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a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"408cee26d3cbfb50f16555a301aebd1461521ef2c8d52894aeef802848216b34","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-fbf962df53148c52","role":"sequence","name":"basis_map_A013609 of 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:              41,               40;\n  2:            1681,             3280,              1600;\n  3:           68921,           201720,            196800,             64000;\n  4:         2825761,         11027360,          16137600,          10496000,           2560000;\n  5:       115856201,        565152200,        1102736000,        1075840000,         524800000,         102400000;\n  6:      4750104241,      27805488240,       67818264000,       88218880000,       64550400000,       25190400000,        4096000000;\n  7:    194754273881,    1330029187480,     3892768353600,     6329704640000,     6175321600000,     3614822400000,     1175552000000,     163840000000;\n  8:   7984925229121,   62321367641920,   212804669996800,   415228624384000,   506376371200000,   395220582400000,   192790528000000,   53739520000000,    6553600000000;\n  9: 327381934393961, 2874573082483560, 11217846175545600, 25536560399616000, 37370576194560000, 36459098726400000, 23713234944000000, 9914941440000000, 2418278400000000, 262144000000000;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (41+40*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(41 + 40*x)*P[n - 1, 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] = (41 + 40*x)*v[i - 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-e1d1088f7d1ff894","steps":["basis A013609","basis A013609"],"edges":["e-25d25e50279a57da","e-34a2b32a8ccc4d9c"],"text":"A038324 \u2192 basis A013609 \u2192 basis A013609"},"headline":"A038324 \u2192 basis A013609 \u2192 basis A013609","chain":null}