{"id":"n-9cc88df18a45b6a2","revision":0,"first_index":0,"fingerprint":"f6-f3eacc80e11c3732","name":"basis_map_A013609 of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^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":[["1444","3648","2304"],["54872","207936","262656","110592"],["2085136","10535424","19961856","16809984","5308416"],["79235168","500432640","1264250880","1596948480","1008599040","254803968"],["3010936384","22819728384","72062300160","121368084480","114980290560","58095304704","12230590464"],["114415582592","1011674625024","3833714368512","8070977617920","10194919096320","7726675525632","3253337063424","587068342272"],["4347792138496","43935583715328","194241528004608","490715439169536","774813851320320","782969786597376","494507233640448","178468776050688","28179280429056"],["165216101262848","1878246203830272","9490086082510848","27970780032663552","52997267430309888","66943916754075648","56373824635011072","30518160704667648","9637313906737152","1352605460594688"],["6278211847988224","79303728606167040","450779088919265280","1518413773201735680","3356493603919626240","5087737673309749248","5355513340326051840","3865633689257902080","1831089642280058880","513990075025981440","64925062108545024"],["238572050223552512","3314895855737782272","20936184352028098560","79337119649790689280","200430618062629109760","354445724573912530944","447720915251257933824","403958720527450767360","255131823491021537280","107423925680430120960","27138675961371820032","3116402981210161152"],["9065737908494995456","137417500928766246912","954690006452481294336","4019747395589394923520","11424545229569859256320","23089607200814873444352","34026789559095602970624","36841035312103509983232","29085027877976455249920","16328436703425378385920","6187618119192774967296","1421079759431833485312","149587343098087735296"],["344498040522809827328","5657020454900877164544","42874260289775069036544","198575521342116109221888","627080593711945608069120","1425783244650318435188736","2401319148884746838212608","3033245240696522321952768","2873600754344073778692096","2016561932873034230661120","1018894450293743611281408","351006700579662870872064","73896147490455341236224","7180192468708211294208"],["13090925539866773438464","231502683231328203964416","1900758872846694727286784","9603834304909615464185856","33360687585475506349277184","84279631794885489724489728","159687723400835664741138432","230526638292935696468410368","254792600218507875044032512","214562189657690842142343168","135512961889067900300427264","62245188236126882434646016","19656375232461120768835584","3819862393352768408518656","344649238497994142121984"],["497455170514937390661632","9425466388704076875694080","83340965963278153427189760","456182129483206734548828160","1728690174883730783553454080","4803939012308472914295914496","10113555815386258766938767360","16425022978371668373374238720","20747397446364212682156933120","20383408017480630003522600960","15448477655353740634248708096","8869939323648080746937057280","3734711294167612946078760960","1088660782105538996427816960","196450065943856661009530880","16543163447903718821855232"],["18903296479567620845142016","382045570955471916028133376","3619379093262365520266526720","21335287286599207277360578560","87586968860775693033375006720","265526810862141048353810546688","614904193575484533029877055488","1109601552316663819001281904640","1576802205923680163843926917120","1770444582089746148877391626240","1565445735742512384270535753728","1078584621755606618827546165248","567676116713477167803971665920","220635251840055903276037570560","59720820046932424946897387520","10058243376325461043687981056","794071845499378503449051136"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["7396","636056","54700816","4704270176","404567235136","34792782221696","2992179271065856","257327417311663616","22130157888803070976","1903193578437064103936","163674647745587512938496","14076019706120526112710656","1210537694726365245693116416","104106241746467411129608011776","8953136790196197357146289012736"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013609","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1444","3648","2304"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["38","48"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":201192,"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-3c81239e42f7812a"]},{"id":201193,"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-3c81239e42f7812a"]},{"id":201194,"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-3c81239e42f7812a"]}],"region_exact":{},"incoming_edges":[{"id":"e-3c81239e42f7812a","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-9cc88df18a45b6a2","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":"ab13db052b8f18f1fea0a7c990932a765a114dc58a2331e59492b0659660e0a5","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-0b6560aa9867a4e4","role":"sequence","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^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-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-15f7863b88f7dbec","kind":"fingerprint_related","node_a_id":"n-2782359f01ec0f55","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-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-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-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-371d92ae3a358712","kind":"fingerprint_related","node_a_id":"n-587b2e7eeca5ddbb","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-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:               1444,                3648,                 2304;\n  1:              54872,              207936,               262656,               110592;\n  2:            2085136,            10535424,             19961856,             16809984,               5308416;\n  3:           79235168,           500432640,           1264250880,           1596948480,            1008599040,             254803968;\n  4:         3010936384,         22819728384,          72062300160,         121368084480,          114980290560,           58095304704,           12230590464;\n  5:       114415582592,       1011674625024,        3833714368512,        8070977617920,        10194919096320,         7726675525632,         3253337063424,          587068342272;\n  6:      4347792138496,      43935583715328,      194241528004608,      490715439169536,       774813851320320,       782969786597376,       494507233640448,       178468776050688,        28179280429056;\n  7:    165216101262848,    1878246203830272,     9490086082510848,    27970780032663552,     52997267430309888,     66943916754075648,     56373824635011072,     30518160704667648,      9637313906737152,      1352605460594688;\n  8:   6278211847988224,   79303728606167040,   450779088919265280,  1518413773201735680,   3356493603919626240,   5087737673309749248,   5355513340326051840,   3865633689257902080,   1831089642280058880,    513990075025981440,    64925062108545024;\n  9: 238572050223552512, 3314895855737782272, 20936184352028098560, 79337119649790689280, 200430618062629109760, 354445724573912530944, 447720915251257933824, 403958720527450767360, 255131823491021537280, 107423925680430120960, 27138675961371820032, 3116402981210161152;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (38+48*x)*P(n-1,x) for n >= 1, with P(0,x) = 1444 + 3648*x + 2304*x^2.","mathematica":"P[0, x_] := 1444 + 3648*x + 2304*x^2;\nP[n_, x_] := P[n, x] = Expand[(38 + 48*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] = 1444 + 3648*x + 2304*x^2; for(n = 1, N, my(i = n + 1); v[i] = (38 + 48*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-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}