{"id":"n-4d3a36b3f07ad6a5","revision":1,"first_index":0,"fingerprint":"f6-90334921cba1a1bb","name":"basis_map_A007318 of Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, 0<=k<=2n.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":2,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["1"],["8","8","2"],["64","128","96","32","4"],["512","1536","1920","1280","480","96","8"],["4096","16384","28672","28672","17920","7168","1792","256","16"],["32768","163840","368640","491520","430080","258048","107520","30720","5760","640","32"],["262144","1572864","4325376","7208960","8110080","6488064","3784704","1622016","506880","112640","16896","1536","64"],["2097152","14680064","47710208","95420416","131203072","131203072","98402304","56229888","24600576","8200192","2050048","372736","46592","3584","128"],["16777216","134217728","503316480","1174405120","1908408320","2290089984","2099249152","1499463680","843448320","374865920","131203072","35782656","7454720","1146880","122880","8192","256"],["134217728","1207959552","5133828096","13690208256","25669140480","35936796672","38931529728","33369882624","22941794304","12745441280","5735448576","2085617664","608305152","140378112","25067520","3342336","313344","18432","512"],["1073741824","10737418240","51002736640","153008209920","325142446080","520227913728","650284892160","650284892160","528356474880","352237649920","193730707456","88059412480","33022279680","10160701440","2540175360","508035072","79380480","9338880","778240","40960","1024"],["8589934592","94489280512","496068722688","1653562408960","3927210721280","7068979298304","10014387339264","11445014102016","10729700720640","8345322782720","5424459808768","2958796259328","1356114952192","521582673920","167651573760","44707086336","9779675136","1725825024","239697920","25231360","1892352","90112","2048"],["68719476736","824633720832","4741643894784","17386027614208","45638322487296","91276644974592","144521354543104","185813170126848","197426493259776","175490216230912","131617662173184","83756694110208","45368209309696","20939173527552","8226103885824","2742034628608","771197239296","181458173952","35283533824","5571084288","696385536","66322432","4521984","196608","4096"],["549755813888","7146825580544","44667659878400","178670639513600","513678088601600","1130091794923520","1977660641116160","2825229487308800","3354960016179200","3354960016179200","2851716013752320","2073975282728960","1296234551705600","697972450918400","324058637926400","129623455170560","44558062714880","13105312563200","3276328140800","689753292800","120706826240","17243832320","1959526400","170393600","10649600","425984","8192"],["4398046511104","61572651155456","415615395299328","1801000046297088","5628125144678400","13507500347228160","25889375665520640","40683304617246720","53396837310136320","59329819233484800","56363328271810560","46115450404208640","32665110702981120","20101606586449920","10768717814169600","5025401646612480","2041569418936320","720553912565760","220169251061760","57939276595200","13036337233920","2483111854080","395040522240","51527024640","5367398400","429391872","24772608","917504","16384"]],"degrees":[0,2,4,6,8,10,12,14,16,18,20,22,24,26,28],"row_sums":["1","18","324","5832","104976","1889568","34012224","612220032","11019960576","198359290368","3570467226624","64268410079232","1156831381426176","20822964865671168","374813367582081024"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A007318","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["8","8","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":161909,"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-debf4dfb42326c05"]},{"id":161910,"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-debf4dfb42326c05"]},{"id":161911,"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-debf4dfb42326c05"]}],"region_exact":{},"incoming_edges":[{"id":"e-debf4dfb42326c05","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-4d3a36b3f07ad6a5","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":"1c2c42d18cc3a69f71c59adab03989b85f6cf86cf8bd3bc072133472fdf0194f","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-336459bdce627a87","role":"sequence","name":"Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, 0<=k<=2n.","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-0ee277dabb911451","kind":"fingerprint_related","node_a_id":"n-4d3a36b3f07ad6a5","node_b_id":"n-0ca8dc857c80d449","fingerprint":"f6-90334921cba1a1bb","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-90334921cba1a1bb"},"other_id":"n-0ca8dc857c80d449","other":{"id":"n-0ca8dc857c80d449","name":"basis_map_A013611 of Even-numbered rows of Pascal's triangle.","first_index":0,"fingerprint":"f6-90334921cba1a1bb","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","16","16"],["16","128","384","512","256"]],"card_rows":[["1"],["4","16","16"],["16","128","384","512","256"],["64","768","3840","10240","15360","12288","4096"]],"row_sums_preview":["1","36","1296","46656","1679616","60466176","2176782336","78364164096"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-08f49539099949d5","steps":["basis A013611"],"edges":["e-a6b5412d61e1c4e3"],"text":"A034870 \u2192 basis A013611"},"headline":"A034870 \u2192 basis A013611","chain":null}},{"id":"l-21983bdfa02024bf","kind":"fingerprint_related","node_a_id":"n-4d3a36b3f07ad6a5","node_b_id":"n-5c5dbc66494212e8","fingerprint":"f6-90334921cba1a1bb","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-90334921cba1a1bb"},"other_id":"n-5c5dbc66494212e8","other":{"id":"n-5c5dbc66494212e8","name":"basis_map_A013609 of basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A008288) regarded as a triangle. shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-90334921cba1a1bb","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","4","4"],["1","8","24","32","16"]],"card_rows":[["1"],["1","4","4"],["1","8","24","32","16"],["1","12","60","160","240","192","64"]],"row_sums_preview":["1","9","81","729","6561","59049","531441","4782969"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cdb0db4ad473ee28","steps":["basis A013609"],"edges":["e-20c0b01c4fb345ed"],"text":"basis-map coefficients of A297191, shifted \u2192 basis A013609"},"headline":"basis-map coefficients of A297191, shifted \u2192 basis A013609","chain":null}},{"id":"l-584f4584edbc554c","kind":"fingerprint_related","node_a_id":"n-8c0973a753fcbe28","node_b_id":"n-4d3a36b3f07ad6a5","fingerprint":"f6-90334921cba1a1bb","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-90334921cba1a1bb"},"other_id":"n-8c0973a753fcbe28","other":{"id":"n-8c0973a753fcbe28","name":"basis_map_A007318 of Even-numbered rows of Pascal's triangle.","first_index":0,"fingerprint":"f6-90334921cba1a1bb","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","4","1"],["16","32","24","8","1"]],"card_rows":[["1"],["4","4","1"],["16","32","24","8","1"],["64","192","240","160","60","12","1"]],"row_sums_preview":["1","9","81","729","6561","59049","531441","4782969"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-08f49539099949d5","steps":["x \u21a6 x+1"],"edges":["e-cfe8f3b6680af0c3"],"text":"A034870 \u2192 x \u21a6 x+1"},"headline":"A034870 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-8c94ee7e73ed1087","kind":"fingerprint_related","node_a_id":"n-4d3a36b3f07ad6a5","node_b_id":"n-f8ceb867f43c1485","fingerprint":"f6-90334921cba1a1bb","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-90334921cba1a1bb"},"other_id":"n-f8ceb867f43c1485","other":{"id":"n-f8ceb867f43c1485","name":"basis_map_A013611 of Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, 0<=k<=2n.","first_index":0,"fingerprint":"f6-90334921cba1a1bb","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["8","32","32"],["64","512","1536","2048","1024"]],"card_rows":[["1"],["8","32","32"],["64","512","1536","2048","1024"],["512","6144","30720","81920","122880","98304","32768"]],"row_sums_preview":["1","72","5184","373248","26873856","1934917632","139314069504","10030613004288"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-336459bdce627a87","steps":["basis A013611"],"edges":["e-770b6439e58a100f"],"text":"A139548 \u2192 basis A013611"},"headline":"A139548 \u2192 basis A013611","chain":null}},{"id":"l-f1fa392310614834","kind":"fingerprint_related","node_a_id":"n-4d3a36b3f07ad6a5","node_b_id":"n-d66b602a2fc4bda7","fingerprint":"f6-90334921cba1a1bb","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-90334921cba1a1bb"},"other_id":"n-d66b602a2fc4bda7","other":{"id":"n-d66b602a2fc4bda7","name":"mobius_x_over_x_plus_one of Even-numbered rows of Pascal's triangle.","first_index":0,"fingerprint":"f6-90334921cba1a1bb","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","4","4"],["1","8","24","32","16"]],"card_rows":[["1"],["1","4","4"],["1","8","24","32","16"],["1","12","60","160","240","192","64"]],"row_sums_preview":["1","9","81","729","6561","59049","531441","4782969"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-08f49539099949d5","steps":["x \u21a6 x/(x+1)"],"edges":["e-78e2985691766434"],"text":"A034870 \u2192 x \u21a6 x/(x+1)"},"headline":"A034870 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-ec60b6cf743e84a1","kind":"oeis_match","node_a_id":"n-4d3a36b3f07ad6a5","node_b_id":null,"fingerprint":null,"oeis_id":"A001027","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A001027","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A001027","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-4d3a36b3f07ad6a5","oeis_id":"A001027","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":21,"oeis_terms_available":21,"term_count":200,"raw_size":25970}],"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:         8,          8,          2;\n  2:        64,        128,         96,          32,           4;\n  3:       512,       1536,       1920,        1280,         480,          96,           8;\n  4:      4096,      16384,      28672,       28672,       17920,        7168,        1792,         256,          16;\n  5:     32768,     163840,     368640,      491520,      430080,      258048,      107520,       30720,        5760,         640,         32;\n  6:    262144,    1572864,    4325376,     7208960,     8110080,     6488064,     3784704,     1622016,      506880,      112640,      16896,       1536,        64;\n  7:   2097152,   14680064,   47710208,    95420416,   131203072,   131203072,    98402304,    56229888,    24600576,     8200192,    2050048,     372736,     46592,      3584,      128;\n  8:  16777216,  134217728,  503316480,  1174405120,  1908408320,  2290089984,  2099249152,  1499463680,   843448320,   374865920,  131203072,   35782656,   7454720,   1146880,   122880,    8192,    256;\n  9: 134217728, 1207959552, 5133828096, 13690208256, 25669140480, 35936796672, 38931529728, 33369882624, 22941794304, 12745441280, 5735448576, 2085617664, 608305152, 140378112, 25067520, 3342336, 313344, 18432, 512;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (8+8*x+2*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(8 + 8*x + 2*x^2)*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] = (8 + 8*x + 2*x^2)*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-336459bdce627a87","steps":["basis A007318"],"edges":["e-debf4dfb42326c05"],"text":"A139548 \u2192 basis A007318"},"headline":"A139548 \u2192 basis A007318","chain":null}