{"id":"n-c16b8d4cf8f42711","revision":0,"first_index":0,"fingerprint":"f6-50db7ebeb5f2c85c","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.","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":true,"rows":[["729","864","256"],["19683","34992","20736","4096"],["531441","1259712","1119744","442368","65536"],["14348907","42515280","50388480","29859840","8847360","1048576"],["387420489","1377495072","2040733440","1612431360","716636160","169869312","16777216"],["10460353203","43391094768","77139724032","76187381760","45148078080","16052649984","3170893824","268435456"],["282429536481","1338925209984","2777030065152","3291294892032","2437996216320","1155790798848","342456532992","57982058496","4294967296"],["7625597484987","40669853253264","96402615118848","133297443127296","118486616113152","70214291030016","27738979172352","7044820107264","1043677052928","68719476736"],["205891132094649","1220095597597920","3253588260261120","5141472806338560","5331897725091840","3791571715620864","1872381094133760","634033809653760","140896402145280","18554258718720","1099511627776"],["5559060566555523","36236839248658224","107368412588616960","190877177935319040","226224803478896640","187682799923232768","111219436991545344","47077010366791680","13948743812382720","2755307419729920","326554953449472","17592186044416"],["150094635296999121","1067339628778660416","3478736567871189504","6871578405671485440","9162104540895313920","8687032453589630976","6005849597543448576","3050590271768100864","1129848248803000320","297573201330831360","52901902458814464","5699868278390784","281474976710656"],["4052555153018976267","31219684141775817168","111003321392980683264","241192402039069138944","357322077094917242880","381143548901245059072","301150458391107207168","178459530898433900544","79315347065970622464","26112048416780451840","6189522587681292288","1000326882857582592","98797716825440256","4503599627370496"],["109418989131512359209","907772354276250683808","3496604623878891522816","8288247997342557683712","13506774514187871780864","16008029053852292481024","14229359158979815538688","9636814668515430629376","4996866865156149215232","1974070860308602159104","584909884535882121216","126041187240055406592","18672768480008208384","1702360659146047488","72057594037927936"],["2954312706550833698643","26260557391562966210160","108932682513150082056960","279728369910311321825280","497294879840553461022720","648325176681017845481472","640321162154091699240960","487863742593593675612160","289104440055462918881280","133249783070830645739520","47377700647406451818496","12761670208055609917440","2520823744801108131840","344728033477074616320","29183325585360814080","1152921504606846976"],["79766443076872509863361","756304052877013426852608","3361351346120059674900480","9295588907788807002193920","17902615674259924596817920","25461497847836337204363264","27661874205056761407209472","23417459644492496429383680","15611639762994997619589120","8223415183799834137067520","3411194446613264530931712","1102608305976004696866816","272248964438519678238720","49640836820698744750080","6303598326437935841280","498062089990157893632","18446744073709551616"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["1849","79507","3418801","147008443","6321363049","271818611107","11688200277601","502592611936843","21611482313284249","929293739471222707","39959630797262576401","1718264124282290785243","73885357344138503765449","3177070365797955661914307","136614025729312093462315201"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["729","864","256"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["27","16"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":194044,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-8d156bd12b171a65"]},{"id":194045,"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-8d156bd12b171a65","e-9d0f90b89216a153"]},{"id":194046,"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-8d156bd12b171a65","e-9d0f90b89216a153"]},{"id":194047,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-8d156bd12b171a65"]},{"id":194048,"position":4,"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-9d0f90b89216a153"]}],"region_exact":{},"incoming_edges":[{"id":"e-9d0f90b89216a153","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-c16b8d4cf8f42711","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":"8383047100f17120a8f5cbfd426ec0480bf52208a43a12328a3976977a7c4894","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-1600c9e1b40ceecc","role":"sequence","name":"basis_map_A013609 of Triangle read by rows: (i,j)-th entry is binomial(i,j)*3^(i-j)*8^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]"}]},{"id":"e-8d156bd12b171a65","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-c16b8d4cf8f42711","premise_count":1,"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"5efda1f7b91445ce266c91b741c995935b50f7199813b84b736934320c46ba98","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-1600c9e1b40ceecc","role":"sequence","name":"basis_map_A013609 of Triangle read by rows: (i,j)-th entry is binomial(i,j)*3^(i-j)*8^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-40758591998834b0","kind":"fingerprint_related","node_a_id":"n-c16b8d4cf8f42711","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}},{"id":"l-73510fc5e9b480fd","kind":"fingerprint_related","node_a_id":"n-c16b8d4cf8f42711","node_b_id":"n-f3597891e233edcd","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-b0b710f447d722c0","other":{"id":"n-b0b710f447d722c0","name":"basis_map_A013610 of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*9^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"],["48","81"],["2304","7776","6561"]],"card_rows":[["1"],["48","81"],["2304","7776","6561"],["110592","559872","944784","531441"]],"row_sums_preview":["1","129","16641","2146689","276922881","35723051649","4608273662721","594467302491009"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"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}}],"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:              729,               864,                256;\n  1:            19683,             34992,              20736,               4096;\n  2:           531441,           1259712,            1119744,             442368,              65536;\n  3:         14348907,          42515280,           50388480,           29859840,            8847360,            1048576;\n  4:        387420489,        1377495072,         2040733440,         1612431360,          716636160,          169869312,           16777216;\n  5:      10460353203,       43391094768,        77139724032,        76187381760,        45148078080,        16052649984,         3170893824,         268435456;\n  6:     282429536481,     1338925209984,      2777030065152,      3291294892032,      2437996216320,      1155790798848,       342456532992,       57982058496,        4294967296;\n  7:    7625597484987,    40669853253264,     96402615118848,    133297443127296,    118486616113152,     70214291030016,     27738979172352,     7044820107264,     1043677052928,      68719476736;\n  8:  205891132094649,  1220095597597920,   3253588260261120,   5141472806338560,   5331897725091840,   3791571715620864,   1872381094133760,   634033809653760,   140896402145280,   18554258718720,   1099511627776;\n  9: 5559060566555523, 36236839248658224, 107368412588616960, 190877177935319040, 226224803478896640, 187682799923232768, 111219436991545344, 47077010366791680, 13948743812382720, 2755307419729920, 326554953449472, 17592186044416;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (27+16*x)*P(n-1,x) for n >= 1, with P(0,x) = 729 + 864*x + 256*x^2.","mathematica":"P[0, x_] := 729 + 864*x + 256*x^2;\nP[n_, x_] := P[n, x] = Expand[(27 + 16*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] = 729 + 864*x + 256*x^2; for(n = 1, N, my(i = n + 1); v[i] = (27 + 16*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-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}
