{"id":"n-41d9deb44c94abe1","revision":0,"first_index":0,"fingerprint":"f6-60cc3cc9b4461a16","name":"derivative of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*12^j.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["1056","1152"],["34848","76032","41472"],["1022208","3345408","3649536","1327104"],["28110720","122664960","200724480","145981440","39813120"],["742123008","4047943680","8831877120","9634775040","5255331840","1146617856"],["19047823872","124676665344","340027269120","494585118720","404660551680","176579149824","32105299968"],["478916714496","3657182183424","11968959873024","21761745223680","23740085698560","15538965184512","5650532794368","880602513408"],["11853188683776","103446010331136","394975675809792","861765110857728","1175134242078720","1025571702177792","559402746642432","174359297654784","23776267862016"],["289744612270080","2844765284106240","12413521239736320","31598054064783360","51705906651463680","56406443619778560","41022868087111680","19179522742026240","5230778929643520","634033809653760"],["7011819616935936","76492577639301120","375509017502023680","1092389869096796160","2085471568275701760","2730071871197282304","2481883519270256640","1547148167856783360","632924250486865920","153436181936209920","16738492574859264"],["168283670806462464","2019404049677549568","11014931180059361280","36048865680194273280","78652070574969323520","120123162332680421376","131043449817469550592","102111779078547701760","55697334042844200960","20253576015579709440","4418962039762845696","438244169232678912"],["4010760820887355392","52504505291616288768","315027031749697732608","1145552842726173573120","2811811523055153315840","4907889203878085787648","6246404441299381911552","5840793763292928540672","3982359384063360368640","1930840913485265633280","631911571686086934528","125337832400546168832","11394348400049651712"],["95024179448715804672","1347615635818151411712","8820756888991536513024","35283027555966146052096","96226438788998580142080","188953734349306302824448","274841795417172804108288","299827413182370331754496","245313338058302998708224","148674750338365453762560","64876254693104925278208","19302026189684110000128","3509459307215292727296","294500081724360228864"],["2239855658434015395840","34208704601537689681920","242570814447267254108160","1058490826678984381562880","3175472480036953144688640","6928303592807897770229760","11337224060958378169466880","14134720907168887068426240","13492233593206664928952320","9812533522332119948328960","5352291012181156335452160","2123222880865252100014080","579060785690523300003840","97185026969038875525120","7572859244340691599360"],["52561946117918227955712","860104572838661912002560","6568071283495236418928640","31049064249250208525844480","101615119361182500630036480","243876286466838001512087552","443411429939705457294704640","621927719915431031010754560","678466603544106579284459520","575668633310151036968632320","376801287257553406015832064","186843613516142184801239040","67943132187688067200450560","17104564746550842092421120","2665646454007923442974720","193865196655121704943616"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["2208","152352","9344256","537294720","29658668544","1591681878528","83676990185472","4330284242098176","221325639040573440","11199077335453016064","561990062651824078848","28005838122149233262592","1387366134666469709316096","68377330922847435673436160","3355047703947714177043267584"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1056","1152"]],"recurrence":{"denominator":{"coeffs":[["1"],["1"]]},"terms":[{"coeff":{"coeffs":[["44","48"],["22","24"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":195008,"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-2eaf39326b6c0eac"]},{"id":195009,"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-2eaf39326b6c0eac"]},{"id":195010,"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-2eaf39326b6c0eac"]},{"id":195011,"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-2eaf39326b6c0eac"]}],"region_exact":{},"incoming_edges":[{"id":"e-2eaf39326b6c0eac","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"0\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-41d9deb44c94abe1","premise_count":1,"theorem":"Laguerre's theorem on differential operators: if phi is a real polynomial with only real roots, then phi(D) maps every real-rooted polynomial to a real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"fa818942a85b898cc8729d33a4cc0edcae433363f0886289abbd9999caaf7608","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-2a3c7d06bf210d3a","role":"sequence","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*12^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-166d8299f6442086","kind":"fingerprint_related","node_a_id":"n-0dc59a93b8bc259f","node_b_id":"n-41d9deb44c94abe1","fingerprint":"f6-60cc3cc9b4461a16","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-60cc3cc9b4461a16"},"other_id":"n-0dc59a93b8bc259f","other":{"id":"n-0dc59a93b8bc259f","name":"derivative of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*3^j.","first_index":0,"fingerprint":"f6-60cc3cc9b4461a16","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["264","288"],["4356","9504","5184"],["63888","209088","228096","82944"]],"card_rows":[["264","288"],["4356","9504","5184"],["63888","209088","228096","82944"],["878460","3833280","6272640","4561920","1244160"]],"row_sums_preview":["552","19044","584016","16790460","463416696","12435014676","326863242912","8457586410348"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-108b34f10b7468f4","steps":["basis A013611","D"],"edges":["e-8f18b93eb5dfe7b3","e-bcc8b3cdbab80436"],"text":"A038281 \u2192 basis A013611 \u2192 D"},"headline":"A038281 \u2192 basis A013611 \u2192 D","chain":null}},{"id":"l-2622d456087c2c7a","kind":"fingerprint_related","node_a_id":"n-41d9deb44c94abe1","node_b_id":"n-c9b6b72f63c86d9a","fingerprint":"f6-60cc3cc9b4461a16","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-60cc3cc9b4461a16"},"other_id":"n-c9b6b72f63c86d9a","other":{"id":"n-c9b6b72f63c86d9a","name":"reversal of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*12^j.","first_index":0,"fingerprint":"f6-60cc3cc9b4461a16","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["576","1056","484"],["13824","38016","34848","10648"],["331776","1216512","1672704","1022208","234256"]],"card_rows":[["576","1056","484"],["13824","38016","34848","10648"],["331776","1216512","1672704","1022208","234256"],["7962624","36495360","66908160","61332480","28110720","5153632"]],"row_sums_preview":["2116","97336","4477456","205962976","9474296896","435817657216","20047612231936","922190162669056"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-29d7c5a1e19b9d74","steps":["basis A013609","reverse"],"edges":["e-61350e73fec19600","e-30873dcc984978cd"],"text":"A038314 \u2192 basis A013609 \u2192 reverse"},"headline":"A038314 \u2192 basis A013609 \u2192 reverse","chain":null}},{"id":"l-2637f522f0fa0b2c","kind":"fingerprint_related","node_a_id":"n-2a3c7d06bf210d3a","node_b_id":"n-41d9deb44c94abe1","fingerprint":"f6-60cc3cc9b4461a16","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-60cc3cc9b4461a16"},"other_id":"n-2a3c7d06bf210d3a","other":{"id":"n-2a3c7d06bf210d3a","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*12^j.","first_index":0,"fingerprint":"f6-60cc3cc9b4461a16","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["484","1056","576"],["10648","34848","38016","13824"],["234256","1022208","1672704","1216512","331776"]],"card_rows":[["484","1056","576"],["10648","34848","38016","13824"],["234256","1022208","1672704","1216512","331776"],["5153632","28110720","61332480","66908160","36495360","7962624"]],"row_sums_preview":["2116","97336","4477456","205962976","9474296896","435817657216","20047612231936","922190162669056"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-29d7c5a1e19b9d74","steps":["basis A013609"],"edges":["e-61350e73fec19600"],"text":"A038314 \u2192 basis A013609"},"headline":"A038314 \u2192 basis A013609","chain":null}},{"id":"l-3cc4e1a7a223955b","kind":"fingerprint_related","node_a_id":"n-344e4bfa9933088a","node_b_id":"n-41d9deb44c94abe1","fingerprint":"f6-60cc3cc9b4461a16","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-60cc3cc9b4461a16"},"other_id":"n-344e4bfa9933088a","other":{"id":"n-344e4bfa9933088a","name":"Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*11^j.","first_index":0,"fingerprint":"f6-60cc3cc9b4461a16","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["144","264","121"],["1728","4752","4356","1331"],["20736","76032","104544","63888","14641"]],"card_rows":[["144","264","121"],["1728","4752","4356","1331"],["20736","76032","104544","63888","14641"],["248832","1140480","2090880","1916640","878460","161051"]],"row_sums_preview":["529","12167","279841","6436343","148035889","3404825447","78310985281","1801152661463"],"regions":["(-inf, 0)","(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-344e4bfa9933088a","steps":[],"edges":[],"text":"A038337"},"headline":"Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*11^j.","chain":null}},{"id":"l-537714a61dd97f5f","kind":"fingerprint_related","node_a_id":"n-b0b0fff0ab69f594","node_b_id":"n-41d9deb44c94abe1","fingerprint":"f6-60cc3cc9b4461a16","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-60cc3cc9b4461a16"},"other_id":"n-b0b0fff0ab69f594","other":{"id":"n-b0b0fff0ab69f594","name":"reversal of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*12^j.","first_index":0,"fingerprint":"f6-60cc3cc9b4461a16","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["12","11"],["144","264","121"]],"card_rows":[["1"],["12","11"],["144","264","121"],["1728","4752","4356","1331"]],"row_sums_preview":["1","23","529","12167","279841","6436343","148035889","3404825447"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-9e472291b944d56e","steps":["reverse"],"edges":["e-8a7ec150f5f093be"],"text":"A038326 \u2192 reverse"},"headline":"A038326 \u2192 reverse","chain":null}},{"id":"l-8a2ea8b0079032a1","kind":"fingerprint_related","node_a_id":"n-9e472291b944d56e","node_b_id":"n-41d9deb44c94abe1","fingerprint":"f6-60cc3cc9b4461a16","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-60cc3cc9b4461a16"},"other_id":"n-9e472291b944d56e","other":{"id":"n-9e472291b944d56e","name":"Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*12^j.","first_index":0,"fingerprint":"f6-60cc3cc9b4461a16","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["11","12"],["121","264","144"]],"card_rows":[["1"],["11","12"],["121","264","144"],["1331","4356","4752","1728"]],"row_sums_preview":["1","23","529","12167","279841","6436343","148035889","3404825447"],"regions":["(-inf, 0)","(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":["A038326"],"derivation":{"root":"n-9e472291b944d56e","steps":[],"edges":[],"text":"A038326"},"headline":"A038326: Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*12^j.","chain":null}},{"id":"l-9fe1049070074914","kind":"fingerprint_related","node_a_id":"n-faaa8ff869da19e7","node_b_id":"n-41d9deb44c94abe1","fingerprint":"f6-60cc3cc9b4461a16","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-60cc3cc9b4461a16"},"other_id":"n-faaa8ff869da19e7","other":{"id":"n-faaa8ff869da19e7","name":"derivative of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*11^j.","first_index":0,"fingerprint":"f6-60cc3cc9b4461a16","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["264","242"],["4752","8712","3993"],["76032","209088","191664","58564"]],"card_rows":[["264","242"],["4752","8712","3993"],["76032","209088","191664","58564"],["1140480","4181760","5749920","3513840","805255"]],"row_sums_preview":["506","17457","535348","15391255","424798638","11398763453","299624639336","7752787542819"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-344e4bfa9933088a","steps":["D"],"edges":["e-552f58accfe91872"],"text":"A038337 \u2192 D"},"headline":"A038337 \u2192 D","chain":null}},{"id":"l-a53f40fc19782b1b","kind":"fingerprint_related","node_a_id":"n-d798b01f393943bf","node_b_id":"n-41d9deb44c94abe1","fingerprint":"f6-60cc3cc9b4461a16","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-60cc3cc9b4461a16"},"other_id":"n-d798b01f393943bf","other":{"id":"n-d798b01f393943bf","name":"derivative of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*12^j.","first_index":0,"fingerprint":"f6-60cc3cc9b4461a16","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["12"],["264","288"]],"card_rows":[["12"],["264","288"],["4356","9504","5184"],["63888","209088","228096","82944"]],"row_sums_preview":["0","12","552","19044","584016","16790460","463416696","12435014676"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-9e472291b944d56e","steps":["D"],"edges":["e-3ceead603d0777b6"],"text":"A038326 \u2192 D"},"headline":"A038326 \u2192 D","chain":null}},{"id":"l-aea9605b92bbf282","kind":"fingerprint_related","node_a_id":"n-41d9deb44c94abe1","node_b_id":"n-49a9936529864dac","fingerprint":"f6-60cc3cc9b4461a16","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-60cc3cc9b4461a16"},"other_id":"n-49a9936529864dac","other":{"id":"n-49a9936529864dac","name":"basis_map_A013609 of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^j.","first_index":0,"fingerprint":"f6-60cc3cc9b4461a16","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1936","4224","2304"],["85184","278784","304128","110592"],["3748096","16355328","26763264","19464192","5308416"]],"card_rows":[["1936","4224","2304"],["85184","278784","304128","110592"],["3748096","16355328","26763264","19464192","5308416"],["164916224","899543040","1962639360","2141061120","1167851520","254803968"]],"row_sums_preview":["8464","778688","71639296","6590815232","606355001344","55784660123648","5132188731375616","472161363286556672"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-6874f381e9c7e0a2","steps":["basis A013610","basis A013609"],"edges":["e-889e30cc3b49f76a","e-fa097368bf8c7b54"],"text":"A038334 \u2192 basis A013610 \u2192 basis A013609"},"headline":"A038334 \u2192 basis A013610 \u2192 basis A013609","chain":null}},{"id":"l-d2fc6ca61df66782","kind":"fingerprint_related","node_a_id":"n-f55a7ba0b32633bc","node_b_id":"n-41d9deb44c94abe1","fingerprint":"f6-60cc3cc9b4461a16","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-60cc3cc9b4461a16"},"other_id":"n-f55a7ba0b32633bc","other":{"id":"n-f55a7ba0b32633bc","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*3^j.","first_index":0,"fingerprint":"f6-60cc3cc9b4461a16","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["121","264","144"],["1331","4356","4752","1728"],["14641","63888","104544","76032","20736"]],"card_rows":[["121","264","144"],["1331","4356","4752","1728"],["14641","63888","104544","76032","20736"],["161051","878460","1916640","2090880","1140480","248832"]],"row_sums_preview":["529","12167","279841","6436343","148035889","3404825447","78310985281","1801152661463"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-108b34f10b7468f4","steps":["basis A013611"],"edges":["e-8f18b93eb5dfe7b3"],"text":"A038281 \u2192 basis A013611"},"headline":"A038281 \u2192 basis A013611","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:             1056,              1152;\n  1:            34848,             76032,              41472;\n  2:          1022208,           3345408,            3649536,             1327104;\n  3:         28110720,         122664960,          200724480,           145981440,            39813120;\n  4:        742123008,        4047943680,         8831877120,          9634775040,          5255331840,          1146617856;\n  5:      19047823872,      124676665344,       340027269120,        494585118720,        404660551680,        176579149824,         32105299968;\n  6:     478916714496,     3657182183424,     11968959873024,      21761745223680,      23740085698560,      15538965184512,       5650532794368,        880602513408;\n  7:   11853188683776,   103446010331136,    394975675809792,     861765110857728,    1175134242078720,    1025571702177792,     559402746642432,     174359297654784,     23776267862016;\n  8:  289744612270080,  2844765284106240,  12413521239736320,   31598054064783360,   51705906651463680,   56406443619778560,   41022868087111680,   19179522742026240,   5230778929643520,    634033809653760;\n  9: 7011819616935936, 76492577639301120, 375509017502023680, 1092389869096796160, 2085471568275701760, 2730071871197282304, 2481883519270256640, 1547148167856783360, 632924250486865920, 153436181936209920, 16738492574859264;\n","recurrence_sentence":"The row polynomials satisfy (n+1)*P(n,x) = (22*n+44+(24*n+48)*x)*P(n-1,x) for n >= 1, with P(0,x) = 1056 + 1152*x.","mathematica":"P[0, x_] := 1056 + 1152*x;\nP[n_, x_] := P[n, x] = Expand[Together[((22*n + 44 + (24*n + 48)*x)*P[n - 1, x])/(n + 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1056 + 1152*x; for(n = 1, N, my(i = n + 1); v[i] = ((22*n + 44 + (24*n + 48)*x)*v[i - 1])/(n + 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-29d7c5a1e19b9d74","steps":["basis A013609","D"],"edges":["e-61350e73fec19600","e-2eaf39326b6c0eac"],"text":"A038314 \u2192 basis A013609 \u2192 D"},"headline":"A038314 \u2192 basis A013609 \u2192 D","chain":null}