{"id":"n-a48b231db9ea7677","revision":0,"first_index":0,"fingerprint":"f6-f3eacc80e11c3732","name":"reversal of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*8^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"],["24","19"],["576","912","361"],["13824","32832","25992","6859"],["331776","1050624","1247616","658464","130321"],["7962624","31518720","49904640","39507840","15638520","2476099"],["191102976","907739136","1796567040","1896376320","1125973440","356558256","47045881"],["4586471424","25416695808","60364652544","79647805440","63054512640","29950893504","7903708008","893871739"],["110075314176","697143656448","1931668881408","3058475728896","3026616606720","1916857184256","758755968768","171623373888","16983563041"],["2641807540224","18822878724096","59605782626304","110105126240256","130749837410304","103510287949824","54630429751296","18535324379904","3668449616856","322687697779"],["63403380965376","501943432642560","1788173478789120","3775032899665920","5229993496412160","4968493821591552","3277825785077760","1482825950392320","440213954022720","77445047466960","6131066257801"],["1521681143169024","13251306621763584","52453088711147520","124576085688975360","197245469007544320","218613728150028288","173069201452105728","97866512725893120","38738827953999360","10222746265638720","1618601492059464","116490258898219"],["36520347436056576","346943300642537472","1510648954881048576","3986434742047211520","7100836884271595520","8994393386744020992","8307321669701074944","5637111133011443712","2789195612687953920","981383641501317120","233078614856562816","33549194562687072","2213314919066161"],["876488338465357824","9020525816705974272","42847497629353377792","124376763951872999424","246162345321415311360","350781342083016818688","370269194420962197504","293129778916595073024","174045806231728324608","76547924037102735360","24240175945082532864","5233674351779183232","690554254748642232","42052983462257059"],["21035720123168587776","233145898031785181184","1199729933621894578176","3799144789802666164224","8271054802799554461696","13095836771099294564352","15551306165680412295168","14070229387996563505152","9746565148976786178048","5144020495293303816192","2036174779386932760576","586171527399268521984","116013114797771894976","14129802443318371824","799006685782884121"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","43","1849","79507","3418801","147008443","6321363049","271818611107","11688200277601","502592611936843","21611482313284249","929293739471222707","39959630797262576401","1718264124282290785243","73885357344138503765449"],"nonn":true,"definitions":[{"position":0,"source":"derived:reversal","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["24","19"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":193230,"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-dd797f7eb576b37a"]},{"id":193231,"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-dd797f7eb576b37a"]},{"id":193232,"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-dd797f7eb576b37a"]}],"region_exact":{},"incoming_edges":[{"id":"e-dd797f7eb576b37a","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"reversal\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":1}}"},"operation_detail":"reversal, shift 0","conclusion_id":"n-a48b231db9ea7677","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":"reversed","lean_declarations":["RealRooted.X_pow_mul_reverse_eq_zero_or_splits","RealRooted.reflect_eq_zero_or_splits"],"certificate_hash":"1f608c3144c6cd0ce23aa7a168b0499fb3fb1e86898ec661f36f29486d5fbed5","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-587b2e7eeca5ddbb","role":"sequence","name":"basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(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-284b317b782cedcc","kind":"fingerprint_related","node_a_id":"n-a48b231db9ea7677","node_b_id":"n-6f599efa913bc710","fingerprint":"f6-f3eacc80e11c3732","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f3eacc80e11c3732"},"other_id":"n-6f599efa913bc710","other":{"id":"n-6f599efa913bc710","name":"basis_map_A013610 of basis_map_A007318 of Triangle read by rows: (i,j)-th entry is binomial(i,j)*3^(i-j)*8^j.","first_index":0,"fingerprint":"f6-f3eacc80e11c3732","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["361","912","576"],["6859","25992","32832","13824"],["130321","658464","1247616","1050624","331776"]],"card_rows":[["361","912","576"],["6859","25992","32832","13824"],["130321","658464","1247616","1050624","331776"],["2476099","15638520","39507840","49904640","31518720","7962624"]],"row_sums_preview":["1849","79507","3418801","147008443","6321363049","271818611107","11688200277601","502592611936843"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-26633b980c3508e8","steps":["basis A007318","basis A013610"],"edges":["e-0566c972c7be0ca0","e-054e82d1e0138074"],"text":"A038226 \u2192 basis A007318 \u2192 basis A013610"},"headline":"A038226 \u2192 basis A007318 \u2192 basis A013610","chain":null}},{"id":"l-334de7043a6ad4f4","kind":"fingerprint_related","node_a_id":"n-a48b231db9ea7677","node_b_id":"n-9cc88df18a45b6a2","fingerprint":"f6-f3eacc80e11c3732","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f3eacc80e11c3732"},"other_id":"n-9cc88df18a45b6a2","other":{"id":"n-9cc88df18a45b6a2","name":"basis_map_A013609 of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^j.","first_index":0,"fingerprint":"f6-f3eacc80e11c3732","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1444","3648","2304"],["54872","207936","262656","110592"],["2085136","10535424","19961856","16809984","5308416"]],"card_rows":[["1444","3648","2304"],["54872","207936","262656","110592"],["2085136","10535424","19961856","16809984","5308416"],["79235168","500432640","1264250880","1596948480","1008599040","254803968"]],"row_sums_preview":["7396","636056","54700816","4704270176","404567235136","34792782221696","2992179271065856","257327417311663616"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c11cb6bca0e3e7dc","steps":["basis A013611","basis A013609"],"edges":["e-06104b6211fa5aaa","e-3c81239e42f7812a"],"text":"A038284 \u2192 basis A013611 \u2192 basis A013609"},"headline":"A038284 \u2192 basis A013611 \u2192 basis A013609","chain":null}},{"id":"l-c5321618166c36f8","kind":"fingerprint_related","node_a_id":"n-2782359f01ec0f55","node_b_id":"n-a48b231db9ea7677","fingerprint":"f6-f3eacc80e11c3732","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f3eacc80e11c3732"},"other_id":"n-2782359f01ec0f55","other":{"id":"n-2782359f01ec0f55","name":"derivative of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*8^j.","first_index":0,"fingerprint":"f6-f3eacc80e11c3732","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["24"],["912","1152"]],"card_rows":[["24"],["912","1152"],["25992","65664","41472"],["658464","2495232","3151872","1327104"]],"row_sums_preview":["0","24","2064","133128","7632672","410256120","21169215792","1061988992232"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2f02b2035f8f4541","steps":["basis A013610","D"],"edges":["e-f9ff39da4c6bf2a9","e-f0a56ac400d3f245"],"text":"A038322 \u2192 basis A013610 \u2192 D"},"headline":"A038322 \u2192 basis A013610 \u2192 D","chain":null}},{"id":"l-e6067cdb891afcae","kind":"fingerprint_related","node_a_id":"n-587b2e7eeca5ddbb","node_b_id":"n-a48b231db9ea7677","fingerprint":"f6-f3eacc80e11c3732","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f3eacc80e11c3732"},"other_id":"n-587b2e7eeca5ddbb","other":{"id":"n-587b2e7eeca5ddbb","name":"basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*8^j.","first_index":0,"fingerprint":"f6-f3eacc80e11c3732","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["19","24"],["361","912","576"]],"card_rows":[["1"],["19","24"],["361","912","576"],["6859","25992","32832","13824"]],"row_sums_preview":["1","43","1849","79507","3418801","147008443","6321363049","271818611107"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2f02b2035f8f4541","steps":["basis A013610"],"edges":["e-f9ff39da4c6bf2a9"],"text":"A038322 \u2192 basis A013610"},"headline":"A038322 \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:            24,             19;\n  2:           576,            912,            361;\n  3:         13824,          32832,          25992,            6859;\n  4:        331776,        1050624,        1247616,          658464,          130321;\n  5:       7962624,       31518720,       49904640,        39507840,        15638520,         2476099;\n  6:     191102976,      907739136,     1796567040,      1896376320,      1125973440,       356558256,       47045881;\n  7:    4586471424,    25416695808,    60364652544,     79647805440,     63054512640,     29950893504,     7903708008,      893871739;\n  8:  110075314176,   697143656448,  1931668881408,   3058475728896,   3026616606720,   1916857184256,   758755968768,   171623373888,   16983563041;\n  9: 2641807540224, 18822878724096, 59605782626304, 110105126240256, 130749837410304, 103510287949824, 54630429751296, 18535324379904, 3668449616856, 322687697779;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (24+19*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[(24 + 19*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] = (24 + 19*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-2f02b2035f8f4541","steps":["basis A013610","reverse"],"edges":["e-f9ff39da4c6bf2a9","e-dd797f7eb576b37a"],"text":"A038322 \u2192 basis A013610 \u2192 reverse"},"headline":"A038322 \u2192 basis A013610 \u2192 reverse","chain":null}
