{"id":"n-c6d0c3e6cdaf94a0","revision":1,"first_index":0,"fingerprint":"f6-33f617d4f0cbaf1f","name":"basis_map_A008277 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*11^j.","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"],["18","11"],["324","517","121"],["5832","18557","10527","1331"],["104976","602305","625207","183678","14641"],["1889568","18621581","31633635","16244855","2928200","161051"],["34012224","560811097","1470182791","1176457590","360241805","43966923","1771561"],["612220032","16637656277","64976233707","76171347791","35276025400","7172245234","632447277","19487171"],["11019960576","489226452385","2782063567327","4599477308118","3024998400501","911610181482","132020268842","8808201292","214358881"],["198359290368","14308786685501","116664033669315","265176041954615","238144151220360","99822525653697","21117414579030","2289002080002","119612255598","2357947691"],["3570467226624","417136766073577","4823958000313111","14807282510047830","17681873837163085","9906630036143841","2871910606829187","449746757969520","37857921973410","1591614691425","25937424601"],["64268410079232","12136241355626597","197546824439333787","808234946015883551","1259156285514637400","917684604847294328","350213421371233959","74316958682225457","8960154066846180","602656060604235","20827751954603","285311670611"],["1156831381426176","352657951824042865","8035371632485244047","43392997315642732758","86958274108082237861","80841695294513497344","39512458048503890164","10912458709894991964","1767262876590175107","169072453826003475","9295168916835769","268763593715562","3138428376721"],["20822964865671168","10239805748092931021","325294102769474233395","2301431951055117055175","5868735965173168807720","6857984771688389922583","4208305124313975244560","1471320615973567028384","307366910727403472946","39221368740134332752","3049578613441016657","139604997611656577","3427163787379332","34522712143931"],["374813367582081024","297183419308217382457","13124401974007991577031","120951264634275186381270","389177381302342753685565","565188983950157321233479","428935462930946209691453","186066814884942595386640","48765419312814005444500","7969936160597155134390","821781118661927792368","52950459415871447430","2049729541835122147","43256958316345543","379749833583241"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","29","962","36247","1530807","71478890","3647443991","201497662889","11959438699404","757842285336177","51001584455856211","3628937473827389940","271952972293683169813","21393114084451740896201","1761354048327086802288538"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A008277","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["18","11"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","11"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":166112,"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-8cea3676de7ce3cf"]},{"id":166113,"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-8cea3676de7ce3cf"]},{"id":166114,"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-8cea3676de7ce3cf"]}],"region_exact":{},"incoming_edges":[{"id":"e-8cea3676de7ce3cf","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-c6d0c3e6cdaf94a0","premise_count":1,"theorem":"ray basis maps: for L f = (a + b t) f + kappa (t - h) f', b > 0, kappa >= 0, and g real-rooted with every root <= h, of multiplicity m at h, T(u^k) = L^k g sends every real polynomial with all roots in S to one with all roots in (-inf, h] if and only if S lies in (-inf, a + b h + kappa m] (RealRooted rayBasisMap_maps_realRootedIn_iff; the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"a4c43a1bd7fcbfcf6b7a8b170ecafac7c38d558eb8f15f9115a07154925941ce","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-b7cf68162d40a956","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*11^j.","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:           18,             11;\n  2:          324,            517,             121;\n  3:         5832,          18557,           10527,            1331;\n  4:       104976,         602305,          625207,          183678,           14641;\n  5:      1889568,       18621581,        31633635,        16244855,         2928200,         161051;\n  6:     34012224,      560811097,      1470182791,      1176457590,       360241805,       43966923,        1771561;\n  7:    612220032,    16637656277,     64976233707,     76171347791,     35276025400,     7172245234,      632447277,      19487171;\n  8:  11019960576,   489226452385,   2782063567327,   4599477308118,   3024998400501,   911610181482,   132020268842,    8808201292,    214358881;\n  9: 198359290368, 14308786685501, 116664033669315, 265176041954615, 238144151220360, 99822525653697, 21117414579030, 2289002080002, 119612255598, 2357947691;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (18+11*x)*P(n-1,x) + 11*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(18 + 11*x)*P[n - 1, x] + 11*x*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (18 + 11*x)*v[i - 1] + 11*x*deriv(v[i - 1], x)); 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-b7cf68162d40a956","steps":["basis A008277"],"edges":["e-8cea3676de7ce3cf"],"text":"A038277 \u2192 basis A008277"},"headline":"A038277 \u2192 basis A008277","chain":null}