{"id":"n-25591ee324abb3de","revision":0,"first_index":0,"fingerprint":"f6-8fad4560d38877fb","name":"basis_map_A008277 of basis_map_A007318 of Pascal-(1,9,1) array.","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"],["3","1"],["27","16","1"],["135","145","30","1"],["891","1390","475","45","1"],["5103","11581","6150","1060","61","1"],["31347","96976","71941","18810","1946","78","1"],["185895","777865","782310","287851","45266","3182","96","1"],["1121931","6197830","8152075","4003335","879087","93150","4820","115","1"],["6711903","48588661","82122150","52166170","15191967","2232147","172080","6915","135","1"],["40330467","378505336","808454581","648637920","242010652","46829376","4966257","293925","9525","156","1"],["241805655","2926652785","7814473590","7786859101","3632574232","893998624","124004772","10014342","473085","12711","178","1"],["1451365371","22531035070","74509933675","91000425825","52121195213","15912909780","2789076460","292652430","18720852","726786","16537","201","1"],["8706597903","172771381741","702741627750","1041251410480","722089532753","268439554853","57988929120","7642805410","631288530","32958438","1075390","21070","225","1"],["52244370387","1321192360096","6571616448421","11715858655830","9729426513678","4341210044634","1134369610073","183223518190","18905727790","1267229268","55265578","1542720","26380","250","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","44","311","2802","23956","221099","2082466","20452344","207192129","2170038196","23430869076","260628058201","2982297183664","35069371313296"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A008277","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["3","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["18","9"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","9"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["3","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":196948,"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-514e4fd25a99f57f"]},{"id":196949,"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-514e4fd25a99f57f"]},{"id":196950,"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-514e4fd25a99f57f"]}],"region_exact":{},"incoming_edges":[{"id":"e-514e4fd25a99f57f","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-25591ee324abb3de","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":"d772fdc8ba0df846d62920c8808b0f3d6d4375cefc96e7f3954ce9c109614429","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-2ee89feef605b3fb","role":"sequence","name":"basis_map_A007318 of Pascal-(1,9,1) array.","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:       3,        1;\n  2:      27,       16,        1;\n  3:     135,      145,       30,        1;\n  4:     891,     1390,      475,       45,        1;\n  5:    5103,    11581,     6150,     1060,       61,       1;\n  6:   31347,    96976,    71941,    18810,     1946,      78,      1;\n  7:  185895,   777865,   782310,   287851,    45266,    3182,     96,    1;\n  8: 1121931,  6197830,  8152075,  4003335,   879087,   93150,   4820,  115,   1;\n  9: 6711903, 48588661, 82122150, 52166170, 15191967, 2232147, 172080, 6915, 135, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (18+9*x)*P(n-2,x) + 9*x*d/dx P(n-2,x) + (3+x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + x;\nP[n_, x_] := P[n, x] = Expand[(18 + 9*x)*P[n - 2, x] + 9*x*D[P[n - 2, x], x] + (3 + x)*P[n - 1, x] + 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; v[2] = 3 + x; for(n = 2, N, my(i = n + 1); v[i] = (18 + 9*x)*v[i - 2] + 9*x*deriv(v[i - 2], x) + (3 + x)*v[i - 1] + 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-7ac21ea1b3b7e282","steps":["basis A007318","basis A008277"],"edges":["e-a3fef8dca23d39bb","e-514e4fd25a99f57f"],"text":"A143685 \u2192 basis A007318 \u2192 basis A008277"},"headline":"A143685 \u2192 basis A007318 \u2192 basis A008277","chain":null}