{"id":"n-329498b88a584186","revision":1,"first_index":0,"fingerprint":"f6-d99d39db05a8c355","name":"basis_map_A008277 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["256","144","16"],["4096","3904","960","64"],["65536","94464","38656","5632","256"],["1048576","2151424","1305600","312320","30720","1024"],["16777216","47222784","39940096","13967360","2232320","159744","4096"],["268435456","1011564544","1147453440","550846464","127303680","14680064","802816","16384"],["4294967296","21305032704","31585140736","20013514752","6277103616","1037697024","90832896","3932160","65536"],["68719476736","443280523264","843263508480","686718976000","280921374720","62465507328","7784103936","536346624","18874368","262144"],["1099511627776","9140488372224","22011446296576","22601185361920","11736359895040","3372443762688","561226186752","54735667200","3051356160","89128960","1048576"],["17592186044416","187207813955584","564836664606720","720878975320064","465968258088960","168353415036928","35938822520832","4653274103808","365407764480","16840130560","415236096","4194304"],["281474976710656","3814525023289344","14304911206383616","22443957967388672","17794500160126976","7924595973685248","2110966560980992","348499350650880","36152669110272","2337317847040","90613743616","1912602624","16777216"],["4503599627370496","77416400372629504","358575969046364160","685650467912417280","659199836993617920","356463455693176832","116137046333980672","23777837672562688","3129325519896576","266151204487168","14423641030656","477211131904","8724152320","67108864"],["72057594037927936","1566342405962072064","8915488858603257856","20632516977733140480","23836996655445442560","15469483752928837632","6071335676131934208","1510773042928680960","245318975645286400","26357164712919040","1872328715665408","86327232036864","2467190276096","39460012032","268435456"],["1152921504606846976","31615078495393153024","220237102230326476800","613372430810940964864","845313960885186723840","652249401727219924992","304731362056992718848","90759356593389699072","17818403002688471040","2351848467652935680","210279066928939008","12668948784873472","503209105817600","12552041922560","177167400960","1073741824"],["18446744073709551616","636913255926290448384","5412150767509408055296","18055376471627652923392","29503536471569739022336","26862234305720664195072","14798252089188588453888","5212337138337117634560","1218320770502605406208","193569732328706539520","21183021618632327168","1601253194808164352","82881177911820288","2866375274004480","62964220559360","790273982464","4294967296"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["416","9024","204544","4849664","120303616","3121102848","84608286720","2393708953600","70580538703872","2165812077002752","69062013749297152","2285135000019927040","78348632266995924992","2779825329729500610560","101934007832909898055680"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A008277","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["256","144","16"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["16","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":166172,"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-1e482bc7f83a739a"]},{"id":166173,"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-1e482bc7f83a739a"]},{"id":166174,"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-1e482bc7f83a739a"]}],"region_exact":{},"incoming_edges":[{"id":"e-1e482bc7f83a739a","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-329498b88a584186","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":"0349cbf01e5f9f7914a9fcb83297f867d7e4c2e3a1eea9bda51efe402cbcd0a3","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-a93042cd70c4761e","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^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":[{"id":"l-144b49ffce07014e","kind":"fingerprint_related","node_a_id":"n-41442279a7af7481","node_b_id":"n-329498b88a584186","fingerprint":"f6-d99d39db05a8c355","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-d99d39db05a8c355"},"other_id":"n-41442279a7af7481","other":{"id":"n-41442279a7af7481","name":"basis_map_A008277 of Cube of lower triangular normalized binomial matrix.","first_index":0,"fingerprint":"f6-d99d39db05a8c355","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["4","1"],["16","9","1"]],"card_rows":[["1"],["4","1"],["16","9","1"],["64","61","15","1"]],"row_sums_preview":["1","5","26","141","799","4736","29371","190497"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8d887bfc40d036b9","steps":["basis A008277"],"edges":["e-6d94a10ae1f2ed2c"],"text":"A027465 \u2192 basis A008277"},"headline":"A027465 \u2192 basis A008277","chain":null}},{"id":"l-cd67789993b0e028","kind":"fingerprint_related","node_a_id":"n-1f64b8c87ab9b182","node_b_id":"n-329498b88a584186","fingerprint":"f6-d99d39db05a8c355","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-d99d39db05a8c355"},"other_id":"n-1f64b8c87ab9b182","other":{"id":"n-1f64b8c87ab9b182","name":"one_plus_derivative of Triangle read by rows: 3-Stirling numbers of the second kind.","first_index":1,"fingerprint":"f6-d99d39db05a8c355","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["4","1"],["16","9","1"]],"card_rows":[["1"],["4","1"],["16","9","1"],["64","61","15","1"]],"row_sums_preview":["1","5","26","141","799","4736","29371","190497"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a2b8ff03da0c14a3","steps":["D+1"],"edges":["e-bbdbb484d8de66ad"],"text":"A143495 \u2192 D+1"},"headline":"A143495 \u2192 D+1","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:            256,             144,              16;\n  1:           4096,            3904,             960,              64;\n  2:          65536,           94464,           38656,            5632,             256;\n  3:        1048576,         2151424,         1305600,          312320,           30720,            1024;\n  4:       16777216,        47222784,        39940096,        13967360,         2232320,          159744,           4096;\n  5:      268435456,      1011564544,      1147453440,       550846464,       127303680,        14680064,         802816,         16384;\n  6:     4294967296,     21305032704,     31585140736,     20013514752,      6277103616,      1037697024,       90832896,       3932160,        65536;\n  7:    68719476736,    443280523264,    843263508480,    686718976000,    280921374720,     62465507328,     7784103936,     536346624,     18874368,      262144;\n  8:  1099511627776,   9140488372224,  22011446296576,  22601185361920,  11736359895040,   3372443762688,   561226186752,   54735667200,   3051356160,    89128960,   1048576;\n  9: 17592186044416, 187207813955584, 564836664606720, 720878975320064, 465968258088960, 168353415036928, 35938822520832, 4653274103808, 365407764480, 16840130560, 415236096, 4194304;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (16+4*x)*P(n-1,x) + 4*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 256 + 144*x + 16*x^2.","mathematica":"P[0, x_] := 256 + 144*x + 16*x^2;\nP[n_, x_] := P[n, x] = Expand[(16 + 4*x)*P[n - 1, x] + 4*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] = 256 + 144*x + 16*x^2; for(n = 1, N, my(i = n + 1); v[i] = (16 + 4*x)*v[i - 1] + 4*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-a93042cd70c4761e","steps":["basis A008277"],"edges":["e-1e482bc7f83a739a"],"text":"A038330 \u2192 basis A008277"},"headline":"A038330 \u2192 basis A008277","chain":null}