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the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"17c6fac8666037781deda001830f5a94adb89887d45a8a4b59de4eccd9d65a20","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-f7935c36a3c0ec8c","role":"sequence","name":"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:      2,       1;\n   3:      8,       9,        1;\n   4:     32,      67,       21,        1;\n   5:    128,     449,      263,       38,       1;\n   6:    512,    2851,     2685,      725,      60,       1;\n   7:   2048,   17553,    24655,    10514,    1625,      87,      1;\n   8:   8192,  106051,   212541,   131649,   31808,    3178,    119,    1;\n   9:  32768,  633089,  1759103,  1506118,  520741,   80794,   5642,  156,   1;\n  10: 131072, 3749827, 14162493, 16225789, 7618116, 1675227, 180810, 9318, 198, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(x+x^2)*P(n-2,x) - (x+2*x^2)*d/dx P(n-2,x) - x^2*d^2/dx^2 P(n-2,x) + (4+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) = 2 + x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 2 + x;\nP[n_, x_] := P[n, x] = Expand[-(x + x^2)*P[n - 2, x] - (x + 2*x^2)*D[P[n - 2, x], x] - x^2*D[P[n - 2, x], {x, 2}] + (4 + 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] = 2 + x; for(n = 3, N, my(i = n); v[i] = -(x + x^2)*v[i - 2] - (x + 2*x^2)*deriv(v[i - 2], x) - x^2*deriv(deriv(v[i - 2], x), x) + (4 + 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 A008277"],"edges":["e-7a99d256bfb82b46"],"text":"A086645 \u2192 basis A008277"},"headline":"A086645 \u2192 basis A008277","chain":null}