{"id":"n-eae311e6cee1a668","revision":0,"first_index":0,"fingerprint":"f6-d36c4f1a2e9bd6b7","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).","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":[["3"],["21","12"],["147","168","48"],["1029","1764","1008","192"],["7203","16464","14112","5376","768"],["50421","144060","164640","94080","26880","3072"],["352947","1210104","1728720","1317120","564480","129024","12288"],["2470629","9882516","16941456","16134720","9219840","3161088","602112","49152"],["17294403","79060128","158120256","180708864","129077760","59006976","16859136","2752512","196608"],["121060821","622598508","1423082304","1897443072","1626379776","929359872","354041856","86704128","12386304","786432"],["847425747","4842432840","12451970160","18974430720","18974430720","13011038208","6195732480","2023096320","433520640","55050240","3145728"],["5931980229","37286732868","106533522480","182628895680","208718737920","166974990336","95414280192","38944604160","11127029760","2119434240","242221056","12582912"],["41523861603","284735050992","894881588832","1704536359680","2191546748160","2003699884032","1335799922688","654269349888","233667624960","59344158720","10173284352","1056964608","50331648"],["290667031221","2159240803356","7403111325792","15511280873088","22158972675840","22792086180864","17365398994944","9923085139968","4252750774272","1350079610880","308589625344","48091889664","4580179968","201326592"],["2034669218547","16277353748376","60458742493968","138191411414784","217157932223232","248180493969408","212726137688064","138923191959552","69461595979776","26461560373248","7560445820928","1571001729024","224428818432","19730006016","805306368"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["3","33","363","3993","43923","483153","5314683","58461513","643076643","7073843073","77812273803","855935011833","9415285130163","103568136431793","1139249500749723"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["3"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["7","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":183541,"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-2889745ac766dab5"]},{"id":183542,"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-2889745ac766dab5","e-2a30922a9f201122"]},{"id":183543,"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-2889745ac766dab5","e-2a30922a9f201122"]},{"id":183544,"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-2889745ac766dab5"]},{"id":183545,"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-2a30922a9f201122"]}],"region_exact":{},"incoming_edges":[{"id":"e-2a30922a9f201122","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-eae311e6cee1a668","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":"947233dc814a050cce66d35febc28412942667469a9863a18707165d9dd321d2","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-4a6fce4b90b6e934","role":"sequence","name":"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).","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-2889745ac766dab5","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-eae311e6cee1a668","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":"7c55b82371f94a76e4165237d8767e5928f9fccebc5f1464602ff48b0fb2e576","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-4a6fce4b90b6e934","role":"sequence","name":"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).","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-12bb7bf17d3fe419","kind":"fingerprint_related","node_a_id":"n-eae311e6cee1a668","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-1cb3e22b8fe9b569","kind":"fingerprint_related","node_a_id":"n-eae311e6cee1a668","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-2ebc2849b7c6dc30","kind":"fingerprint_related","node_a_id":"n-eae311e6cee1a668","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-39581562a57a1d68","kind":"fingerprint_related","node_a_id":"n-eae311e6cee1a668","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-6f01e795d9ae3207","kind":"fingerprint_related","node_a_id":"n-eae311e6cee1a668","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-859228d37baf9af8","kind":"fingerprint_related","node_a_id":"n-eae311e6cee1a668","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-89c8dd37dcc5c90a","kind":"fingerprint_related","node_a_id":"n-ade19c5257c8a726","node_b_id":"n-eae311e6cee1a668","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-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-fef6f335ad51810f","other":{"id":"n-fef6f335ad51810f","name":"mobius_x_over_x_plus_one of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*9^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"],["12","21"],["144","504","441"]],"card_rows":[["1"],["12","21"],["144","504","441"],["1728","9072","15876","9261"]],"row_sums_preview":["1","33","1089","35937","1185921","39135393","1291467969","42618442977"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"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}},{"id":"l-9ecee137219bc2d7","kind":"fingerprint_related","node_a_id":"n-eae311e6cee1a668","node_b_id":"n-d7780beb48787c44","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-aa724800f8e5cd5a","kind":"fingerprint_related","node_a_id":"n-8afe6c02ab69bf6d","node_b_id":"n-eae311e6cee1a668","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-b741bd63a1dc5d17","kind":"fingerprint_related","node_a_id":"n-ddf06e5e8d330762","node_b_id":"n-eae311e6cee1a668","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-e99f6e95d4187a79","kind":"fingerprint_related","node_a_id":"n-eae311e6cee1a668","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}},{"id":"l-f8020af644dd7532","kind":"fingerprint_related","node_a_id":"n-eae311e6cee1a668","node_b_id":"n-6e3b5b8a1581542e","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}}],"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:         3;\n  1:        21,        12;\n  2:       147,       168,         48;\n  3:      1029,      1764,       1008,        192;\n  4:      7203,     16464,      14112,       5376,        768;\n  5:     50421,    144060,     164640,      94080,      26880,      3072;\n  6:    352947,   1210104,    1728720,    1317120,     564480,    129024,     12288;\n  7:   2470629,   9882516,   16941456,   16134720,    9219840,   3161088,    602112,    49152;\n  8:  17294403,  79060128,  158120256,  180708864,  129077760,  59006976,  16859136,  2752512,   196608;\n  9: 121060821, 622598508, 1423082304, 1897443072, 1626379776, 929359872, 354041856, 86704128, 12386304, 786432;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (7+4*x)*P(n-1,x) for n >= 1, with P(0,x) = 3.","mathematica":"P[0, x_] := 3;\nP[n_, x_] := P[n, x] = Expand[(7 + 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] = 3; for(n = 1, N, my(i = n + 1); v[i] = (7 + 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-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}