d7b58c607fd7a9d9f9b8f918e523bc52cc05645","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-c1716beb5dbd6d37","role":"sequence","name":"basis_map_A007318 of Triangle read by rows: T(n, k) = binomial(2n, 2k).","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:       3,        1;\n   3:      17,       11,        1;\n   4:      99,      109,       24,        1;\n   5:     577,      975,      377,       42,        1;\n   6:    3363,     8221,     4860,      965,       65,       1;\n   7:   19601,    66863,    56197,    17014,     2060,      93,      1;\n   8:  114243,   531133,   607488,   258881,    47698,    3892,    126,     1;\n   9:  665857,  4151295,  6277457,  3589082,   924711,  114646,   6734,   164,   1;\n  10: 3880899, 32074813, 62839860, 46698253, 15994251, 2760087, 246036, 10902, 207, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(1+3*x+x^2)*P(n-2,x) - (3*x+2*x^2)*d/dx P(n-2,x) - x^2*d^2/dx^2 P(n-2,x) + (6+2*x)*P(n-1,x) + 2*x*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 3 + x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 3 + x;\nP[n_, x_] := P[n, x] = Expand[-(1 + 3*x + x^2)*P[n - 2, x] - (3*x + 2*x^2)*D[P[n - 2, x], x] - x^2*D[P[n - 2, x], {x, 2}] + (6 + 2*x)*P[n - 1, x] + 2*x*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 3 + x; for(n = 3, N, my(i = n); v[i] = -(1 + 3*x + x^2)*v[i - 2] - (3*x + 2*x^2)*deriv(v[i - 2], x) - x^2*deriv(deriv(v[i - 2], x), x) + (6 + 2*x)*v[i - 1] + 2*x*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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-f7935c36a3c0ec8c","steps":["basis A007318","basis A008277"],"edges":["e-219ad3693217b36c","e-e2fa9af9ee7ec898"],"text":"A086645 \u2192 basis A007318 \u2192 basis A008277"},"headline":"A086645 \u2192 basis A007318 \u2192 basis A008277","chain":null}