{"id":"n-f038b923445dac3e","revision":0,"first_index":0,"fingerprint":"f6-8d10fbbacee008e8","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.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["1"],["14","4"],["196","112","16"],["2744","2352","672","64"],["38416","43904","18816","3584","256"],["537824","768320","439040","125440","17920","1024"],["7529536","12907776","9219840","3512320","752640","86016","4096"],["105413504","210827008","180708864","86051840","24586240","4214784","401408","16384"],["1475789056","3373232128","3373232128","1927561216","688414720","157351936","22478848","1835008","65536"],["20661046784","53128406016","60718178304","40478785536","17348050944","4956585984","944111616","115605504","8257536","262144"],["289254654976","826441871360","1062568120320","809575710720","404787855360","138784407552","33043906560","5394923520","578027520","36700160","1048576"],["4049565169664","12727204818944","18181721169920","15584332431360","8905332817920","3562133127168","1017752322048","207704555520","29672079360","2825912320","161480704","4194304"],["56693912375296","194379128143872","305452915654656","290907538718720","187011989176320","85491195052032","28497065017344","6978873065472","1246227333120","158251089920","13564379136","704643072","16777216"],["793714773254144","2948083443515392","5053857331740672","5294517204680704","3781798003343360","1944924687433728","740923690450944","211692482985984","45362674925568","7200424591360","822905667584","64122519552","3053453312","67108864"],["11112006825558016","44448027302232064","82546336418430976","94338670192492544","74123240865529856","42356137637445632","18152630416048128","5927389523607552","1481847380901888","282256643981312","40322377711616","4189337944064","299238424576","13153337344","268435456"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","18","324","5832","104976","1889568","34012224","612220032","11019960576","198359290368","3570467226624","64268410079232","1156831381426176","20822964865671168","374813367582081024"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["14","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":188078,"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-076cd38cb97ec8fe"]},{"id":188079,"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-0128d6c646b3f27d","e-076cd38cb97ec8fe"]},{"id":188080,"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-0128d6c646b3f27d","e-076cd38cb97ec8fe"]},{"id":188081,"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-076cd38cb97ec8fe"]},{"id":188082,"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-0128d6c646b3f27d"]}],"region_exact":{},"incoming_edges":[{"id":"e-0128d6c646b3f27d","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-f038b923445dac3e","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":"14557832145989885f9106b39abe5e212eaa703587743719d336dda057e9d548","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-8afa7b2308002172","role":"sequence","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*1^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-076cd38cb97ec8fe","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-f038b923445dac3e","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":"2216178f11ecac117cec2a122a0e06ced6f7ed7431b644d375f1a18fa253b457","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-8afa7b2308002172","role":"sequence","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*1^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-1ee159e25590e555","kind":"fingerprint_related","node_a_id":"n-2a3c47627a03f4f8","node_b_id":"n-f038b923445dac3e","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}},{"id":"l-20378e703bd6c54d","kind":"fingerprint_related","node_a_id":"n-f038b923445dac3e","node_b_id":"n-ff1da23ecc68394a","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-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-5a4e3575d52df992","other":{"id":"n-5a4e3575d52df992","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.","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":[["441","252","36"],["9261","7938","2268","216"],["194481","222264","95256","18144","1296"]],"card_rows":[["441","252","36"],["9261","7938","2268","216"],["194481","222264","95256","18144","1296"],["4084101","5834430","3333960","952560","136080","7776"]],"row_sums_preview":["729","19683","531441","14348907","387420489","10460353203","282429536481","7625597484987"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"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}},{"id":"l-5bd17183222fcf3b","kind":"fingerprint_related","node_a_id":"n-f038b923445dac3e","node_b_id":"n-4e63782ec2a203f8","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-8d16320d6d787bf8","kind":"fingerprint_related","node_a_id":"n-f038b923445dac3e","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-9b24b98e575011a1","kind":"fingerprint_related","node_a_id":"n-6890f01c78bc6b64","node_b_id":"n-f038b923445dac3e","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-b805bf98ab04a338","kind":"fingerprint_related","node_a_id":"n-f038b923445dac3e","node_b_id":"n-72a51ec1ec4c0613","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-e95ca5d678bb1699","kind":"fingerprint_related","node_a_id":"n-6a4f4798749d2440","node_b_id":"n-f038b923445dac3e","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-f37c999a55bcf2e3","kind":"fingerprint_related","node_a_id":"n-f038b923445dac3e","node_b_id":"n-8c340f191f09d9f8","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-fe726258a3de01bf","kind":"fingerprint_related","node_a_id":"n-3208539f9ba783e8","node_b_id":"n-f038b923445dac3e","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}}],"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:           1;\n  1:          14,           4;\n  2:         196,         112,          16;\n  3:        2744,        2352,         672,          64;\n  4:       38416,       43904,       18816,        3584,         256;\n  5:      537824,      768320,      439040,      125440,       17920,       1024;\n  6:     7529536,    12907776,     9219840,     3512320,      752640,      86016,      4096;\n  7:   105413504,   210827008,   180708864,    86051840,    24586240,    4214784,    401408,     16384;\n  8:  1475789056,  3373232128,  3373232128,  1927561216,   688414720,  157351936,  22478848,   1835008,   65536;\n  9: 20661046784, 53128406016, 60718178304, 40478785536, 17348050944, 4956585984, 944111616, 115605504, 8257536, 262144;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (14+4*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(14 + 4*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] = 1; for(n = 1, N, my(i = n + 1); v[i] = (14 + 4*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-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}