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the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"b9a7f2c0ea209df23ad357921fbae8a8867a717510b7cfe9df8f38b14cf1bd78","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-ba8de4e02acfb6ca","role":"sequence","name":"Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.","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:      4,       3;\n   3:     16,      25,        5;\n   4:     64,     175,       77,        7;\n   5:    256,    1137,      855,      174,        9;\n   6:   1024,    7095,     8217,     2827,      330,      11;\n   7:   4096,   43225,    72839,    37726,     7397,     559,     13;\n   8:  16384,  259423,   613985,   449315,   130930,   16548,    875,    15;\n   9:  65536, 1541985,  5005599,  4974166,  2014517,  375258,  33082,  1292,   17;\n  10: 262144, 9107175, 39878169, 52379675, 28268466, 7241261, 935370, 60762, 1824, 19;\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) = 4 + 3*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 4 + 3*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] = 4 + 3*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-ba8de4e02acfb6ca","steps":["basis A008277"],"edges":["e-ac43a5abaccfbd29"],"text":"A091042 \u2192 basis A008277"},"headline":"A091042 \u2192 basis A008277","chain":null}