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the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"191d8fe0bf289d86b67e81ac0bc7a7f22642d2793436df4b0ece46bb091f9223","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-1be843c964e88f9e","role":"sequence","name":"(7,1) Pascal triangle.","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:    8,     1;\n  2:   16,    11,      1;\n  3:   32,    49,     15,      1;\n  4:   64,   179,    109,     20,      1;\n  5:  128,   601,    615,    209,     26,     1;\n  6:  256,  1931,   3061,   1660,    365,    33,     1;\n  7:  512,  6049,  14175,  11361,   3850,   596,    41,    1;\n  8: 1024, 18659,  62749,  70980,  34461,  8022,   924,   50,  1;\n  9: 2048, 57001, 269655, 417649, 277746, 90615, 15414, 1374, 60, 1;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (2*n-2+(n-1)*x)*P(n-1,x) + (n-1)*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 8 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 8 + x;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n - 2 + (n - 1)*x)*P[n - 1, x] + (n - 1)*x*D[P[n - 1, x], x])/(n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 8 + x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n - 2 + (n - 1)*x)*v[i - 1] + (n - 1)*x*deriv(v[i - 1], x))/(n - 1)); 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-1be843c964e88f9e","steps":["basis A008277"],"edges":["e-cb959bb80f9670c6"],"text":"A093564 \u2192 basis A008277"},"headline":"A093564 \u2192 basis A008277","chain":null}