{"id":"n-3a6170eba9ce35d2","revision":1,"first_index":0,"fingerprint":"f6-7bf0e36386f39b52","name":"basis_map_A013610 of Triangle read by rows: coefficients of the polynomials given by KummerU(-n, 1 - n - x, 3).","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"],["4","3"],["17","27","9"],["78","195","135","27"],["393","1356","1449","594","81"],["2208","9687","14040","8775","2430","243"],["13977","73467","133686","112185","46980","9477","729"],["100026","602235","1304541","1364958","762615","231336","35721","2187"],["806769","5381856","13349250","16572276","11497059","4633524","1071630","131220","6561"],["7280604","52548939","145027584","206156070","169378776","84541401","25902828","4737042","472392","19683"],["72865089","559667907","1682575659","2667660210","2509971705","1482069393","561426957","135747090","20175075","1673055","59049"],["801693126","6477477795","20884235427","36210287817","38061221220","25742644587","11529100737","3442628871","675520560","83357505","5845851","177147"],["9620848953","81098205372","277170968043","517774236606","596720115486","451415042496","231637355235","81747081306","19794917538","3221713440","335811663","20194758","531441"],["125072630712","1093244632191","3926197629432","7812330328413","9730640593674","8086065821262","4645395184716","1878930727245","537641184366","107931297474","14827676292","1324173825","69087330","1594323"],["1751021612937","15797546629875","59215425843240","124395013243683","165585796573743","149049854085132","94196058749793","42689871746379","14004690531975","3322108785996","562767734529","66206388339","5126279886","234365481","4782969"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","7","53","435","3873","37383","390501","4403619","53450145","696065319","9693891189","143914190643","2270557022337","37959672561255","670641727353957"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013610","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["4","3"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["3"],["-3"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["3","3"],["1","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":167390,"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-5617fd17362674c9"]},{"id":167391,"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-5617fd17362674c9"]},{"id":167392,"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-5617fd17362674c9"]},{"id":167393,"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-5617fd17362674c9"]}],"region_exact":{},"incoming_edges":[{"id":"e-5617fd17362674c9","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-3a6170eba9ce35d2","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":"24af9e629df0653f2537fbc059d4eb6273f85331afaa2833bb56fbf2a652c3e3","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-ff69dadb0811f5e6","role":"sequence","name":"Triangle read by rows: coefficients of the polynomials given by KummerU(-n, 1 - n - x, 3).","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:       4,        3;\n  2:      17,       27,         9;\n  3:      78,      195,       135,        27;\n  4:     393,     1356,      1449,       594,        81;\n  5:    2208,     9687,     14040,      8775,      2430,      243;\n  6:   13977,    73467,    133686,    112185,     46980,     9477,      729;\n  7:  100026,   602235,   1304541,   1364958,    762615,   231336,    35721,    2187;\n  8:  806769,  5381856,  13349250,  16572276,  11497059,  4633524,  1071630,  131220,   6561;\n  9: 7280604, 52548939, 145027584, 206156070, 169378776, 84541401, 25902828, 4737042, 472392, 19683;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(3*n-3)*P(n-2,x) + (n+3+3*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 3*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4 + 3*x;\nP[n_, x_] := P[n, x] = Expand[-(3*n - 3)*P[n - 2, x] + (n + 3 + 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] = 1; v[2] = 4 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = -(3*n - 3)*v[i - 2] + (n + 3 + 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-ff69dadb0811f5e6","steps":["basis A013610"],"edges":["e-5617fd17362674c9"],"text":"A327997 \u2192 basis A013610"},"headline":"A327997 \u2192 basis A013610","chain":null}