{"id":"n-b0b710f447d722c0","revision":0,"first_index":0,"fingerprint":"f6-50db7ebeb5f2c85c","name":"basis_map_A013610 of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*9^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":true,"rows":[["1"],["48","81"],["2304","7776","6561"],["110592","559872","944784","531441"],["5308416","35831808","90699264","102036672","43046721"],["254803968","2149908480","7255941120","12244400640","10331213040","3486784401"],["12230590464","123834728448","522427760640","1175462461440","1487694677760","1004193907488","282429536481"],["587068342272","6934744793088","35107145515008","98738846760960","166621803909120","168704576457984","94896324257616","22876792454961"],["28179280429056","380420285792256","2246857312960512","7583143431241728","15995693175275520","21594185786621952","18220094257462272","8784688302705024","1853020188851841"],["1352605460594688","20542695432781824","138663194171277312","545986327049404416","1382027890343804928","2332172064955170816","2623693573074567168","1897492673384285184","800504721583995312","150094635296999121"],["64925062108545024","1095610423081697280","8319791650276638720","37439062426244874240","110562231227504394240","223888518235696398336","314843228768948060160","303598827741485629440","192121133180158874880","72045424942559578080","12157665459056928801"],["3116402981210161152","57848230338713616384","488094443482896138240","2470978120132161699840","8339551155446045736960","19702189604741283053568","33247444958000915152896","40075045261876103086080","33813319439707961978880","19019992184835728613120","6419247362382058406928","984770902183611232881"],["149587343098087735296","3029143697736276639744","28114239944614817562624","158142599688458348789760","600447683192115293061120","1621208744618711291265024","3191754715968087854678016","4616645214168127075516416","4869117999317946524958720","3651838499488459893719040","1848743240366032821195264","567228039657760070139456","79766443076872509863361"],["7180192468708211294208","157515472282286385266688","1594844156858149650825216","9868098220559800964481024","41631039367986660318904320","126454282080259480718671872","284522134680583831617011712","480131102273485215853707264","607665926314879726314848256","569686805920199743420170240","384538593996134826808614912","176975148373221141883510272","49774260479968446154737264","6461081889226673298932241"],["344649238497994142121984","8142338259515111607631872","89311272784056380446212096","602851091292380568011931648","2797605845528703573430370304","9441919728659374560327499776","23899859313169041855828983808","46092585818254580721955897344","68058583747266529347263004672","76565906715674845515670880256","64602483791350650903847305216","39642433235601535781906300928","16724151521269397907991720704","4341847029560324456882465952","523347633027360537213511521"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","129","16641","2146689","276922881","35723051649","4608273662721","594467302491009","76686282021340161","9892530380752880769","1276136419117121619201","164621598066108688876929","21236186150528020865123841","2739468013418114691600975489","353391373730936795216525838081"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013610","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["48","81"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":203913,"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-9a0bf9909e32e625"]},{"id":203914,"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-9a0bf9909e32e625"]},{"id":203915,"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-9a0bf9909e32e625"]}],"region_exact":{},"incoming_edges":[{"id":"e-9a0bf9909e32e625","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-b0b710f447d722c0","premise_count":1,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"664e56411dac9df5dde2e13c5836ddef42d72e07f1fe85ca8c75df344ad9b07b","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-504bafdebe53d9d7","role":"sequence","name":"basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*9^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-b4cff32605c8ab48","kind":"fingerprint_related","node_a_id":"n-f3597891e233edcd","node_b_id":"n-b0b710f447d722c0","fingerprint":"f6-50db7ebeb5f2c85c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-50db7ebeb5f2c85c"},"other_id":"n-f3597891e233edcd","other":{"id":"n-f3597891e233edcd","name":"basis_map_A013609 of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*8^j.","first_index":0,"fingerprint":"f6-50db7ebeb5f2c85c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["27","16"],["729","864","256"]],"card_rows":[["1"],["27","16"],["729","864","256"],["19683","34992","20736","4096"]],"row_sums_preview":["1","43","1849","79507","3418801","147008443","6321363049","271818611107"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2f02b2035f8f4541","steps":["basis A007318","basis A013609"],"edges":["e-c22af348c5e552a2","e-0bd30d7eb0266a52"],"text":"A038322 \u2192 basis A007318 \u2192 basis A013609"},"headline":"A038322 \u2192 basis A007318 \u2192 basis A013609","chain":null}},{"id":"l-d3979bf03548fb77","kind":"fingerprint_related","node_a_id":"n-c16b8d4cf8f42711","node_b_id":"n-b0b710f447d722c0","fingerprint":"f6-50db7ebeb5f2c85c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-50db7ebeb5f2c85c"},"other_id":"n-c16b8d4cf8f42711","other":{"id":"n-c16b8d4cf8f42711","name":"shift_x_plus_one of basis_map_A013609 of Triangle read by rows: (i,j)-th entry is binomial(i,j)*3^(i-j)*8^j.","first_index":0,"fingerprint":"f6-50db7ebeb5f2c85c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["729","864","256"],["19683","34992","20736","4096"],["531441","1259712","1119744","442368","65536"]],"card_rows":[["729","864","256"],["19683","34992","20736","4096"],["531441","1259712","1119744","442368","65536"],["14348907","42515280","50388480","29859840","8847360","1048576"]],"row_sums_preview":["1849","79507","3418801","147008443","6321363049","271818611107","11688200277601","502592611936843"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-26633b980c3508e8","steps":["basis A013609","x \u21a6 x+1"],"edges":["e-c8a2ee9cf24a53ed","e-8d156bd12b171a65"],"text":"A038226 \u2192 basis A013609 \u2192 x \u21a6 x+1"},"headline":"A038226 \u2192 basis A013609 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-fa32260748a5f341","kind":"fingerprint_related","node_a_id":"n-b0b710f447d722c0","node_b_id":"n-d4c4d98e7be818e1","fingerprint":"f6-50db7ebeb5f2c85c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-50db7ebeb5f2c85c"},"other_id":"n-d4c4d98e7be818e1","other":{"id":"n-d4c4d98e7be818e1","name":"basis_map_A013610 of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^j.","first_index":0,"fingerprint":"f6-50db7ebeb5f2c85c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1024","3456","2916"],["32768","165888","279936","157464"],["1048576","7077888","17915904","20155392","8503056"]],"card_rows":[["1024","3456","2916"],["32768","165888","279936","157464"],["1048576","7077888","17915904","20155392","8503056"],["33554432","283115520","955514880","1612431360","1360488960","459165024"]],"row_sums_preview":["7396","636056","54700816","4704270176","404567235136","34792782221696","2992179271065856","257327417311663616"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c11cb6bca0e3e7dc","steps":["basis A013610","basis A013610"],"edges":["e-89b2e37c644537dd","e-171f61666f2fcb5c"],"text":"A038284 \u2192 basis A013610 \u2192 basis A013610"},"headline":"A038284 \u2192 basis A013610 \u2192 basis A013610","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:                1;\n  1:               48,                81;\n  2:             2304,              7776,               6561;\n  3:           110592,            559872,             944784,             531441;\n  4:          5308416,          35831808,           90699264,          102036672,            43046721;\n  5:        254803968,        2149908480,         7255941120,        12244400640,         10331213040,          3486784401;\n  6:      12230590464,      123834728448,       522427760640,      1175462461440,       1487694677760,       1004193907488,        282429536481;\n  7:     587068342272,     6934744793088,     35107145515008,     98738846760960,     166621803909120,     168704576457984,      94896324257616,      22876792454961;\n  8:   28179280429056,   380420285792256,   2246857312960512,   7583143431241728,   15995693175275520,   21594185786621952,   18220094257462272,    8784688302705024,   1853020188851841;\n  9: 1352605460594688, 20542695432781824, 138663194171277312, 545986327049404416, 1382027890343804928, 2332172064955170816, 2623693573074567168, 1897492673384285184, 800504721583995312, 150094635296999121;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (48+81*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(48 + 81*x)*P[n - 1, 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] = (48 + 81*x)*v[i - 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-f485cd65a8f8089f","steps":["basis A013610","basis A013610"],"edges":["e-b7b3e04e23eb02df","e-9a0bf9909e32e625"],"text":"A038335 \u2192 basis A013610 \u2192 basis A013610"},"headline":"A038335 \u2192 basis A013610 \u2192 basis A013610","chain":null}
