{"id":"n-faaa8ff869da19e7","revision":0,"first_index":0,"fingerprint":"f6-60cc3cc9b4461a16","name":"derivative of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*11^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":[["264","242"],["4752","8712","3993"],["76032","209088","191664","58564"],["1140480","4181760","5749920","3513840","805255"],["16422912","75271680","137998080","126498240","57978360","10629366"],["229920768","1264564224","2897959680","3541950720","2435091120","892866744","136410197"],["3153199104","20233027584","55640825856","85006817280","77922915840","42857603712","13095378912","1714871048"],["42568187904","312166711296","1001534865408","1836147253248","2103918727680","1542873733632","707150461248","185206073184","21221529219"],["567575838720","4682500669440","17169169121280","36722945064960","50494049464320","46286212008960","28286018449920","11112364391040","2546583506280","259374246010"],["7492001071104","68676676485120","283291290501120","692489821224960","1110869088215040","1221955997036544","933438608847360","488944033205760","168074511414480","34237400473320","3138428376721"],["98077104930816","988944141385728","4532660648017920","12464816782049280","22852164100423680","29326943928877056","26883031934803968","17601985195407360","8067576547895040","2465092834079040","451933686247824","37661140520652"],["1275002364100608","14025026005106688","70709506109079552","216056824222187520","445617199958261760","653571893272117248","698958830304903168","549181938096709632","314635485367906560","128184827372110080","35250827527330272","5875137921221712","448795257871103"],["16476953628377088","196350364071493632","1079927002393214976","3629754646932750336","8318187732554219520","13725009758714462208","16775011927317676032","15377094266707869696","10571752308361660416","5383762749628623360","1974046341530495232","493511585382623808","75397603322345304","5316497670165374"],["211846546650562560","2718697348682219520","16198905035898224640","59395985131626823680","149727379185975951360","274500195174289244160","377437768364647710720","395410995429630935040","317152569250849812480","193815458986630440960","88832085368872285440","29610695122957428480","6785784299011077360","956969580629767320","62658722541234765"],["2711635797127200768","37284992210499010560","239245366684035317760","950335762106029178880","2613423345791580241920","5270403747346353487872","8052005725112484495360","9489863890311142440960","8699041899451880570880","6202094687572174110720","3411152078164695760896","1421313365901956567040","434290195136708951040","91869079740457662720","12030474727917074880","735195677817154576"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["506","17457","535348","15391255","424798638","11398763453","299624639336","7752787542819","198126792760930","5012607856851529","125770888044638364","3133791293778905903","77621599738215976982","1912817993548893718485","46927801441732859226832"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["264","242"]],"recurrence":{"denominator":{"coeffs":[["1"],["1"]]},"terms":[{"coeff":{"coeffs":[["24","22"],["12","11"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":193763,"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-552f58accfe91872"]},{"id":193764,"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-552f58accfe91872"]},{"id":193765,"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-552f58accfe91872"]},{"id":193766,"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-552f58accfe91872"]}],"region_exact":{},"incoming_edges":[{"id":"e-552f58accfe91872","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-faaa8ff869da19e7","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":"4143de4a9b7caecac7688fe9d104d29c8f0c9f5e73becea7ab489ffaa855b19c","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-344e4bfa9933088a","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*11^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-09822992dc53d3fb","kind":"fingerprint_related","node_a_id":"n-faaa8ff869da19e7","node_b_id":"n-0dc59a93b8bc259f","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-15b81773b5b1c9cd","kind":"fingerprint_related","node_a_id":"n-faaa8ff869da19e7","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-21bffb1efd817bd1","kind":"fingerprint_related","node_a_id":"n-faaa8ff869da19e7","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-4b2e658f32874366","kind":"fingerprint_related","node_a_id":"n-b0b0fff0ab69f594","node_b_id":"n-faaa8ff869da19e7","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-5e65ecb1b3692a83","kind":"fingerprint_related","node_a_id":"n-f55a7ba0b32633bc","node_b_id":"n-faaa8ff869da19e7","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}},{"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-41d9deb44c94abe1","other":{"id":"n-41d9deb44c94abe1","name":"derivative 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":[["1056","1152"],["34848","76032","41472"],["1022208","3345408","3649536","1327104"]],"card_rows":[["1056","1152"],["34848","76032","41472"],["1022208","3345408","3649536","1327104"],["28110720","122664960","200724480","145981440","39813120"]],"row_sums_preview":["2208","152352","9344256","537294720","29658668544","1591681878528","83676990185472","4330284242098176"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"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}},{"id":"l-aa816fbde18fb600","kind":"fingerprint_related","node_a_id":"n-9e472291b944d56e","node_b_id":"n-faaa8ff869da19e7","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-b07c0dd493c080f5","kind":"fingerprint_related","node_a_id":"n-2a3c7d06bf210d3a","node_b_id":"n-faaa8ff869da19e7","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-b2ea9d1b54d5a91a","kind":"fingerprint_related","node_a_id":"n-344e4bfa9933088a","node_b_id":"n-faaa8ff869da19e7","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-f1f45f3bf3f9b65a","kind":"fingerprint_related","node_a_id":"n-d798b01f393943bf","node_b_id":"n-faaa8ff869da19e7","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}}],"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:           264,            242;\n  1:          4752,           8712,            3993;\n  2:         76032,         209088,          191664,           58564;\n  3:       1140480,        4181760,         5749920,         3513840,           805255;\n  4:      16422912,       75271680,       137998080,       126498240,         57978360,         10629366;\n  5:     229920768,     1264564224,      2897959680,      3541950720,       2435091120,        892866744,       136410197;\n  6:    3153199104,    20233027584,     55640825856,     85006817280,      77922915840,      42857603712,     13095378912,      1714871048;\n  7:   42568187904,   312166711296,   1001534865408,   1836147253248,    2103918727680,    1542873733632,    707150461248,    185206073184,     21221529219;\n  8:  567575838720,  4682500669440,  17169169121280,  36722945064960,   50494049464320,   46286212008960,  28286018449920,  11112364391040,   2546583506280,   259374246010;\n  9: 7492001071104, 68676676485120, 283291290501120, 692489821224960, 1110869088215040, 1221955997036544, 933438608847360, 488944033205760, 168074511414480, 34237400473320, 3138428376721;\n","recurrence_sentence":"The row polynomials satisfy (n+1)*P(n,x) = (12*n+24+(11*n+22)*x)*P(n-1,x) for n >= 1, with P(0,x) = 264 + 242*x.","mathematica":"P[0, x_] := 264 + 242*x;\nP[n_, x_] := P[n, x] = Expand[Together[((12*n + 24 + (11*n + 22)*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] = 264 + 242*x; for(n = 1, N, my(i = n + 1); v[i] = ((12*n + 24 + (11*n + 22)*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-344e4bfa9933088a","steps":["D"],"edges":["e-552f58accfe91872"],"text":"A038337 \u2192 D"},"headline":"A038337 \u2192 D","chain":null}
