{"id":"n-32aabcb5e6797d69","revision":0,"first_index":0,"fingerprint":"f6-8e9037e0ccb64a62","name":"reversal of basis_map_A013609 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"],["18","21"],["324","756","441"],["5832","20412","23814","9261"],["104976","489888","857304","666792","194481"],["1889568","11022480","25719120","30005640","17503290","4084101"],["34012224","238085568","694416240","1080203040","945177660","441082908","85766121"],["612220032","4999796928","17499289248","34026395760","39697461720","27788223204","10806531246","1801088541"],["11019960576","102852965376","419982941952","979960197888","1429108621920","1333834713792","778070249712","259356749904","37822859361"],["198359290368","2082772548864","9719605228032","26458925342976","46303119350208","54020305908576","42015793484448","21007896742224","6127303216482","794280046581"],["3570467226624","41655450977280","218691117630720","680372365962240","1389093580506240","1944731012708736","1890710706800160","1260473804533440","551457289483380","142970408384580","16679880978201"],["64268410079232","824777929350144","4811204587875840","16839216057565440","39291504134319360","64176123419388288","74872143989286336","62393453324405280","36396181105903080","14154070430073420","3302616433683798","350277500542221"],["1156831381426176","16195639339966464","103922019098118144","404141185381570560","1060870611626622720","1980291808369695744","2695397183614308096","2695397183614308096","1965393779718766320","1019093070965286240","356682574837850184","75659940117119736","7355827511386641"],["20822964865671168","315814967129346048","2210704769905422336","9456903737928751104","27582635902292190720","57923535394813600512","90103277280821156352","105120490160958015744","91980428890838263776","59616944651469245040","27821240837352314352","8852212993703009112","1721263637664473994","154472377739119461"],["374813367582081024","6121951670507323392","46424800168013869056","216649067450731388928","695082424737763206144","1621858991054780814336","2838253234345866425088","3784337645794488566784","3863178013415207078592","3004694010434049950016","1752738172753195804176","743585891471052765408","216879218345723723244","38927039190258104172","3243919932521508681"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","39","1521","59319","2313441","90224199","3518743761","137231006679","5352009260481","208728361158759","8140406085191601","317475837322472439","12381557655576425121","482880748567480579719","18832349194131742609041"],"nonn":true,"definitions":[{"position":0,"source":"derived:reversal","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["18","21"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":193138,"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-f240e739fb22bf96"]},{"id":193139,"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-f240e739fb22bf96"]},{"id":193140,"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-f240e739fb22bf96"]}],"region_exact":{},"incoming_edges":[{"id":"e-f240e739fb22bf96","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"reversal\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":1}}"},"operation_detail":"reversal, shift 0","conclusion_id":"n-32aabcb5e6797d69","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":"reversed","lean_declarations":["RealRooted.X_pow_mul_reverse_eq_zero_or_splits","RealRooted.reflect_eq_zero_or_splits"],"certificate_hash":"4994b220562e3bfa1735e2baf52f2a0694a26f356d53a60b017bb4adc6a839d0","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-c73133912dd2cb81","role":"sequence","name":"basis_map_A013609 of 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-0869224a57729d09","kind":"fingerprint_related","node_a_id":"n-bc97c3236a99a08c","node_b_id":"n-32aabcb5e6797d69","fingerprint":"f6-8e9037e0ccb64a62","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8e9037e0ccb64a62"},"other_id":"n-bc97c3236a99a08c","other":{"id":"n-bc97c3236a99a08c","name":"basis_map_A007318 of Triangle of coefficients in expansion of (1+6x)^n.","first_index":0,"fingerprint":"f6-8e9037e0ccb64a62","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["7","6"],["49","84","36"]],"card_rows":[["1"],["7","6"],["49","84","36"],["343","882","756","216"]],"row_sums_preview":["1","13","169","2197","28561","371293","4826809","62748517"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-efce1b6c07b70e1a","steps":["x \u21a6 x+1"],"edges":["e-3efa57e183892da5"],"text":"A013613 \u2192 x \u21a6 x+1"},"headline":"A013613 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-092a95b0d5885f8c","kind":"fingerprint_related","node_a_id":"n-06c3ea6d255a8d34","node_b_id":"n-32aabcb5e6797d69","fingerprint":"f6-8e9037e0ccb64a62","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8e9037e0ccb64a62"},"other_id":"n-06c3ea6d255a8d34","other":{"id":"n-06c3ea6d255a8d34","name":"derivative of basis_map_A007318 of Triangle of coefficients in expansion of (1+6x)^n.","first_index":0,"fingerprint":"f6-8e9037e0ccb64a62","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["6"],["84","72"]],"card_rows":[["6"],["84","72"],["882","1512","648"],["8232","21168","18144","5184"]],"row_sums_preview":["0","6","156","3042","52728","856830","13366548","202725978"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-efce1b6c07b70e1a","steps":["x \u21a6 x+1","D"],"edges":["e-3efa57e183892da5","e-6c62a54648270cef"],"text":"A013613 \u2192 x \u21a6 x+1 \u2192 D"},"headline":"A013613 \u2192 x \u21a6 x+1 \u2192 D","chain":null}},{"id":"l-133594d477355bfc","kind":"fingerprint_related","node_a_id":"n-c73133912dd2cb81","node_b_id":"n-32aabcb5e6797d69","fingerprint":"f6-8e9037e0ccb64a62","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8e9037e0ccb64a62"},"other_id":"n-c73133912dd2cb81","other":{"id":"n-c73133912dd2cb81","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*9^j.","first_index":0,"fingerprint":"f6-8e9037e0ccb64a62","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["21","18"],["441","756","324"]],"card_rows":[["1"],["21","18"],["441","756","324"],["9261","23814","20412","5832"]],"row_sums_preview":["1","39","1521","59319","2313441","90224199","3518743761","137231006679"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f485cd65a8f8089f","steps":["basis A013609"],"edges":["e-f8a7b8e38f7708ee"],"text":"A038335 \u2192 basis A013609"},"headline":"A038335 \u2192 basis A013609","chain":null}},{"id":"l-1b32d38bc4e144fb","kind":"fingerprint_related","node_a_id":"n-32aabcb5e6797d69","node_b_id":"n-5beb2fe033521602","fingerprint":"f6-8e9037e0ccb64a62","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8e9037e0ccb64a62"},"other_id":"n-5beb2fe033521602","other":{"id":"n-5beb2fe033521602","name":"derivative of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^j.","first_index":0,"fingerprint":"f6-8e9037e0ccb64a62","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["336","288"],["7056","12096","5184"],["131712","338688","290304","82944"]],"card_rows":[["336","288"],["7056","12096","5184"],["131712","338688","290304","82944"],["2304960","7902720","10160640","5806080","1244160"]],"row_sums_preview":["624","24336","843648","27418560","855459072","25948925184","771053776896","22553322974208"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c11cb6bca0e3e7dc","steps":["basis A013609","D"],"edges":["e-65e466871b01ed5e","e-d7981b09feda9007"],"text":"A038284 \u2192 basis A013609 \u2192 D"},"headline":"A038284 \u2192 basis A013609 \u2192 D","chain":null}},{"id":"l-2ac5ccaae4c3c645","kind":"fingerprint_related","node_a_id":"n-32aabcb5e6797d69","node_b_id":"n-9fa7dedda2858777","fingerprint":"f6-8e9037e0ccb64a62","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8e9037e0ccb64a62"},"other_id":"n-9fa7dedda2858777","other":{"id":"n-9fa7dedda2858777","name":"basis_map_A013610 of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*1^j.","first_index":0,"fingerprint":"f6-8e9037e0ccb64a62","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["14","12"],["196","336","144"]],"card_rows":[["1"],["14","12"],["196","336","144"],["2744","7056","6048","1728"]],"row_sums_preview":["1","26","676","17576","456976","11881376","308915776","8031810176"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3dd184aba99388f7","steps":["basis A013611","basis A013610"],"edges":["e-8c5f302cdd82df95","e-14b3dc58248d0b82"],"text":"A038291 \u2192 basis A013611 \u2192 basis A013610"},"headline":"A038291 \u2192 basis A013611 \u2192 basis A013610","chain":null}},{"id":"l-313c2f5680585385","kind":"fingerprint_related","node_a_id":"n-32aabcb5e6797d69","node_b_id":"n-f1a4988ad8573671","fingerprint":"f6-8e9037e0ccb64a62","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8e9037e0ccb64a62"},"other_id":"n-f1a4988ad8573671","other":{"id":"n-f1a4988ad8573671","name":"derivative of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^j.","first_index":0,"fingerprint":"f6-8e9037e0ccb64a62","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["756","882"],["20412","47628","27783"],["489888","1714608","2000376","777924"]],"card_rows":[["756","882"],["20412","47628","27783"],["489888","1714608","2000376","777924"],["11022480","51438240","90016920","70013160","20420505"]],"row_sums_preview":["1638","95823","4982796","242911305","11368249074","517255332867","23054809122072","1011529750230909"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7dcf11eddf2b9a8a","steps":["basis A013610","D"],"edges":["e-632cc4d34c6e419b","e-4b041894d1c3ebf3"],"text":"A038321 \u2192 basis A013610 \u2192 D"},"headline":"A038321 \u2192 basis A013610 \u2192 D","chain":null}},{"id":"l-32298038bb8d0084","kind":"fingerprint_related","node_a_id":"n-332d43f68e0cd00a","node_b_id":"n-32aabcb5e6797d69","fingerprint":"f6-8e9037e0ccb64a62","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8e9037e0ccb64a62"},"other_id":"n-332d43f68e0cd00a","other":{"id":"n-332d43f68e0cd00a","name":"mobius_x_over_x_plus_one of Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j).","first_index":0,"fingerprint":"f6-8e9037e0ccb64a62","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["6","7"],["36","84","49"]],"card_rows":[["1"],["6","7"],["36","84","49"],["216","756","882","343"]],"row_sums_preview":["1","13","169","2197","28561","371293","4826809","62748517"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-22693abd431a71f2","steps":["x \u21a6 x/(x+1)"],"edges":["e-9ce5d31ba1886028"],"text":"A038255 \u2192 x \u21a6 x/(x+1)"},"headline":"A038255 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-3acad387574acd64","kind":"fingerprint_related","node_a_id":"n-32aabcb5e6797d69","node_b_id":"n-bb7cd1e2fa0622c2","fingerprint":"f6-8e9037e0ccb64a62","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8e9037e0ccb64a62"},"other_id":"n-bb7cd1e2fa0622c2","other":{"id":"n-bb7cd1e2fa0622c2","name":"reversal of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^j.","first_index":0,"fingerprint":"f6-8e9037e0ccb64a62","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["441","756","324"],["9261","23814","20412","5832"],["194481","666792","857304","489888","104976"]],"card_rows":[["441","756","324"],["9261","23814","20412","5832"],["194481","666792","857304","489888","104976"],["4084101","17503290","30005640","25719120","11022480","1889568"]],"row_sums_preview":["1521","59319","2313441","90224199","3518743761","137231006679","5352009260481","208728361158759"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-615df6ea54d3bbef","steps":["basis A013609","basis A013610"],"edges":["e-f3e6c91e6b1bdc2d","e-484970742987dfcd"],"text":"A038329 \u2192 basis A013609 \u2192 basis A013610"},"headline":"A038329 \u2192 basis A013609 \u2192 basis A013610","chain":null}},{"id":"l-5d1b1c10ad12114e","kind":"fingerprint_related","node_a_id":"n-86b3443645d7d7be","node_b_id":"n-32aabcb5e6797d69","fingerprint":"f6-8e9037e0ccb64a62","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8e9037e0ccb64a62"},"other_id":"n-86b3443645d7d7be","other":{"id":"n-86b3443645d7d7be","name":"derivative of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*9^j.","first_index":0,"fingerprint":"f6-8e9037e0ccb64a62","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["18"],["756","648"]],"card_rows":[["18"],["756","648"],["23814","40824","17496"],["666792","1714608","1469664","419904"]],"row_sums_preview":["0","18","1404","82134","4270968","208209690","9744213492","443361713886"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f485cd65a8f8089f","steps":["basis A013609","D"],"edges":["e-f8a7b8e38f7708ee","e-e33e4749ba9b0451"],"text":"A038335 \u2192 basis A013609 \u2192 D"},"headline":"A038335 \u2192 basis A013609 \u2192 D","chain":null}},{"id":"l-67f7efc2754b8bf1","kind":"fingerprint_related","node_a_id":"n-0649e5ac47163e39","node_b_id":"n-32aabcb5e6797d69","fingerprint":"f6-8e9037e0ccb64a62","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8e9037e0ccb64a62"},"other_id":"n-0649e5ac47163e39","other":{"id":"n-0649e5ac47163e39","name":"mobius_x_over_x_plus_one of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.","first_index":0,"fingerprint":"f6-8e9037e0ccb64a62","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["144","336","196"],["1728","6048","7056","2744"],["20736","96768","169344","131712","38416"]],"card_rows":[["144","336","196"],["1728","6048","7056","2744"],["20736","96768","169344","131712","38416"],["248832","1451520","3386880","3951360","2304960","537824"]],"row_sums_preview":["676","17576","456976","11881376","308915776","8031810176","208827064576","5429503678976"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ea9b82d727693fb6","steps":["x \u21a6 x/(x+1)"],"edges":["e-c7fee94f42a4e124"],"text":"A038328 \u2192 x \u21a6 x/(x+1)"},"headline":"A038328 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-bce752c350d19c25","kind":"fingerprint_related","node_a_id":"n-fa9216f6263c78e4","node_b_id":"n-32aabcb5e6797d69","fingerprint":"f6-8e9037e0ccb64a62","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8e9037e0ccb64a62"},"other_id":"n-fa9216f6263c78e4","other":{"id":"n-fa9216f6263c78e4","name":"reversal of basis_map_A007318 of Triangle of coefficients in expansion of (1+6x)^n.","first_index":0,"fingerprint":"f6-8e9037e0ccb64a62","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["6","7"],["36","84","49"]],"card_rows":[["1"],["6","7"],["36","84","49"],["216","756","882","343"]],"row_sums_preview":["1","13","169","2197","28561","371293","4826809","62748517"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-efce1b6c07b70e1a","steps":["x \u21a6 x+1","reverse"],"edges":["e-3efa57e183892da5","e-45e8e5a256280bb4"],"text":"A013613 \u2192 x \u21a6 x+1 \u2192 reverse"},"headline":"A013613 \u2192 x \u21a6 x+1 \u2192 reverse","chain":null}},{"id":"l-cbdb1396ab98e06f","kind":"fingerprint_related","node_a_id":"n-a36d781ea0b4816b","node_b_id":"n-32aabcb5e6797d69","fingerprint":"f6-8e9037e0ccb64a62","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8e9037e0ccb64a62"},"other_id":"n-a36d781ea0b4816b","other":{"id":"n-a36d781ea0b4816b","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^j.","first_index":0,"fingerprint":"f6-8e9037e0ccb64a62","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["196","336","144"],["2744","7056","6048","1728"],["38416","131712","169344","96768","20736"]],"card_rows":[["196","336","144"],["2744","7056","6048","1728"],["38416","131712","169344","96768","20736"],["537824","2304960","3951360","3386880","1451520","248832"]],"row_sums_preview":["676","17576","456976","11881376","308915776","8031810176","208827064576","5429503678976"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c11cb6bca0e3e7dc","steps":["basis A013609"],"edges":["e-65e466871b01ed5e"],"text":"A038284 \u2192 basis A013609"},"headline":"A038284 \u2192 basis A013609","chain":null}},{"id":"l-e047840ef1b6c3ec","kind":"fingerprint_related","node_a_id":"n-63d702ebcae1deea","node_b_id":"n-32aabcb5e6797d69","fingerprint":"f6-8e9037e0ccb64a62","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8e9037e0ccb64a62"},"other_id":"n-63d702ebcae1deea","other":{"id":"n-63d702ebcae1deea","name":"basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^j.","first_index":0,"fingerprint":"f6-8e9037e0ccb64a62","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["324","756","441"],["5832","20412","23814","9261"],["104976","489888","857304","666792","194481"]],"card_rows":[["324","756","441"],["5832","20412","23814","9261"],["104976","489888","857304","666792","194481"],["1889568","11022480","25719120","30005640","17503290","4084101"]],"row_sums_preview":["1521","59319","2313441","90224199","3518743761","137231006679","5352009260481","208728361158759"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7dcf11eddf2b9a8a","steps":["basis A013610"],"edges":["e-632cc4d34c6e419b"],"text":"A038321 \u2192 basis A013610"},"headline":"A038321 \u2192 basis A013610","chain":null}},{"id":"l-ead3b85adb95b457","kind":"fingerprint_related","node_a_id":"n-32aabcb5e6797d69","node_b_id":"n-9ca6bd6fc8b00a8f","fingerprint":"f6-8e9037e0ccb64a62","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8e9037e0ccb64a62"},"other_id":"n-9ca6bd6fc8b00a8f","other":{"id":"n-9ca6bd6fc8b00a8f","name":"basis_map_A013609 of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.","first_index":0,"fingerprint":"f6-8e9037e0ccb64a62","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["784","1344","576"],["21952","56448","48384","13824"],["614656","2107392","2709504","1548288","331776"]],"card_rows":[["784","1344","576"],["21952","56448","48384","13824"],["614656","2107392","2709504","1548288","331776"],["17210368","73758720","126443520","108380160","46448640","7962624"]],"row_sums_preview":["2704","140608","7311616","380204032","19770609664","1028071702528","53459728531456","2779905883635712"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a93042cd70c4761e","steps":["basis A013610","basis A013609"],"edges":["e-77aaa200c8a95097","e-83c3257804ef6073"],"text":"A038330 \u2192 basis A013610 \u2192 basis A013609"},"headline":"A038330 \u2192 basis A013610 \u2192 basis A013609","chain":null}},{"id":"l-fe8d17b78d0d1fc0","kind":"fingerprint_related","node_a_id":"n-32aabcb5e6797d69","node_b_id":"n-fc34b13bea5f9d2b","fingerprint":"f6-8e9037e0ccb64a62","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8e9037e0ccb64a62"},"other_id":"n-fc34b13bea5f9d2b","other":{"id":"n-fc34b13bea5f9d2b","name":"reversal of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^j.","first_index":0,"fingerprint":"f6-8e9037e0ccb64a62","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["144","336","196"],["1728","6048","7056","2744"],["20736","96768","169344","131712","38416"]],"card_rows":[["144","336","196"],["1728","6048","7056","2744"],["20736","96768","169344","131712","38416"],["248832","1451520","3386880","3951360","2304960","537824"]],"row_sums_preview":["676","17576","456976","11881376","308915776","8031810176","208827064576","5429503678976"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c11cb6bca0e3e7dc","steps":["basis A013609","reverse"],"edges":["e-65e466871b01ed5e","e-bdeb4934af2a58a9"],"text":"A038284 \u2192 basis A013609 \u2192 reverse"},"headline":"A038284 \u2192 basis A013609 \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:            1;\n  1:           18,            21;\n  2:          324,           756,           441;\n  3:         5832,         20412,         23814,           9261;\n  4:       104976,        489888,        857304,         666792,         194481;\n  5:      1889568,      11022480,      25719120,       30005640,       17503290,        4084101;\n  6:     34012224,     238085568,     694416240,     1080203040,      945177660,      441082908,       85766121;\n  7:    612220032,    4999796928,   17499289248,    34026395760,    39697461720,    27788223204,    10806531246,     1801088541;\n  8:  11019960576,  102852965376,  419982941952,   979960197888,  1429108621920,  1333834713792,   778070249712,   259356749904,   37822859361;\n  9: 198359290368, 2082772548864, 9719605228032, 26458925342976, 46303119350208, 54020305908576, 42015793484448, 21007896742224, 6127303216482, 794280046581;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (18+21*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[(18 + 21*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] = (18 + 21*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-f485cd65a8f8089f","steps":["basis A013609","reverse"],"edges":["e-f8a7b8e38f7708ee","e-f240e739fb22bf96"],"text":"A038335 \u2192 basis A013609 \u2192 reverse"},"headline":"A038335 \u2192 basis A013609 \u2192 reverse","chain":null}