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a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"e92997e3645a860cef6c6f85a041b78001411fad7bd7c6c4682babb35362f79f","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-6aadffa116a44288","role":"sequence","name":"basis_map_A013610 of Triangle T(n,k), 0 <= k <= n, read by rows defined by: T(n,k) = (binomial(2*n,2*k) + binomial(2*n+1,2*k))/2.","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:         9,         24;\n  2:        81,        384,         432;\n  3:       729,       4920,       10512,         6912;\n  4:      6561,      57408,      180000,       235008,       103680;\n  5:     59049,     634776,     2623968,      5142528,      4665600,       1492992;\n  6:    531441,    6775872,    34776144,     91058688,    126302976,      85598208,      20901888;\n  7:   4782969,   70548792,   432559728,   1418598144,   2658334464,    2797369344,    1484034048,     286654464;\n  8:  43046721,  721032576,  5141392704,  20268278784,  47928983040,   68797071360,   57438388224,   24652283904,   3869835264;\n  9: 387420489, 7264134168, 59057410752, 272005502976, 777375843840, 1417411768320, 1627946532864, 1112792629248, 396013142016, 51597803520;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(9+72*x+144*x^2)*P(n-2,x) + (10+24*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 9 + 24*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 9 + 24*x;\nP[n_, x_] := P[n, x] = Expand[-(9 + 72*x + 144*x^2)*P[n - 2, x] + (10 + 24*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; v[2] = 9 + 24*x; for(n = 2, N, my(i = n + 1); v[i] = -(9 + 72*x + 144*x^2)*v[i - 2] + (10 + 24*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-c85abc56078671cc","steps":["basis A013610","basis A013611"],"edges":["e-12f73d6f0ea24cf5","e-fdb7155c6fdc6b35"],"text":"A232535 \u2192 basis A013610 \u2192 basis A013611"},"headline":"A232535 \u2192 basis A013610 \u2192 basis A013611","chain":null}