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the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"391d44e0d709e5cce7929bdf9d7d583e6828b15d6a4c7421bf0b91e87312ecb5","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-e1d1088f7d1ff894","role":"sequence","name":"Triangle read by rows whose (i,j)-th entry is binomial(i,j)*11^(i-j)*10^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:           21,             10;\n  2:          441,            520,             100;\n  3:         9261,          20530,            9300,            1000;\n  4:       194481,         729040,          586600,          144000,           10000;\n  5:      4084101,       24545050,        31341000,        13210000,         2050000,         100000;\n  6:     85766121,      801737560,      1530431500,       987120000,       257150000,       27600000,        1000000;\n  7:   1801088541,    25711525570,     70765067100,     65647435000,     25557350000,     4531100000,      357000000,      10000000;\n  8:  37822859361,   815068178080,   3158483006800,   4055669856000,   2215472700000,   577281600000,    74228000000,    4480000000,   100000000;\n  9: 794280046581, 25645342114090, 137648485059600, 238423992724000, 175700533260000, 63141720600000, 11785284000000, 1149960000000, 54900000000, 1000000000;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (21+10*x)*P(n-1,x) + 10*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[(21 + 10*x)*P[n - 1, x] + 10*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] = (21 + 10*x)*v[i - 1] + 10*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-e1d1088f7d1ff894","steps":["basis A008277"],"edges":["e-6a3d937c421a3f39"],"text":"A038324 \u2192 basis A008277"},"headline":"A038324 \u2192 basis A008277","chain":null}