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the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"ba033308ffb8572d7b6f6b6b27a32792fe7d2cf7654c12ce877dc66cdb0fe830","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-a920b56c9b3ef4d1","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*1^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:         11,          1;\n  2:        121,         23,          1;\n  3:       1331,        397,         36,         1;\n  4:      14641,       6095,        865,        50,        1;\n  5:     161051,      87781,      17340,      1565,       65,       1;\n  6:    1771561,    1214423,     313201,     39250,     2540,      81,      1;\n  7:   19487171,   16344637,    5286036,    862701,    77350,    3836,     98,    1;\n  8:  214358881,  215622815,   85063105,  17363850,  2022951,  138726,   5502,  116,   1;\n  9: 2357947691, 2801832661, 1321443180, 328157005, 47708115, 4242567, 232260, 7590, 135, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (11+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[(11 + 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] = (11 + 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-a920b56c9b3ef4d1","steps":["basis A008277"],"edges":["e-697c46cc77488b97"],"text":"A038303 \u2192 basis A008277"},"headline":"A038303 \u2192 basis A008277","chain":null}
