{"id":"n-6f599efa913bc710","revision":0,"first_index":0,"fingerprint":"f6-f3eacc80e11c3732","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.","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":[["361","912","576"],["6859","25992","32832","13824"],["130321","658464","1247616","1050624","331776"],["2476099","15638520","39507840","49904640","31518720","7962624"],["47045881","356558256","1125973440","1896376320","1796567040","907739136","191102976"],["893871739","7903708008","29950893504","63054512640","79647805440","60364652544","25416695808","4586471424"],["16983563041","171623373888","758755968768","1916857184256","3026616606720","3058475728896","1931668881408","697143656448","110075314176"],["322687697779","3668449616856","18535324379904","54630429751296","103510287949824","130749837410304","110105126240256","59605782626304","18822878724096","2641807540224"],["6131066257801","77445047466960","440213954022720","1482825950392320","3277825785077760","4968493821591552","5229993496412160","3775032899665920","1788173478789120","501943432642560","63403380965376"],["116490258898219","1618601492059464","10222746265638720","38738827953999360","97866512725893120","173069201452105728","218613728150028288","197245469007544320","124576085688975360","52453088711147520","13251306621763584","1521681143169024"],["2213314919066161","33549194562687072","233078614856562816","981383641501317120","2789195612687953920","5637111133011443712","8307321669701074944","8994393386744020992","7100836884271595520","3986434742047211520","1510648954881048576","346943300642537472","36520347436056576"],["42052983462257059","690554254748642232","5233674351779183232","24240175945082532864","76547924037102735360","174045806231728324608","293129778916595073024","370269194420962197504","350781342083016818688","246162345321415311360","124376763951872999424","42847497629353377792","9020525816705974272","876488338465357824"],["799006685782884121","14129802443318371824","116013114797771894976","586171527399268521984","2036174779386932760576","5144020495293303816192","9746565148976786178048","14070229387996563505152","15551306165680412295168","13095836771099294564352","8271054802799554461696","3799144789802666164224","1199729933621894578176","233145898031785181184","21035720123168587776"],["15181127029874798299","287642406881838283560","2543364439797306928320","13921573775732627397120","52755437465934166978560","146604584115859158761472","308641229717598228971520","501251921947377574871040","633160322459845357731840","622052246627216491806720","471450123759574604316672","270689066273439964200960","113974343694079984926720","33223290469529388318720","5995180235103047516160","504857282956046106624"],["288441413567621167681","5829552779471922546816","55227342121312950443520","325550648294055286824960","1336471082470332230123520","4051617597383744023953408","9382693383414986160734208","16931176030222531417866240","24060092253474123593809920","27014840424953401929891840","23886806270485113285378048","16457895229425149823418368","8662050120750078854430720","3366626767578978016296960","911267395735663222456320","153476614018638016413696","12116574790945106558976"]],"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:basis_map_A013610","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["361","912","576"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["19","24"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":203863,"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-054e82d1e0138074"]},{"id":203864,"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-054e82d1e0138074"]},{"id":203865,"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-054e82d1e0138074"]}],"region_exact":{},"incoming_edges":[{"id":"e-054e82d1e0138074","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-6f599efa913bc710","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":"22b29a560c957c337349539d042193de105fc8460afc2f1be7fbce65b48a6e88","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-2e0a8214849adcd6","role":"sequence","name":"basis_map_A007318 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-0e0a43acc8003220","kind":"fingerprint_related","node_a_id":"n-9cc88df18a45b6a2","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-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-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-a48b231db9ea7677","other":{"id":"n-a48b231db9ea7677","name":"reversal 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":[["1"],["24","19"],["576","912","361"]],"card_rows":[["1"],["24","19"],["576","912","361"],["13824","32832","25992","6859"]],"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","reverse"],"edges":["e-f9ff39da4c6bf2a9","e-dd797f7eb576b37a"],"text":"A038322 \u2192 basis A013610 \u2192 reverse"},"headline":"A038322 \u2192 basis A013610 \u2192 reverse","chain":null}},{"id":"l-5764f1d9a6f8885e","kind":"fingerprint_related","node_a_id":"n-587b2e7eeca5ddbb","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-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}},{"id":"l-9f654531e8fd18b4","kind":"fingerprint_related","node_a_id":"n-2782359f01ec0f55","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-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}}],"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:             361,              912,               576;\n  1:            6859,            25992,             32832,             13824;\n  2:          130321,           658464,           1247616,           1050624,            331776;\n  3:         2476099,         15638520,          39507840,          49904640,          31518720,            7962624;\n  4:        47045881,        356558256,        1125973440,        1896376320,        1796567040,          907739136,          191102976;\n  5:       893871739,       7903708008,       29950893504,       63054512640,       79647805440,        60364652544,        25416695808,         4586471424;\n  6:     16983563041,     171623373888,      758755968768,     1916857184256,     3026616606720,      3058475728896,      1931668881408,       697143656448,       110075314176;\n  7:    322687697779,    3668449616856,    18535324379904,    54630429751296,   103510287949824,    130749837410304,    110105126240256,     59605782626304,     18822878724096,     2641807540224;\n  8:   6131066257801,   77445047466960,   440213954022720,  1482825950392320,  3277825785077760,   4968493821591552,   5229993496412160,   3775032899665920,   1788173478789120,   501943432642560,    63403380965376;\n  9: 116490258898219, 1618601492059464, 10222746265638720, 38738827953999360, 97866512725893120, 173069201452105728, 218613728150028288, 197245469007544320, 124576085688975360, 52453088711147520, 13251306621763584, 1521681143169024;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (19+24*x)*P(n-1,x) for n >= 1, with P(0,x) = 361 + 912*x + 576*x^2.","mathematica":"P[0, x_] := 361 + 912*x + 576*x^2;\nP[n_, x_] := P[n, x] = Expand[(19 + 24*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] = 361 + 912*x + 576*x^2; for(n = 1, N, my(i = n + 1); v[i] = (19 + 24*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 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}
