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the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"76a013e9d58f11c580103f0009e5739a8ccfb9e2b1f8b83f8c7b0cec143a6a08","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-7b91a7a037fb3513","role":"sequence","name":"Triangle whose (i,j)-th entry is 5^(i-j)*binomial(i,j).","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:        6,        1;\n  2:       36,       13,        1;\n  3:      216,      127,       21,        1;\n  4:     1296,     1105,      295,       30,       1;\n  5:     7776,     9031,     3465,      565,      40,      1;\n  6:    46656,    70993,    36751,     8550,     965,     51,     1;\n  7:   279936,   543607,   365001,   113701,   18200,   1526,    63,    1;\n  8:  1679616,  4085185,  3463615,  1388310,  295701,  34986,  2282,   76,  1;\n  9: 10077696, 30275911, 31794105, 15958405, 4345320, 680547, 62370, 3270, 90, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (6+x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(6 + x)*P[n - 1, x] + 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; for(n = 1, N, my(i = n + 1); v[i] = (6 + x)*v[i - 1] + 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-7b91a7a037fb3513","steps":["basis A008277"],"edges":["e-9e3373425bf2889e"],"text":"A038243 \u2192 basis A008277"},"headline":"A038243 \u2192 basis A008277","chain":null}