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a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"f5e4f9696eb0e3271ed0b679117bf7b399a02fc53d522d3943f4e0cb8e3811de","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,             40;\n  2:          441,           1680,            1600;\n  3:         9261,          52920,          100800,           64000;\n  4:       194481,        1481760,         4233600,         5376000,          2560000;\n  5:      4084101,       38896200,       148176000,       282240000,        268800000,        102400000;\n  6:     85766121,      980184240,      4667544000,     11854080000,      16934400000,      12902400000,       4096000000;\n  7:   1801088541,    24014513880,    137225793600,    435637440000,     829785600000,     948326400000,     602112000000,     163840000000;\n  8:  37822859361,   576348333120,   3842322220800,  14637417984000,   34850995200000,   53106278400000,   50577408000000,   27525120000000,    6553600000000;\n  9: 794280046581, 13616229369960, 103742699961600, 461078666496000, 1317367618560000, 2509271654400000, 3186376704000000, 2601123840000000, 1238630400000000, 262144000000000;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (21+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[(21 + 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] = (21 + 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 A013611"],"edges":["e-bf7e65720bb019eb"],"text":"A038324 \u2192 basis A013611"},"headline":"A038324 \u2192 basis A013611","chain":null}