{"id":"n-5a4e3575d52df992","revision":0,"first_index":0,"fingerprint":"f6-8d10fbbacee008e8","name":"shift_x_plus_one of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^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":[["441","252","36"],["9261","7938","2268","216"],["194481","222264","95256","18144","1296"],["4084101","5834430","3333960","952560","136080","7776"],["85766121","147027636","105019740","40007520","8573040","979776","46656"],["1801088541","3602177082","3087580356","1470276360","420078960","72013536","6858432","279936"],["37822859361","86452249968","86452249968","49401285696","17643316320","4032758016","576108288","47029248","1679616"],["794280046581","2042434405494","2334210749136","1556140499424","666917356896","190547816256","36294822144","4444263936","317447424","10077696"],["16679880978201","47656802794860","61273032164820","46684214982720","23342107491360","8003008282752","1905478162560","311098475520","33331979520","2116316160","60466176"],["350277500542221","1100872144561266","1572674492230380","1348006707626040","770289547214880","308115818885952","88033091110272","17965936961280","2566562423040","244434516480","13967686656","362797056"],["7355827511386641","25219980039039912","39631397204205576","37744187813529120","24264120737268720","11092169479894272","3697389826631424","905483222848512","161693432651520","20532499384320","1759928518656","91424858112","2176782336"],["154472377739119461","573754545888157998","983579221522556568","1030416327309344976","736011662363817840","378520283501392032","144198203238625536","41199486639607296","8828461422772992","1401343082979840","160153495197696","12479493132288","594261577728","13060694016"],["3243919932521508681","12975679730086034724","24097690927302635916","27540218202631583904","21638742873496244496","12364995927712139712","5299283969019488448","1730378438863506432","432594609715876608","82398973279214592","11771281897030656","1222990326964224","87356451926016","3839844040704","78364164096"],["68122318582951682301","291952793926935781290","583905587853871562580","722930727819079077480","619654909559210637840","389497371722932400928","185474938915682095680","68133651030250565760","19466757437214447360","4325946097158766080","741590759512931328","96310488248432640","9172427452231680","604775436410880","24684711690240","470184984576"],["1430568690241985328321","6539742583963361500896","14013734108492917501920","18684978811323890002560","17350337467657897859520","11897374263536844246528","6231957947566918414848","2543656305129354455040","817603812363006789120","207645412663620771840","41529082532724154368","6472064810294673408","770483905987461120","67734848878018560","4147031563960320","157982154817536","2821109907456"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["729","19683","531441","14348907","387420489","10460353203","282429536481","7625597484987","205891132094649","5559060566555523","150094635296999121","4052555153018976267","109418989131512359209","2954312706550833698643","79766443076872509863361"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["441","252","36"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["21","6"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":193556,"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-369f3efc1225378f"]},{"id":193557,"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-0c773e6bfb9f702d","e-369f3efc1225378f"]},{"id":193558,"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-0c773e6bfb9f702d","e-369f3efc1225378f"]},{"id":193559,"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-369f3efc1225378f"]},{"id":193560,"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-0c773e6bfb9f702d"]}],"region_exact":{},"incoming_edges":[{"id":"e-0c773e6bfb9f702d","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-5a4e3575d52df992","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":"95b24f7c5b74a12095a1c487009ce290d96a5188bff1db51068979679c1ba5a3","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-4ef858fd88b37fab","role":"sequence","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^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-369f3efc1225378f","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-5a4e3575d52df992","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":"e179f25616071ec793953c2110e03c8e95f8e17ece490b465c191c6012b928f5","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-4ef858fd88b37fab","role":"sequence","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^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-03300078195bf2d9","kind":"fingerprint_related","node_a_id":"n-6890f01c78bc6b64","node_b_id":"n-5a4e3575d52df992","fingerprint":"f6-8d10fbbacee008e8","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8d10fbbacee008e8"},"other_id":"n-6890f01c78bc6b64","other":{"id":"n-6890f01c78bc6b64","name":"derivative of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j).","first_index":0,"fingerprint":"f6-8d10fbbacee008e8","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["2"],["28","8"]],"card_rows":[["2"],["28","8"],["294","168","24"],["2744","2352","672","64"]],"row_sums_preview":["0","2","36","486","5832","65610","708588","7440174"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-22693abd431a71f2","steps":["basis A013609","D"],"edges":["e-4c464713c83c5f6b","e-6e1fc1f5bbccb636"],"text":"A038255 \u2192 basis A013609 \u2192 D"},"headline":"A038255 \u2192 basis A013609 \u2192 D","chain":null}},{"id":"l-0e3fe215ae66b9b1","kind":"fingerprint_related","node_a_id":"n-6a4f4798749d2440","node_b_id":"n-5a4e3575d52df992","fingerprint":"f6-8d10fbbacee008e8","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8d10fbbacee008e8"},"other_id":"n-6a4f4798749d2440","other":{"id":"n-6a4f4798749d2440","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j).","first_index":0,"fingerprint":"f6-8d10fbbacee008e8","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["7","2"],["49","28","4"]],"card_rows":[["1"],["7","2"],["49","28","4"],["343","294","84","8"]],"row_sums_preview":["1","9","81","729","6561","59049","531441","4782969"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-22693abd431a71f2","steps":["basis A013609"],"edges":["e-4c464713c83c5f6b"],"text":"A038255 \u2192 basis A013609"},"headline":"A038255 \u2192 basis A013609","chain":null}},{"id":"l-381385261e38524c","kind":"fingerprint_related","node_a_id":"n-f038b923445dac3e","node_b_id":"n-5a4e3575d52df992","fingerprint":"f6-8d10fbbacee008e8","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8d10fbbacee008e8"},"other_id":"n-f038b923445dac3e","other":{"id":"n-f038b923445dac3e","name":"shift_x_plus_one of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*1^j.","first_index":0,"fingerprint":"f6-8d10fbbacee008e8","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["14","4"],["196","112","16"]],"card_rows":[["1"],["14","4"],["196","112","16"],["2744","2352","672","64"]],"row_sums_preview":["1","18","324","5832","104976","1889568","34012224","612220032"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3dd184aba99388f7","steps":["basis A013611","basis A007318"],"edges":["e-8c5f302cdd82df95","e-0128d6c646b3f27d"],"text":"A038291 \u2192 basis A013611 \u2192 basis A007318"},"headline":"A038291 \u2192 basis A013611 \u2192 basis A007318","chain":null}},{"id":"l-40eca54fcc7f8730","kind":"fingerprint_related","node_a_id":"n-5a4e3575d52df992","node_b_id":"n-01f9460eada72015","fingerprint":"f6-8d10fbbacee008e8","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8d10fbbacee008e8"},"other_id":"n-01f9460eada72015","other":{"id":"n-01f9460eada72015","name":"shift_x_plus_one of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^j.","first_index":0,"fingerprint":"f6-8d10fbbacee008e8","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["784","448","64"],["21952","18816","5376","512"],["614656","702464","301056","57344","4096"]],"card_rows":[["784","448","64"],["21952","18816","5376","512"],["614656","702464","301056","57344","4096"],["17210368","24586240","14049280","4014080","573440","32768"]],"row_sums_preview":["1296","46656","1679616","60466176","2176782336","78364164096","2821109907456","101559956668416"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-6874f381e9c7e0a2","steps":["x \u21a6 x+1","x \u21a6 x+1"],"edges":["e-4a3b4bf6a80699a3","e-b097c5c47d554a10"],"text":"A038334 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x+1"},"headline":"A038334 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-540c877ce7f6faaf","kind":"fingerprint_related","node_a_id":"n-ff1da23ecc68394a","node_b_id":"n-5a4e3575d52df992","fingerprint":"f6-8d10fbbacee008e8","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8d10fbbacee008e8"},"other_id":"n-ff1da23ecc68394a","other":{"id":"n-ff1da23ecc68394a","name":"derivative of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.","first_index":0,"fingerprint":"f6-8d10fbbacee008e8","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["112","32"],["2352","1344","192"],["43904","37632","10752","1024"]],"card_rows":[["112","32"],["2352","1344","192"],["43904","37632","10752","1024"],["768320","878080","376320","71680","5120"]],"row_sums_preview":["144","3888","93312","2099520","45349632","952342272","19591041024","396718580736"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ea9b82d727693fb6","steps":["basis A013609","D"],"edges":["e-ccf7b0982517b62a","e-9ffb1eb7f95ae355"],"text":"A038328 \u2192 basis A013609 \u2192 D"},"headline":"A038328 \u2192 basis A013609 \u2192 D","chain":null}},{"id":"l-8d5f402aee8f1cad","kind":"fingerprint_related","node_a_id":"n-8c340f191f09d9f8","node_b_id":"n-5a4e3575d52df992","fingerprint":"f6-8d10fbbacee008e8","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8d10fbbacee008e8"},"other_id":"n-8c340f191f09d9f8","other":{"id":"n-8c340f191f09d9f8","name":"mobius_x_over_x_plus_one of basis_map_A013611 of Triangle of coefficients in expansion of (3+5x)^n.","first_index":0,"fingerprint":"f6-8d10fbbacee008e8","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["8","28"],["64","448","784"]],"card_rows":[["1"],["8","28"],["64","448","784"],["512","5376","18816","21952"]],"row_sums_preview":["1","36","1296","46656","1679616","60466176","2176782336","78364164096"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c44f7114310405ad","steps":["basis A013611","x \u21a6 x/(x+1)"],"edges":["e-96f2019a633e1196","e-7f720102102753d9"],"text":"A013622 \u2192 basis A013611 \u2192 x \u21a6 x/(x+1)"},"headline":"A013622 \u2192 basis A013611 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-9044cb70fcfe6f09","kind":"fingerprint_related","node_a_id":"n-4e63782ec2a203f8","node_b_id":"n-5a4e3575d52df992","fingerprint":"f6-8d10fbbacee008e8","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8d10fbbacee008e8"},"other_id":"n-4e63782ec2a203f8","other":{"id":"n-4e63782ec2a203f8","name":"reversal of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.","first_index":0,"fingerprint":"f6-8d10fbbacee008e8","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["16","112","196"],["64","672","2352","2744"],["256","3584","18816","43904","38416"]],"card_rows":[["16","112","196"],["64","672","2352","2744"],["256","3584","18816","43904","38416"],["1024","17920","125440","439040","768320","537824"]],"row_sums_preview":["324","5832","104976","1889568","34012224","612220032","11019960576","198359290368"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ea9b82d727693fb6","steps":["basis A013609","reverse"],"edges":["e-ccf7b0982517b62a","e-64c98336971fc9ef"],"text":"A038328 \u2192 basis A013609 \u2192 reverse"},"headline":"A038328 \u2192 basis A013609 \u2192 reverse","chain":null}},{"id":"l-a4ca00ceda7e0e98","kind":"fingerprint_related","node_a_id":"n-72a51ec1ec4c0613","node_b_id":"n-5a4e3575d52df992","fingerprint":"f6-8d10fbbacee008e8","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8d10fbbacee008e8"},"other_id":"n-72a51ec1ec4c0613","other":{"id":"n-72a51ec1ec4c0613","name":"mobius_x_over_x_plus_one of basis_map_A013610 of Triangle of coefficients in expansion of (1+5x)^n.","first_index":0,"fingerprint":"f6-8d10fbbacee008e8","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["36","252","441"],["216","2268","7938","9261"],["1296","18144","95256","222264","194481"]],"card_rows":[["36","252","441"],["216","2268","7938","9261"],["1296","18144","95256","222264","194481"],["7776","136080","952560","3333960","5834430","4084101"]],"row_sums_preview":["729","19683","531441","14348907","387420489","10460353203","282429536481","7625597484987"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-bbfc5a54ff13a785","steps":["basis A013610","x \u21a6 x/(x+1)"],"edges":["e-46de8f4df4fd1cb3","e-3f57f2d00eaeeeb9"],"text":"A013612 \u2192 basis A013610 \u2192 x \u21a6 x/(x+1)"},"headline":"A013612 \u2192 basis A013610 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-ba22586da80937b7","kind":"fingerprint_related","node_a_id":"n-3208539f9ba783e8","node_b_id":"n-5a4e3575d52df992","fingerprint":"f6-8d10fbbacee008e8","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8d10fbbacee008e8"},"other_id":"n-3208539f9ba783e8","other":{"id":"n-3208539f9ba783e8","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.","first_index":0,"fingerprint":"f6-8d10fbbacee008e8","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["196","112","16"],["2744","2352","672","64"],["38416","43904","18816","3584","256"]],"card_rows":[["196","112","16"],["2744","2352","672","64"],["38416","43904","18816","3584","256"],["537824","768320","439040","125440","17920","1024"]],"row_sums_preview":["324","5832","104976","1889568","34012224","612220032","11019960576","198359290368"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ea9b82d727693fb6","steps":["basis A013609"],"edges":["e-ccf7b0982517b62a"],"text":"A038328 \u2192 basis A013609"},"headline":"A038328 \u2192 basis A013609","chain":null}},{"id":"l-d33692963967edfd","kind":"fingerprint_related","node_a_id":"n-2a3c47627a03f4f8","node_b_id":"n-5a4e3575d52df992","fingerprint":"f6-8d10fbbacee008e8","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8d10fbbacee008e8"},"other_id":"n-2a3c47627a03f4f8","other":{"id":"n-2a3c47627a03f4f8","name":"reversal of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j).","first_index":0,"fingerprint":"f6-8d10fbbacee008e8","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","7"],["4","28","49"]],"card_rows":[["1"],["2","7"],["4","28","49"],["8","84","294","343"]],"row_sums_preview":["1","9","81","729","6561","59049","531441","4782969"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-22693abd431a71f2","steps":["basis A013609","reverse"],"edges":["e-4c464713c83c5f6b","e-04563bea8e03ff94"],"text":"A038255 \u2192 basis A013609 \u2192 reverse"},"headline":"A038255 \u2192 basis A013609 \u2192 reverse","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:             441,              252,               36;\n  1:            9261,             7938,             2268,              216;\n  2:          194481,           222264,            95256,            18144,            1296;\n  3:         4084101,          5834430,          3333960,           952560,          136080,            7776;\n  4:        85766121,        147027636,        105019740,         40007520,         8573040,          979776,          46656;\n  5:      1801088541,       3602177082,       3087580356,       1470276360,       420078960,        72013536,        6858432,         279936;\n  6:     37822859361,      86452249968,      86452249968,      49401285696,     17643316320,      4032758016,      576108288,       47029248,       1679616;\n  7:    794280046581,    2042434405494,    2334210749136,    1556140499424,    666917356896,    190547816256,    36294822144,     4444263936,     317447424,     10077696;\n  8:  16679880978201,   47656802794860,   61273032164820,   46684214982720,  23342107491360,   8003008282752,  1905478162560,   311098475520,   33331979520,   2116316160,    60466176;\n  9: 350277500542221, 1100872144561266, 1572674492230380, 1348006707626040, 770289547214880, 308115818885952, 88033091110272, 17965936961280, 2566562423040, 244434516480, 13967686656, 362797056;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (21+6*x)*P(n-1,x) for n >= 1, with P(0,x) = 441 + 252*x + 36*x^2.","mathematica":"P[0, x_] := 441 + 252*x + 36*x^2;\nP[n_, x_] := P[n, x] = Expand[(21 + 6*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] = 441 + 252*x + 36*x^2; for(n = 1, N, my(i = n + 1); v[i] = (21 + 6*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-615df6ea54d3bbef","steps":["basis A013609","basis A007318"],"edges":["e-f3e6c91e6b1bdc2d","e-0c773e6bfb9f702d"],"text":"A038329 \u2192 basis A013609 \u2192 basis A007318"},"headline":"A038329 \u2192 basis A013609 \u2192 basis A007318","chain":null}