{"id":"n-fef6f335ad51810f","revision":0,"first_index":0,"fingerprint":"f6-d36c4f1a2e9bd6b7","name":"mobius_x_over_x_plus_one of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*9^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"],["12","21"],["144","504","441"],["1728","9072","15876","9261"],["20736","145152","381024","444528","194481"],["248832","2177280","7620480","13335840","11668860","4084101"],["2985984","31352832","137168640","320060160","420078960","294055272","85766121"],["35831808","438939648","2304433152","6721263360","11762210880","12350321424","7204354164","1801088541"],["429981696","6019743744","36870930432","129048256512","282293061120","395210285568","345808999872","172904499936","37822859361"],["5159780352","81266540544","568865783808","2322868617216","6097530120192","10670677710336","12449123995392","9336842996544","4084868810988","794280046581"],["61917364224","1083553873920","8532986757120","39820604866560","121950602403840","256096265048064","373473719861760","373473719861760","245092128659280","95313605589720","16679880978201"],["743008370688","14302911135744","125150472437760","657039980298240","2299639931043840","5634117831057408","9859706204350464","12324632755438080","10784053661008320","6290697968921520","2201744289122532","350277500542221"],["8916100448256","187238109413376","1802166803103744","10512639684771840","41393518758789120","115901852524609536","236632948904411136","354949423356616704","388225931796299520","301953502508232960","158525588816822304","50439960078079824","7355827511386641"],["106993205379072","2434095422373888","25558001934925824","163997179082440704","717487658485678080","2260086124229885952","5273534289869733888","9228685007272034304","12112649072044545024","11776186597821085440","8243330618474759808","3934316886090226272","1147509091776315996","154472377739119461"],["1283918464548864","31456002381447168","357812027088961536","2504684189622730752","12053792662559391744","42188274318957871104","110744220087264411648","221488440174528823296","339154174017247260672","395679869686788470784","346219885975939911936","220321745621052671232","96390763709210543664","25951359460172069448","3243919932521508681"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","33","1089","35937","1185921","39135393","1291467969","42618442977","1406408618241","46411484401953","1531578985264449","50542106513726817","1667889514952984961","55040353993448503713","1816331681783800622529"],"nonn":true,"definitions":[{"position":0,"source":"derived:mobius_x_over_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["1","1"]]},"terms":[{"coeff":{"coeffs":[["12","33","21"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":191259,"position":0,"claim":"real_rooted","from_index":0,"region_key":"[-1, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in [-1, 0]","region_exact":null,"edges":["e-688743e75672a01f"]},{"id":191260,"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-688743e75672a01f"]},{"id":191261,"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-688743e75672a01f"]}],"region_exact":{},"incoming_edges":[{"id":"e-688743e75672a01f","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"mobius\",\"a\":\"1\",\"b\":\"0\",\"c\":\"1\",\"e\":\"1\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":1}}"},"operation_detail":"mobius, shift 0","conclusion_id":"n-fef6f335ad51810f","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":"fTransform","lean_declarations":["RealRooted.splits_fPolynomial"],"certificate_hash":"09b31ad171e253457749a98f8904c1c0b4d7f6b53be6da478256951b51e78334","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-f485cd65a8f8089f","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*9^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-304b990d03d08bfb","kind":"fingerprint_related","node_a_id":"n-ddf06e5e8d330762","node_b_id":"n-fef6f335ad51810f","fingerprint":"f6-d36c4f1a2e9bd6b7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-d36c4f1a2e9bd6b7"},"other_id":"n-ddf06e5e8d330762","other":{"id":"n-ddf06e5e8d330762","name":"mobius_x_over_x_plus_one of basis_map_A007318 of Triangle of coefficients in expansion of (1+3*x)^n.","first_index":0,"fingerprint":"f6-d36c4f1a2e9bd6b7","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","7"],["16","56","49"]],"card_rows":[["1"],["4","7"],["16","56","49"],["64","336","588","343"]],"row_sums_preview":["1","11","121","1331","14641","161051","1771561","19487171"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-6a5af5e1168bb679","steps":["basis A007318","x \u21a6 x/(x+1)"],"edges":["e-5bade392cbb9f615","e-26d42dbf393ae5f6"],"text":"A013610 \u2192 basis A007318 \u2192 x \u21a6 x/(x+1)"},"headline":"A013610 \u2192 basis A007318 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-3767bb0ac2abb606","kind":"fingerprint_related","node_a_id":"n-ade19c5257c8a726","node_b_id":"n-fef6f335ad51810f","fingerprint":"f6-d36c4f1a2e9bd6b7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-d36c4f1a2e9bd6b7"},"other_id":"n-ade19c5257c8a726","other":{"id":"n-ade19c5257c8a726","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j).","first_index":0,"fingerprint":"f6-d36c4f1a2e9bd6b7","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["7","4"],["49","56","16"]],"card_rows":[["1"],["7","4"],["49","56","16"],["343","588","336","64"]],"row_sums_preview":["1","11","121","1331","14641","161051","1771561","19487171"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-22693abd431a71f2","steps":["basis A013611"],"edges":["e-1b27bff9bafb5517"],"text":"A038255 \u2192 basis A013611"},"headline":"A038255 \u2192 basis A013611","chain":null}},{"id":"l-63c82253bc4785cc","kind":"fingerprint_related","node_a_id":"n-fef6f335ad51810f","node_b_id":"n-5fcf1e679a2a1727","fingerprint":"f6-d36c4f1a2e9bd6b7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-d36c4f1a2e9bd6b7"},"other_id":"n-5fcf1e679a2a1727","other":{"id":"n-5fcf1e679a2a1727","name":"basis_map_A013609 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-d36c4f1a2e9bd6b7","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["441","504","144"],["9261","15876","9072","1728"],["194481","444528","381024","145152","20736"]],"card_rows":[["441","504","144"],["9261","15876","9072","1728"],["194481","444528","381024","145152","20736"],["4084101","11668860","13335840","7620480","2177280","248832"]],"row_sums_preview":["1089","35937","1185921","39135393","1291467969","42618442977","1406408618241","46411484401953"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-615df6ea54d3bbef","steps":["basis A013609","basis A013609"],"edges":["e-f3e6c91e6b1bdc2d","e-09fc2301440d61ef"],"text":"A038329 \u2192 basis A013609 \u2192 basis A013609"},"headline":"A038329 \u2192 basis A013609 \u2192 basis A013609","chain":null}},{"id":"l-6c7fc0534fb07fd5","kind":"fingerprint_related","node_a_id":"n-fef6f335ad51810f","node_b_id":"n-34e46ec237b67647","fingerprint":"f6-d36c4f1a2e9bd6b7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-d36c4f1a2e9bd6b7"},"other_id":"n-34e46ec237b67647","other":{"id":"n-34e46ec237b67647","name":"derivative of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.","first_index":0,"fingerprint":"f6-d36c4f1a2e9bd6b7","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["224","128"],["4704","5376","1536"],["87808","150528","86016","16384"]],"card_rows":[["224","128"],["4704","5376","1536"],["87808","150528","86016","16384"],["1536640","3512320","3010560","1146880","163840"]],"row_sums_preview":["352","11616","340736","9370240","247374336","6349274624","159638904832","3951062894592"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ea9b82d727693fb6","steps":["basis A013611","D"],"edges":["e-0eb3975e7d8d74fc","e-81551fffb5775f04"],"text":"A038328 \u2192 basis A013611 \u2192 D"},"headline":"A038328 \u2192 basis A013611 \u2192 D","chain":null}},{"id":"l-8ef151bf8ea3e7a4","kind":"fingerprint_related","node_a_id":"n-fef6f335ad51810f","node_b_id":"n-8b06684614b7f078","fingerprint":"f6-d36c4f1a2e9bd6b7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-d36c4f1a2e9bd6b7"},"other_id":"n-8b06684614b7f078","other":{"id":"n-8b06684614b7f078","name":"mobius_x_over_x_plus_one of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.","first_index":0,"fingerprint":"f6-d36c4f1a2e9bd6b7","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["256","896","784"],["4096","21504","37632","21952"],["65536","458752","1204224","1404928","614656"]],"card_rows":[["256","896","784"],["4096","21504","37632","21952"],["65536","458752","1204224","1404928","614656"],["1048576","9175040","32112640","56197120","49172480","17210368"]],"row_sums_preview":["1936","85184","3748096","164916224","7256313856","319277809664","14048223625216","618121839509504"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a93042cd70c4761e","steps":["basis A013610","x \u21a6 x/(x+1)"],"edges":["e-77aaa200c8a95097","e-f5610a9e130b3ad4"],"text":"A038330 \u2192 basis A013610 \u2192 x \u21a6 x/(x+1)"},"headline":"A038330 \u2192 basis A013610 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-8f662591c9bd60cc","kind":"fingerprint_related","node_a_id":"n-6e3b5b8a1581542e","node_b_id":"n-fef6f335ad51810f","fingerprint":"f6-d36c4f1a2e9bd6b7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-d36c4f1a2e9bd6b7"},"other_id":"n-6e3b5b8a1581542e","other":{"id":"n-6e3b5b8a1581542e","name":"reversal of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j).","first_index":0,"fingerprint":"f6-d36c4f1a2e9bd6b7","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","7"],["16","56","49"]],"card_rows":[["1"],["4","7"],["16","56","49"],["64","336","588","343"]],"row_sums_preview":["1","11","121","1331","14641","161051","1771561","19487171"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-22693abd431a71f2","steps":["basis A013611","reverse"],"edges":["e-1b27bff9bafb5517","e-b27b8d0469de5c2d"],"text":"A038255 \u2192 basis A013611 \u2192 reverse"},"headline":"A038255 \u2192 basis A013611 \u2192 reverse","chain":null}},{"id":"l-9107ff0ca8be6d69","kind":"fingerprint_related","node_a_id":"n-eae311e6cee1a668","node_b_id":"n-fef6f335ad51810f","fingerprint":"f6-d36c4f1a2e9bd6b7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-d36c4f1a2e9bd6b7"},"other_id":"n-eae311e6cee1a668","other":{"id":"n-eae311e6cee1a668","name":"shift_x_plus_one of basis_map_A013609 of Triangle read by rows: T(n,k) is the number of ternary words of length n having k runs (i.e., subwords of maximal length) of identical letters (1 <= k <= n).","first_index":0,"fingerprint":"f6-d36c4f1a2e9bd6b7","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["3"],["21","12"],["147","168","48"]],"card_rows":[["3"],["21","12"],["147","168","48"],["1029","1764","1008","192"]],"row_sums_preview":["3","33","363","3993","43923","483153","5314683","58461513"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-51b4f5e7901c2b50","steps":["basis A013611","x \u21a6 x+1"],"edges":["e-1a05d30902084644","e-2889745ac766dab5"],"text":"A120910 \u2192 basis A013611 \u2192 x \u21a6 x+1"},"headline":"A120910 \u2192 basis A013611 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-aa6e3262a17fdd9c","kind":"fingerprint_related","node_a_id":"n-fef6f335ad51810f","node_b_id":"n-021042e647efbf92","fingerprint":"f6-d36c4f1a2e9bd6b7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-d36c4f1a2e9bd6b7"},"other_id":"n-021042e647efbf92","other":{"id":"n-021042e647efbf92","name":"basis_map_A013609 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-d36c4f1a2e9bd6b7","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["784","896","256"],["21952","37632","21504","4096"],["614656","1404928","1204224","458752","65536"]],"card_rows":[["784","896","256"],["21952","37632","21504","4096"],["614656","1404928","1204224","458752","65536"],["17210368","49172480","56197120","32112640","9175040","1048576"]],"row_sums_preview":["1936","85184","3748096","164916224","7256313856","319277809664","14048223625216","618121839509504"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-6874f381e9c7e0a2","steps":["x \u21a6 x+1","basis A013609"],"edges":["e-4a3b4bf6a80699a3","e-00134e7c04415ba4"],"text":"A038334 \u2192 x \u21a6 x+1 \u2192 basis A013609"},"headline":"A038334 \u2192 x \u21a6 x+1 \u2192 basis A013609","chain":null}},{"id":"l-ac552315279d79cf","kind":"fingerprint_related","node_a_id":"n-d7780beb48787c44","node_b_id":"n-fef6f335ad51810f","fingerprint":"f6-d36c4f1a2e9bd6b7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-d36c4f1a2e9bd6b7"},"other_id":"n-d7780beb48787c44","other":{"id":"n-d7780beb48787c44","name":"derivative of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j).","first_index":0,"fingerprint":"f6-d36c4f1a2e9bd6b7","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["4"],["56","32"]],"card_rows":[["4"],["56","32"],["588","672","192"],["5488","9408","5376","1024"]],"row_sums_preview":["0","4","88","1452","21296","292820","3865224","49603708"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-22693abd431a71f2","steps":["basis A013611","D"],"edges":["e-1b27bff9bafb5517","e-07891fc1478baa0e"],"text":"A038255 \u2192 basis A013611 \u2192 D"},"headline":"A038255 \u2192 basis A013611 \u2192 D","chain":null}},{"id":"l-adca00763b7ef636","kind":"fingerprint_related","node_a_id":"n-8afe6c02ab69bf6d","node_b_id":"n-fef6f335ad51810f","fingerprint":"f6-d36c4f1a2e9bd6b7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-d36c4f1a2e9bd6b7"},"other_id":"n-8afe6c02ab69bf6d","other":{"id":"n-8afe6c02ab69bf6d","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.","first_index":0,"fingerprint":"f6-d36c4f1a2e9bd6b7","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["196","224","64"],["2744","4704","2688","512"],["38416","87808","75264","28672","4096"]],"card_rows":[["196","224","64"],["2744","4704","2688","512"],["38416","87808","75264","28672","4096"],["537824","1536640","1756160","1003520","286720","32768"]],"row_sums_preview":["484","10648","234256","5153632","113379904","2494357888","54875873536","1207269217792"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ea9b82d727693fb6","steps":["basis A013611"],"edges":["e-0eb3975e7d8d74fc"],"text":"A038328 \u2192 basis A013611"},"headline":"A038328 \u2192 basis A013611","chain":null}},{"id":"l-b0c8940df6dfba2b","kind":"fingerprint_related","node_a_id":"n-fef6f335ad51810f","node_b_id":"n-69d1a6e742732aa5","fingerprint":"f6-d36c4f1a2e9bd6b7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-d36c4f1a2e9bd6b7"},"other_id":"n-69d1a6e742732aa5","other":{"id":"n-69d1a6e742732aa5","name":"reversal of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.","first_index":0,"fingerprint":"f6-d36c4f1a2e9bd6b7","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["64","224","196"],["512","2688","4704","2744"],["4096","28672","75264","87808","38416"]],"card_rows":[["64","224","196"],["512","2688","4704","2744"],["4096","28672","75264","87808","38416"],["32768","286720","1003520","1756160","1536640","537824"]],"row_sums_preview":["484","10648","234256","5153632","113379904","2494357888","54875873536","1207269217792"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ea9b82d727693fb6","steps":["basis A013611","reverse"],"edges":["e-0eb3975e7d8d74fc","e-e27ff744fe86f928"],"text":"A038328 \u2192 basis A013611 \u2192 reverse"},"headline":"A038328 \u2192 basis A013611 \u2192 reverse","chain":null}},{"id":"l-b52aa803de72103d","kind":"fingerprint_related","node_a_id":"n-fef6f335ad51810f","node_b_id":"n-a70ed65b45a0249a","fingerprint":"f6-d36c4f1a2e9bd6b7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-d36c4f1a2e9bd6b7"},"other_id":"n-a70ed65b45a0249a","other":{"id":"n-a70ed65b45a0249a","name":"basis_map_A013609 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-d36c4f1a2e9bd6b7","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["14","8"],["196","224","64"]],"card_rows":[["1"],["14","8"],["196","224","64"],["2744","4704","2688","512"]],"row_sums_preview":["1","22","484","10648","234256","5153632","113379904","2494357888"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3dd184aba99388f7","steps":["basis A013611","basis A013609"],"edges":["e-8c5f302cdd82df95","e-69ebf8f06f7e77b6"],"text":"A038291 \u2192 basis A013611 \u2192 basis A013609"},"headline":"A038291 \u2192 basis A013611 \u2192 basis A013609","chain":null}},{"id":"l-dc7ac764c328a2dd","kind":"fingerprint_related","node_a_id":"n-fef6f335ad51810f","node_b_id":"n-d1304848dcec90e3","fingerprint":"f6-d36c4f1a2e9bd6b7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-d36c4f1a2e9bd6b7"},"other_id":"n-d1304848dcec90e3","other":{"id":"n-d1304848dcec90e3","name":"mobius_x_over_x_plus_one of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^j.","first_index":0,"fingerprint":"f6-d36c4f1a2e9bd6b7","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["64","224","196"],["512","2688","4704","2744"],["4096","28672","75264","87808","38416"]],"card_rows":[["64","224","196"],["512","2688","4704","2744"],["4096","28672","75264","87808","38416"],["32768","286720","1003520","1756160","1536640","537824"]],"row_sums_preview":["484","10648","234256","5153632","113379904","2494357888","54875873536","1207269217792"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c11cb6bca0e3e7dc","steps":["x \u21a6 x/(x+1)"],"edges":["e-aed80537f4191068"],"text":"A038284 \u2192 x \u21a6 x/(x+1)"},"headline":"A038284 \u2192 x \u21a6 x/(x+1)","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:         12,          21;\n  2:        144,         504,          441;\n  3:       1728,        9072,        15876,          9261;\n  4:      20736,      145152,       381024,        444528,        194481;\n  5:     248832,     2177280,      7620480,      13335840,      11668860,        4084101;\n  6:    2985984,    31352832,    137168640,     320060160,     420078960,      294055272,       85766121;\n  7:   35831808,   438939648,   2304433152,    6721263360,   11762210880,    12350321424,     7204354164,    1801088541;\n  8:  429981696,  6019743744,  36870930432,  129048256512,  282293061120,   395210285568,   345808999872,  172904499936,   37822859361;\n  9: 5159780352, 81266540544, 568865783808, 2322868617216, 6097530120192, 10670677710336, 12449123995392, 9336842996544, 4084868810988, 794280046581;\n","recurrence_sentence":"The row polynomials satisfy (1+x)*P(n,x) = (12+33*x+21*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[Together[((12 + 33*x + 21*x^2)*P[n - 1, x])/(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] = ((12 + 33*x + 21*x^2)*v[i - 1])/(1 + x)); 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-f485cd65a8f8089f","steps":["x \u21a6 x/(x+1)"],"edges":["e-688743e75672a01f"],"text":"A038335 \u2192 x \u21a6 x/(x+1)"},"headline":"A038335 \u2192 x \u21a6 x/(x+1)","chain":null}
