{"id":"n-ce8c753db7a6f446","revision":1,"first_index":0,"fingerprint":"f6-fcf9c3d1dfe3ab62","name":"basis_map_A008277 of Triangle whose (n,k)-th entry is 15^(n-k)*binomial(n,k).","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"],["16","1"],["256","33","1"],["4096","817","51","1"],["65536","17985","1735","70","1"],["1048576","371281","49215","3065","90","1"],["16777216","7360353","1257151","107450","4865","111","1"],["268435456","141903217","29989071","3298701","204750","7196","133","1"],["4294967296","2680790145","681706495","92664390","7393701","355866","10122","156","1"],["68719476736","49868399761","14951507055","2442329905","240538410","14866887","578550","13710","180","1"],["1099511627776","916482272673","318995526751","61355775250","7253098105","552743037","27594987","893880","18030","205","1"],["17592186044416","16679710263217","6658401754191","1484755256501","206417737350","18860701882","1159832751","48154227","1326600","23155","231","1"],["281474976710656","301147260519105","136530941838655","34868751627710","5613110003501","602492476872","44377022404","2267379972","79992627","1905475","29161","258","1"],["4503599627370496","5400978405535441","2758704213614895","799037222765145","147130951697730","18265452017813","1578786969760","96526761760","4187203020","127629502","2663661","36127","286","1"],["72057594037927936","96320232521472993","55057654250603551","17940411446152650","3741656256719745","530705444071803","52998765352533","3798902490240","197019634240","7377940570","196884688","3639090","44135","315","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","17","290","4965","85327","1472228","25507147","443838525","7757888172","136237711195","2404179550695","42641541094522","760284259506397","13629395504265637","245705256222934490"],"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":[["16","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":166118,"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-64407099b774cdf3"]},{"id":166119,"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-64407099b774cdf3"]},{"id":166120,"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-64407099b774cdf3"]}],"region_exact":{},"incoming_edges":[{"id":"e-64407099b774cdf3","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-ce8c753db7a6f446","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":"73e1f46fda279b082da997013056789094c3542160ee2bdaa98704248eb2792e","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-ff0214c333bfdcca","role":"sequence","name":"Triangle whose (n,k)-th entry is 15^(n-k)*binomial(n,k).","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:          16,           1;\n  2:         256,          33,           1;\n  3:        4096,         817,          51,          1;\n  4:       65536,       17985,        1735,         70,         1;\n  5:     1048576,      371281,       49215,       3065,        90,        1;\n  6:    16777216,     7360353,     1257151,     107450,      4865,      111,      1;\n  7:   268435456,   141903217,    29989071,    3298701,    204750,     7196,    133,     1;\n  8:  4294967296,  2680790145,   681706495,   92664390,   7393701,   355866,  10122,   156,   1;\n  9: 68719476736, 49868399761, 14951507055, 2442329905, 240538410, 14866887, 578550, 13710, 180, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (16+x)*P(n-1,x) + 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[(16 + 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; for(n = 1, N, my(i = n + 1); v[i] = (16 + 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-ff0214c333bfdcca","steps":["basis A008277"],"edges":["e-64407099b774cdf3"],"text":"A027467 \u2192 basis A008277"},"headline":"A027467 \u2192 basis A008277","chain":null}