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a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"077ac631128b0c6c4241e6eb784ef6e52fd1ed9836d5d8dca39f8ce0f2bed59b","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-84c761192e0db3c0","role":"sequence","name":"basis_map_A013611 of Riordan array (1-x,x/(1-x)).","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:        25,         80,          64;\n  1:       150,        680,        1024,          512;\n  2:       900,       5280,       11584,        11264,         4096;\n  3:      5400,      38880,      111744,       160256,       114688,        32768;\n  4:     32400,     276480,      981504,      1855488,      1970176,      1114112,       262144;\n  5:    194400,    1918080,     8100864,     18984960,     26664960,     22446080,     10485760,      2097152;\n  6:   1166400,   13063680,    63949824,    178716672,    311869440,    347996160,    242483200,     96468992,     16777216;\n  7:   6998400,   87713280,   488208384,   1583898624,   3300950016,   4582932480,   4238868480,   2518679552,    872415232,    134217728;\n  8:  41990400,  582266880,  3630956544,  13409058816,  32476889088,  53905195008,  62096670720,  49023025152,  25383927808,   7784628224,  1073741824;\n  9: 251942400, 3829524480, 26443874304, 109502005248, 302133805056, 583246282752, 803821584384, 790911516672, 544487768064, 249779191808, 68719476736, 8589934592;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (6+8*x)*P(n-1,x) for n >= 1, with P(0,x) = 25 + 80*x + 64*x^2.","mathematica":"P[0, x_] := 25 + 80*x + 64*x^2;\nP[n_, x_] := P[n, x] = Expand[(6 + 8*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] = 25 + 80*x + 64*x^2; for(n = 1, N, my(i = n + 1); v[i] = (6 + 8*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-7717563017cb9e5f","steps":["basis A013611","basis A013609"],"edges":["e-d3bf3843e93b142d","e-b045b4757da86ee8"],"text":"A159854 \u2192 basis A013611 \u2192 basis A013609"},"headline":"A159854 \u2192 basis A013611 \u2192 basis A013609","chain":null}