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the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"eb0f36183a0281cba9e3dc874383270d65184303419bccf2945a8f3ba339b223","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-0836bf76d9ba1ba5","role":"sequence","name":"basis_map_A013609 of Triangle read by rows: T(n,k) = binomial(k,n-k).","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:      4,       2;\n  2:     15,      20,        4;\n  3:     57,     148,       72,        8;\n  4:    216,     984,      868,      224,       16;\n  5:    819,    6206,     8844,     3968,      640,       32;\n  6:   3105,   38020,    82364,    57152,    15600,     1728,      64;\n  7:  11772,  228978,   726908,   732496,   300880,    55360,    4480,    128;\n  8:  44631, 1364744,  6198900,  8725376,  5060576,  1375104,  182336,  11264,   256;\n  9: 169209, 8081416, 51668100, 98972464, 77891760, 29318944, 5663168, 567296, 27648, 512;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3+2*x)*P(n-2,x) + 2*x*d/dx P(n-2,x) + (3+2*x)*P(n-1,x) + 2*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4 + 2*x;\nP[n_, x_] := P[n, x] = Expand[(3 + 2*x)*P[n - 2, x] + 2*x*D[P[n - 2, x], x] + (3 + 2*x)*P[n - 1, x] + 2*x*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; v[2] = 4 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (3 + 2*x)*v[i - 2] + 2*x*deriv(v[i - 2], x) + (3 + 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 - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-21854f4257effb01","steps":["basis A013609","basis A008277"],"edges":["e-38aa524f6e49a0d8","e-121b3a0cd5968a76"],"text":"A030528 \u2192 basis A013609 \u2192 basis A008277"},"headline":"A030528 \u2192 basis A013609 \u2192 basis A008277","chain":null}