{"id":"n-fa5dee57fd93f326","revision":0,"first_index":0,"fingerprint":"f6-c7f57c7d3a795bc5","name":"basis_map_A008277 of derivative of Triangle of coefficients in expansion of (1+4x)^n.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[[],["4"],["40","32"],["300","672","192"],["2000","9664","6912","1024"],["12500","118720","160640","56320","5120"],["75000","1342176","3075840","1920000","399360","24576"],["437500","14442848","52913728","52433920","18744320","2580480","114688"],["2500000","150555008","852171264","1260607488","689561600","159416320","15597568","524288"],["14062500","1535619456","13140502272","27943888896","21963626496","7586611200","1226244096","89653248","2359296"],["78125000","15418694560","196632230400","586231244800","636679680000","308355317760","73230581760","8737259520","495452160","10485760"],["429687500","152988826144","2879683150784","11827706091520","17286718085120","11281161830400","3692818956288","639377080320","58608844800","2652897280","46137344"],["2343750000","1503947111232","41506730470656","231916075809792","447634804531200","383101001564160","166054251331584","39131693973504","5155675176960","374404546560","13841203200","201326592"],["12695312500","14673640584512","591070224943744","4450983561536512","11188661464927232","12315402945331200","6876975402778624","2118526481989632","376227320561664","38971061043200","2297178357760","70665633792","872415232"],["68359375000","142276127011424","8338192677884160","84033386172503040","272209350010357760","379765851761680384","267824198343589888","104913065941139456","24117171926335488","3341393768480768","279200137216000","13624508678144","354200584192","3758096384"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","4","72","1164","19600","353300","6836952","141667484","3130933536","73502567460","1825869071720","47822191587500","1316394970795440","37973873515415604","1144978137692932088"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A008277","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["4"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0"],["5","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0"],["0","4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":197029,"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-d3e29ae485b33b6e"]},{"id":197030,"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-d3e29ae485b33b6e"]},{"id":197031,"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-d3e29ae485b33b6e"]}],"region_exact":{},"incoming_edges":[{"id":"e-d3e29ae485b33b6e","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-fa5dee57fd93f326","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":"758670a031322464e35e69ccc1d378453683c7085a42c2e93aaedd7c21c2d44e","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-8d4f9cf28f37366a","role":"sequence","name":"derivative of Triangle of coefficients in expansion of (1+4x)^n.","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":[{"id":"l-1a57bf120d62ce8a","kind":"fingerprint_related","node_a_id":"n-e6ac71558d9546a6","node_b_id":"n-fa5dee57fd93f326","fingerprint":"f6-c7f57c7d3a795bc5","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-c7f57c7d3a795bc5"},"other_id":"n-e6ac71558d9546a6","other":{"id":"n-e6ac71558d9546a6","name":"basis_map_A008277 of Triangle of coefficients in expansion of (1+4x)^n.","first_index":0,"fingerprint":"f6-c7f57c7d3a795bc5","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["5","4"],["25","56","16"]],"card_rows":[["1"],["5","4"],["25","56","16"],["125","604","432","64"]],"row_sums_preview":["1","9","97","1225","17665","284873","5059553","97841673"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4c2a0f8cb422ddaa","steps":["D+1"],"edges":["e-41e37891330eb8cc"],"text":"A285061 \u2192 D+1"},"headline":"A285061 \u2192 D+1","chain":null}},{"id":"l-8f7e2719bbb027d7","kind":"fingerprint_related","node_a_id":"n-fa5dee57fd93f326","node_b_id":"n-33e1acaf6d61c325","fingerprint":"f6-c7f57c7d3a795bc5","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-c7f57c7d3a795bc5"},"other_id":"n-33e1acaf6d61c325","other":{"id":"n-33e1acaf6d61c325","name":"basis_map_A008277 of basis_map_A013611 of basis-map coefficients of A scaled Hermite triangle.","first_index":1,"fingerprint":"f6-c7f57c7d3a795bc5","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["25","56","16"],["125","604","432","64"],["625","5936","8032","2816","256"]],"card_rows":[["25","56","16"],["125","604","432","64"],["625","5936","8032","2816","256"],["3125","55924","128160","80000","16640","1024"]],"row_sums_preview":["97","1225","17665","284873","5059553","97841673","2041737985","45646726793"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-5fe240122e98ffb3","steps":["basis A013611","basis A008277"],"edges":["e-b91bf4c52fd06745","e-7ca2c4cfb981192c"],"text":"basis-map coefficients of A112227 \u2192 basis A013611 \u2192 basis A008277"},"headline":"basis-map coefficients of A112227 \u2192 basis A013611 \u2192 basis A008277","chain":null}},{"id":"l-b4126746568d0e72","kind":"fingerprint_related","node_a_id":"n-57992f4258de839d","node_b_id":"n-fa5dee57fd93f326","fingerprint":"f6-c7f57c7d3a795bc5","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-c7f57c7d3a795bc5"},"other_id":"n-57992f4258de839d","other":{"id":"n-57992f4258de839d","name":"basis_map_A008277 of basis_map_A013611 of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","first_index":1,"fingerprint":"f6-c7f57c7d3a795bc5","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["5","4"],["25","56","16"]],"card_rows":[["1"],["5","4"],["25","56","16"],["125","604","432","64"]],"row_sums_preview":["1","9","97","1225","17665","284873","5059553","97841673"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1bf99ad40f056d33","steps":["basis A013611","basis A008277"],"edges":["e-f93ad378f9476e41","e-7c4b628cf4f48c09"],"text":"basis-map coefficients of A026729 \u2192 basis A013611 \u2192 basis A008277"},"headline":"basis-map coefficients of A026729 \u2192 basis A013611 \u2192 basis A008277","chain":null}},{"id":"l-ca99cd5011622ff6","kind":"fingerprint_related","node_a_id":"n-fa5dee57fd93f326","node_b_id":"n-7f838cb81c71df94","fingerprint":"f6-c7f57c7d3a795bc5","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-c7f57c7d3a795bc5"},"other_id":"n-7f838cb81c71df94","other":{"id":"n-7f838cb81c71df94","name":"basis_map_A008277 of basis_map_A013611 of Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros followed by the term 2^k. Triangle by columns, (2^k, 0, 0, 0, ...).","first_index":0,"fingerprint":"f6-c7f57c7d3a795bc5","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["10","8"],["100","224","64"]],"card_rows":[["1"],["10","8"],["100","224","64"],["1000","4832","3456","512"]],"row_sums_preview":["1","18","388","9800","282640","9115936","323811392","12523734144"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ae7910bbe46f8a41","steps":["basis A013611","basis A008277"],"edges":["e-d7d2422771947afc","e-392a278c3a0e5481"],"text":"A130123 \u2192 basis A013611 \u2192 basis A008277"},"headline":"A130123 \u2192 basis A013611 \u2192 basis A008277","chain":null}},{"id":"l-e304a16e5564b09a","kind":"fingerprint_related","node_a_id":"n-fa5dee57fd93f326","node_b_id":"n-2ccf2fd4c3930c32","fingerprint":"f6-c7f57c7d3a795bc5","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-c7f57c7d3a795bc5"},"other_id":"n-2ccf2fd4c3930c32","other":{"id":"n-2ccf2fd4c3930c32","name":"basis_map_A008277 of basis_map_A013611 of Triangle read by rows: row n consists of n zeros followed by n!.","first_index":0,"fingerprint":"f6-c7f57c7d3a795bc5","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["5","4"],["50","112","32"]],"card_rows":[["1"],["5","4"],["50","112","32"],["750","3624","2592","384"]],"row_sums_preview":["1","9","194","7350","423960","34184760","3642878160","493122031920"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-fbb4b9430d9de872","steps":["basis A013611","basis A008277"],"edges":["e-3656191e13afb7f3","e-01c85482b8798924"],"text":"A136572 \u2192 basis A013611 \u2192 basis A008277"},"headline":"A136572 \u2192 basis A013611 \u2192 basis A008277","chain":null}}],"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:        0;\n  1:        4;\n  2:       40,         32;\n  3:      300,        672,         192;\n  4:     2000,       9664,        6912,        1024;\n  5:    12500,     118720,      160640,       56320,        5120;\n  6:    75000,    1342176,     3075840,     1920000,      399360,      24576;\n  7:   437500,   14442848,    52913728,    52433920,    18744320,    2580480,     114688;\n  8:  2500000,  150555008,   852171264,  1260607488,   689561600,  159416320,   15597568,   524288;\n  9: 14062500, 1535619456, 13140502272, 27943888896, 21963626496, 7586611200, 1226244096, 89653248, 2359296;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (5*n+4*n*x)*P(n-1,x) + 4*n*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 4;\nP[n_, x_] := P[n, x] = Expand[Together[((5*n + 4*n*x)*P[n - 1, x] + 4*n*x*D[P[n - 1, x], x])/(n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 4; for(n = 2, N, my(i = n + 1); v[i] = ((5*n + 4*n*x)*v[i - 1] + 4*n*x*deriv(v[i - 1], x))/(n - 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-2e0409c0014ec240","steps":["D","basis A008277"],"edges":["e-b04b02754d5cebbf","e-d3e29ae485b33b6e"],"text":"A013611 \u2192 D \u2192 basis A008277"},"headline":"A013611 \u2192 D \u2192 basis A008277","chain":null}