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the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"900d1e15dfeab9e5d921e36bf64f8fac69dbd6287a73f4a5da8dfeaa3378e21e","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-2feb6dc01f424b0a","role":"sequence","name":"Triangle read by rows: coefficients of 1; 1(X+2); 1(X+2)(X+3); 1(X+2)(X+3)(X+4); ....","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:        3,          2;\n  2:       12,         23,           6;\n  3:       60,        255,         170,           24;\n  4:      360,       3105,        3995,         1354,          120;\n  5:     2520,      42525,       94535,        57820,        11844,          720;\n  6:    20160,     655515,     2377445,      2338525,       831124,       113868,        5040;\n  7:   181440,   11305035,    64680245,     96190885,     52784284,     12228524,     1198224,      40320;\n  8:  1814400,  216428625,  1912792175,   4141184950,   3293795029,   1147627376,   186743052,   13727376,    362880;\n  9: 19958400, 4563145125, 61460843675, 188916504700, 209395395979, 102943220226, 24735030452, 2979347976, 170295840, 3628800;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (n+2+(n+1)*x)*P(n-1,x) + (n+1)*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[(n + 2 + (n + 1)*x)*P[n - 1, x] + (n + 1)*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] = (n + 2 + (n + 1)*x)*v[i - 1] + (n + 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-2feb6dc01f424b0a","steps":["basis A008277"],"edges":["e-d5c3213d2b3c5934"],"text":"A145324 \u2192 basis A008277"},"headline":"A145324 \u2192 basis A008277","chain":null}