{"id":"n-2342641336502fe1","revision":0,"first_index":0,"fingerprint":"f6-6d253706f4f62393","name":"basis_map_A013610 of shift_x_plus_one of Riordan array ((1+x)^3,x).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":3,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["27","81","81","27"],["54","243","405","297","81"],["108","648","1539","1809","1053","243"],["216","1620","5022","8235","7533","3645","729"],["432","3888","14904","31536","39771","29889","12393","2187"],["864","9072","41472","107784","174150","179091","114453","41553","6561"],["1728","20736","110160","339984","671652","880632","766179","426465","137781","19683"],["3456","46656","282528","1010448","2363256","3776220","4174254","3151467","1554957","452709","59049"],["6912","103680","705024","2868480","7757856","14642208","19677168","18825696","12564315","5570289","1476225","177147"],["13824","228096","1721088","7852032","24121152","52557984","83280960","96682896","81605718","48833523","19663317","4782969","531441"],["27648","497664","4126464","20867328","71798400","177479424","324235872","443208672","453260124","342484200","185827203","68555889","15411789","1594323"],["55296","1078272","9745920","54114048","206198784","570354048","1180910016","1859124960","2236146264","2044748772","1399107006","694593387","236491245","49424013","4782969"],["110592","2322432","22726656","137465856","574739712","1759304448","4072882176","7260979968","10049667408","10797936336","8932460328","5586507792","2556762651","808321761","157837977","14348907"],["221184","4976640","52420608","343111680","1561876992","5242828032","13423677696","26740606464","41882274720","51744874896","50258729664","37970396568","21873048678","9286931475","2740641237","502211745","43046721"],["442368","10616832","119771136","843485184","4153089024","15171287040","42575839488","93752246016","163986368832","229136573952","255752084016","226716982128","157657287060","84193008984","33342076899","9226347201","1592728677","129140163"]],"degrees":[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],"row_sums":["216","1080","5400","27000","135000","675000","3375000","16875000","84375000","421875000","2109375000","10546875000","52734375000","263671875000","1318359375000"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013610","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["27","81","81","27"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","3"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":201787,"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-e2f49af397920dd3"]},{"id":201788,"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-e2f49af397920dd3"]},{"id":201789,"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-e2f49af397920dd3"]}],"region_exact":{},"incoming_edges":[{"id":"e-e2f49af397920dd3","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-2342641336502fe1","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":"8356ec5ea0104fce127794744b5eb20b430207ac1b931172777ccdc555221294","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-9aa9e3dd2403bca2","role":"sequence","name":"shift_x_plus_one of Riordan array ((1+x)^3,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:    27,     81,      81,      27;\n  1:    54,    243,     405,     297,       81;\n  2:   108,    648,    1539,    1809,     1053,      243;\n  3:   216,   1620,    5022,    8235,     7533,     3645,      729;\n  4:   432,   3888,   14904,   31536,    39771,    29889,    12393,     2187;\n  5:   864,   9072,   41472,  107784,   174150,   179091,   114453,    41553,     6561;\n  6:  1728,  20736,  110160,  339984,   671652,   880632,   766179,   426465,   137781,    19683;\n  7:  3456,  46656,  282528, 1010448,  2363256,  3776220,  4174254,  3151467,  1554957,   452709,    59049;\n  8:  6912, 103680,  705024, 2868480,  7757856, 14642208, 19677168, 18825696, 12564315,  5570289,  1476225,  177147;\n  9: 13824, 228096, 1721088, 7852032, 24121152, 52557984, 83280960, 96682896, 81605718, 48833523, 19663317, 4782969, 531441;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2+3*x)*P(n-1,x) for n >= 1, with P(0,x) = 27 + 81*x + 81*x^2 + 27*x^3.","mathematica":"P[0, x_] := 27 + 81*x + 81*x^2 + 27*x^3;\nP[n_, x_] := P[n, x] = Expand[(2 + 3*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] = 27 + 81*x + 81*x^2 + 27*x^3; for(n = 1, N, my(i = n + 1); v[i] = (2 + 3*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-f48bed27dbc29c5f","steps":["P\u2099 + P\u2099\u208a\u2081","basis A013610"],"edges":["e-0331d381cbbd8a41","e-e2f49af397920dd3"],"text":"A391533 \u2192 P\u2099 + P\u2099\u208a\u2081 \u2192 basis A013610"},"headline":"A391533 \u2192 P\u2099 + P\u2099\u208a\u2081 \u2192 basis A013610","chain":null}