{"id":"n-3121cc0e9141d5d1","revision":0,"first_index":0,"fingerprint":"f6-bf9e494f8c7b5418","name":"basis_map_A013610 of Triangle of coefficients of polynomials u(n,x) jointly generated with A112351; see the Formula section.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["2","3"],["6","18","9"],["18","84","108","27"],["54","342","720","540","81"],["162","1296","3798","4860","2430","243"],["486","4698","17658","32130","28350","10206","729"],["1458","16524","75816","179280","229068","149688","40824","2187"],["4374","56862","308124","897480","1507572","1449252","734832","157464","6561"],["13122","192456","1202850","4165020","8687412","11092464","8389332","3411720","590490","19683"],["39366","642978","4553334","18276030","45586800","72801828","73806876","45323388","15155910","2165130","59049"],["118098","2125764","16822404","76807440","223095870","428378868","546767496","453811248","231668910","64953900","7794468","177147"],["354294","6967782","60938568","311909940","1034324154","2322467154","3581889012","3765401640","2617642170","1131496938","270208224","27634932","531441"],["1062882","22674816","217182222","1231762140","4592608146","11812791816","21387470526","27291826692","24162995142","14321035872","5319724410","1096185636","96722262","1594323"],["3188646","73338858","763503570","4752854010","19684220346","57083862402","118762843626","178552431810","192706739550","146199537198","74932118118","24212806254","4352501790","334807830","4782969"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","33","237","1737","12789","94257","694845","5122521","37764549","278410689","2052521613","15131766249","111555636885","822419536977"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013610","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["2","3"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","-3","-9"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["3","6"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":201824,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-50ca6587bb81d58f"]},{"id":201825,"position":1,"claim":"one_signed_coefficients","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":0},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 0: the coefficients of P_n have one sign","region_exact":null,"edges":["e-50ca6587bb81d58f"]},{"id":201826,"position":2,"claim":"hadamard_kernel","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":0},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-50ca6587bb81d58f"]},{"id":201827,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-50ca6587bb81d58f"]}],"region_exact":{},"incoming_edges":[{"id":"e-50ca6587bb81d58f","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-3121cc0e9141d5d1","premise_count":1,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"00d7e62c485815e1264cce0c0f3debb148ec46dbf0852ac83c676dd5d6f904ab","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-522def1e8ebf49cd","role":"sequence","name":"Triangle of coefficients of polynomials u(n,x) jointly generated with A112351; see the Formula section.","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:     2,      3;\n  2:     6,     18,       9;\n  3:    18,     84,     108,      27;\n  4:    54,    342,     720,     540,      81;\n  5:   162,   1296,    3798,    4860,    2430,      243;\n  6:   486,   4698,   17658,   32130,   28350,    10206,     729;\n  7:  1458,  16524,   75816,  179280,  229068,   149688,   40824,    2187;\n  8:  4374,  56862,  308124,  897480, 1507572,  1449252,  734832,  157464,   6561;\n  9: 13122, 192456, 1202850, 4165020, 8687412, 11092464, 8389332, 3411720, 590490, 19683;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(3*x+9*x^2)*P(n-2,x) + (3+6*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 3*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 3*x;\nP[n_, x_] := P[n, x] = Expand[-(3*x + 9*x^2)*P[n - 2, x] + (3 + 6*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] = 2 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = -(3*x + 9*x^2)*v[i - 2] + (3 + 6*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-522def1e8ebf49cd","steps":["basis A013610"],"edges":["e-50ca6587bb81d58f"],"text":"A209414 \u2192 basis A013610"},"headline":"A209414 \u2192 basis A013610","chain":null}