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a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"7f3a62b95142f2f6997cd0892efbdd80b707a906586d018bc12cc0278e04490e","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-8fd7b2e367875124","role":"sequence","name":"basis_map_A013609 of Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:       1;\n   2:       5,        8;\n   3:      24,       80,        64;\n   4:     115,      584,       960,       512;\n   5:     551,     3760,      9408,     10240,       4096;\n   6:    2640,    22624,     76160,    125952,     102400,      32768;\n   7:   12649,   130480,    552384,   1228800,    1515520,     983040,     262144;\n   8:   60605,   730968,   3729600,  10437120,   17305600,   17006592,    9175040,    2097152;\n   9:  290376,  4009200,  23943360,  80793600,  168509440,  222494720,  181665792,   83886080,  16777216;\n  10: 1391275, 21638040, 148060800, 585077760, 1471590400, 2443542528, 2679111680, 1870659584, 754974720, 134217728;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -P(n-2,x) + (5+8*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 5 + 8*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 5 + 8*x;\nP[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (5 + 8*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 5 + 8*x; for(n = 3, N, my(i = n); v[i] = -v[i - 2] + (5 + 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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-976cdb6e4a693ec8","steps":["basis A013609","basis A013611"],"edges":["e-567d6f79aa6a87e5","e-292fccded59db922"],"text":"A078812 \u2192 basis A013609 \u2192 basis A013611"},"headline":"A078812 \u2192 basis A013609 \u2192 basis A013611","chain":null}