{"id":"n-bd16f85c006e2508","revision":0,"first_index":0,"fingerprint":"f6-dca555f310728b4d","name":"basis_map_A013610 of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["3600","12960","11664"],["216000","1166400","2099520","1259712"],["12960000","93312000","251942400","302330880","136048896"],["777600000","6998400000","25194240000","45349632000","40814668800","14693280768"],["46656000000","503884800000","2267481600000","5441955840000","7346640384000","5289581076480","1586874322944"],["2799360000000","35271936000000","190468454400000","571405363200000","1028529653760000","1110812026060800","666487215636480","171382426877952"],["167961600000000","2418647040000000","15237476352000000","54854914867200000","123423558451200000","177729924169728000","159956931752755200","82263564901416960","18509302102818816"],["10077696000000000","163258675200000000","1175462461440000000","4936942338048000000","13329744312729600000","23993539762913280000","28792247715495936000","22211162523382579200","9995023135522160640","1999004627104432128"],["604661760000000000","10883911680000000000","88159684608000000000","423166486118400000000","1332974431272960000000","2879224771549593600000","4318837157324390400000","4442232504676515840000","2998506940656648192000","1199402776262659276800","215892499727278669824"],["36279705600000000000","718338170880000000000","6465043537920000000000","34911235104768000000000","125680446377164800000000","316714724870455296000000","570086504766819532800000","732968363271625113600000","659671526944462602240000","395802916166677561344000","142489049820003922083840","23316389970546096340992"],["2176782336000000000000","47018498457600000000000","465483134730240000000000","2792898808381440000000000","11311240173944832000000000","32576371700961116160000000","68410380572018343936000000","105547444311114016358400000","118740874850003268403200000","94992699880002614722560000","51296057935201411950182400","16787800778793189365514240","2518170116818978404827136"],["130606940160000000000000","3056202399744000000000000","33006985917235200000000000","217846107053752320000000000","980307481741885440000000000","3176196240843708825600000000","7622870978024901181440000000","13721167760444822126592000000","18523576476600509870899200000","18523576476600509870899200000","13336975063152367107047424000","6547242303729343852550553600","1964172691118803155765166080","271962372616449667721330688"],["7836416409600000000000000","197477693521920000000000000","2310489014206464000000000000","16635520902286540800000000000","82345828466318376960000000000","296444982478746157056000000000","800401452692614624051200000000","1646540131253378655191040000000","2593300706724071381925888000000","3111960848068885658311065600000","2800764763261997092479959040000","1833227845044216278714155008000","824952530269897325421369753600","228448392997817720885917777920","29371936242576564113903714304"],["470184984576000000000000000","12694994583552000000000000000","159956931752755200000000000000","1247664067671490560000000000000","6737385965426049024000000000000","26680048423087154135040000000000","80040145269261462405120000000000","185235764766005098708992000000000","333424376578809177676185600000000","466794127210332848746659840000000","504137657387159476646392627200000","412476265134948662710684876800000","247485759080969197626410926080000","102801776849017974398663000064000","26434742618318907702513342873600","3172169114198268924301601144832"],["28211099074560000000000000000","812479653347328000000000000000","10968475320188928000000000000000","92135192689586995200000000000000","538990877234083921920000000000000","2328440589651242542694400000000000","7683853945849100390891520000000000","19758481575040543862292480000000000","40010925189457101321142272000000000","64017480303131362113827635200000000","80662025181945516263422820352000000","79195442905910143240451496345600000","59396582179432607430338622259200000","32896568591685751807572160020480000","12688676456793075697206404579328000","3045282349630338167329537099038720","342594264333413043824572923641856"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["28224","4741632","796594176","133827821568","22483074023424","3777156435935232","634562281237118976","106606463247835987968","17909885825636445978624","3008860818706922924408832","505488617542763051300683776","84922087747184192618514874368","14266910741526944359910498893824","2396841004576526652464963814162432","402669288768856477614113920779288576"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013610","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["3600","12960","11664"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["60","108"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":204070,"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-b3602fe9ead223e6"]},{"id":204071,"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-b3602fe9ead223e6"]},{"id":204072,"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-b3602fe9ead223e6"]}],"region_exact":{},"incoming_edges":[{"id":"e-b3602fe9ead223e6","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-bd16f85c006e2508","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":"b2fed802c264336d14ad838d04a06913ec061867888b5cf75545e52d71bd787b","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-67e5bcee85c7fcd9","role":"sequence","name":"basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^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-0903dce112a30cc8","kind":"fingerprint_related","node_a_id":"n-8fb4a561853504a4","node_b_id":"n-bd16f85c006e2508","fingerprint":"f6-dca555f310728b4d","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-dca555f310728b4d"},"other_id":"n-8fb4a561853504a4","other":{"id":"n-8fb4a561853504a4","name":"basis_map_A013610 of basis_map_A013610 of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).","first_index":0,"fingerprint":"f6-dca555f310728b4d","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["2"],["10","18"],["50","180","162"]],"card_rows":[["2"],["10","18"],["50","180","162"],["250","1350","2430","1458"]],"row_sums_preview":["2","28","392","5488","76832","1075648","15059072","210827008"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-089a9b4f824a1c20","steps":["basis A013610","basis A013610"],"edges":["e-28b777b3c0b25332","e-6c693a9651656922"],"text":"A028326 \u2192 basis A013610 \u2192 basis A013610"},"headline":"A028326 \u2192 basis A013610 \u2192 basis A013610","chain":null}},{"id":"l-091fdafdc56172bd","kind":"fingerprint_related","node_a_id":"n-4547a8b17f528145","node_b_id":"n-bd16f85c006e2508","fingerprint":"f6-dca555f310728b4d","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-dca555f310728b4d"},"other_id":"n-4547a8b17f528145","other":{"id":"n-4547a8b17f528145","name":"basis_map_A013610 of basis_map_A013610 of Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.","first_index":0,"fingerprint":"f6-dca555f310728b4d","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["10","18"],["150","540","486"]],"card_rows":[["1"],["10","18"],["150","540","486"],["3000","16200","29160","17496"]],"row_sums_preview":["1","28","1176","65856","4609920","387233280","37948861440","4250272481280"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-75a6475b3e851dc5","steps":["basis A013610","basis A013610"],"edges":["e-1eb90284df2d6b2d","e-689234fd11c25b0c"],"text":"A156992 \u2192 basis A013610 \u2192 basis A013610"},"headline":"A156992 \u2192 basis A013610 \u2192 basis A013610","chain":null}},{"id":"l-0eb90e3ced75d999","kind":"fingerprint_related","node_a_id":"n-411e78f82a532dd1","node_b_id":"n-bd16f85c006e2508","fingerprint":"f6-dca555f310728b4d","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-dca555f310728b4d"},"other_id":"n-411e78f82a532dd1","other":{"id":"n-411e78f82a532dd1","name":"basis_map_A013610 of basis_map_A013610 of Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-dca555f310728b4d","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["25","45"],["375","1350","1215"]],"card_rows":[["1"],["25","45"],["375","1350","1215"],["4375","23625","42525","25515"]],"row_sums_preview":["1","70","2940","96040","2689120","67765824","1581202560","34786456320"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-79986ab5023c2764","steps":["basis A013610","basis A013610"],"edges":["e-0c010b1c5e20b0cb","e-b9cadc9baa1b079d"],"text":"A178821 \u2192 basis A013610 \u2192 basis A013610"},"headline":"A178821 \u2192 basis A013610 \u2192 basis A013610","chain":null}},{"id":"l-2725a30166ddd1f1","kind":"fingerprint_related","node_a_id":"n-6d9baf9da524d2a0","node_b_id":"n-bd16f85c006e2508","fingerprint":"f6-dca555f310728b4d","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-dca555f310728b4d"},"other_id":"n-6d9baf9da524d2a0","other":{"id":"n-6d9baf9da524d2a0","name":"shift_x_plus_one of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*5^j.","first_index":0,"fingerprint":"f6-dca555f310728b4d","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["27","15"],["729","810","225"]],"card_rows":[["1"],["27","15"],["729","810","225"],["19683","32805","18225","3375"]],"row_sums_preview":["1","42","1764","74088","3111696","130691232","5489031744","230539333248"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a6a633b989da1879","steps":["basis A013610","basis A007318"],"edges":["e-cc1debec4a411a9c","e-8d37d71f22412f85"],"text":"A038271 \u2192 basis A013610 \u2192 basis A007318"},"headline":"A038271 \u2192 basis A013610 \u2192 basis A007318","chain":null}},{"id":"l-438e8b7dd3a84d50","kind":"fingerprint_related","node_a_id":"n-3d8880ae4f67526d","node_b_id":"n-bd16f85c006e2508","fingerprint":"f6-dca555f310728b4d","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-dca555f310728b4d"},"other_id":"n-3d8880ae4f67526d","other":{"id":"n-3d8880ae4f67526d","name":"shift_x_plus_one of basis_map_A013609 of Triangle of coefficients in expansion of (3+5x)^n.","first_index":0,"fingerprint":"f6-dca555f310728b4d","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["18","10"],["324","360","100"]],"card_rows":[["1"],["18","10"],["324","360","100"],["5832","9720","5400","1000"]],"row_sums_preview":["1","28","784","21952","614656","17210368","481890304","13492928512"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c44f7114310405ad","steps":["basis A013609","basis A007318"],"edges":["e-c5caeaebd0a50686","e-148d1827675110ad"],"text":"A013622 \u2192 basis A013609 \u2192 basis A007318"},"headline":"A013622 \u2192 basis A013609 \u2192 basis A007318","chain":null}},{"id":"l-5b3a0e6c5abeeb78","kind":"fingerprint_related","node_a_id":"n-2090bc852fe1ed0c","node_b_id":"n-bd16f85c006e2508","fingerprint":"f6-dca555f310728b4d","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-dca555f310728b4d"},"other_id":"n-2090bc852fe1ed0c","other":{"id":"n-2090bc852fe1ed0c","name":"shift_x_plus_one of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*5^j.","first_index":0,"fingerprint":"f6-dca555f310728b4d","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1296","1440","400"],["46656","77760","43200","8000"],["1679616","3732480","3110400","1152000","160000"]],"card_rows":[["1296","1440","400"],["46656","77760","43200","8000"],["1679616","3732480","3110400","1152000","160000"],["60466176","167961600","186624000","103680000","28800000","3200000"]],"row_sums_preview":["3136","175616","9834496","550731776","30840979456","1727094849536","96717311574016","5416169448144896"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2bc1923905556bf6","steps":["basis A013611","x \u21a6 x+1"],"edges":["e-fd91849598d32af1","e-cef54db26bf72db9"],"text":"A038319 \u2192 basis A013611 \u2192 x \u21a6 x+1"},"headline":"A038319 \u2192 basis A013611 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-609908807386bf5d","kind":"fingerprint_related","node_a_id":"n-38edd6a4bb3bb62b","node_b_id":"n-bd16f85c006e2508","fingerprint":"f6-dca555f310728b4d","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-dca555f310728b4d"},"other_id":"n-38edd6a4bb3bb62b","other":{"id":"n-38edd6a4bb3bb62b","name":"reversal of basis_map_A013610 of Triangle of coefficients in expansion of (2+3x)^n.","first_index":0,"fingerprint":"f6-dca555f310728b4d","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["9","5"],["81","90","25"]],"card_rows":[["1"],["9","5"],["81","90","25"],["729","1215","675","125"]],"row_sums_preview":["1","14","196","2744","38416","537824","7529536","105413504"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-5733d79f45a14084","steps":["basis A013610","reverse"],"edges":["e-4eef4a156684ec8f","e-215be35d185d0513"],"text":"A013620 \u2192 basis A013610 \u2192 reverse"},"headline":"A013620 \u2192 basis A013610 \u2192 reverse","chain":null}},{"id":"l-741dd4f9e3ff058c","kind":"fingerprint_related","node_a_id":"n-1273b38f23038cfd","node_b_id":"n-bd16f85c006e2508","fingerprint":"f6-dca555f310728b4d","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-dca555f310728b4d"},"other_id":"n-1273b38f23038cfd","other":{"id":"n-1273b38f23038cfd","name":"basis_map_A013610 of basis_map_A013609 of Triangle of coefficients in expansion of (1+3*x)^n.","first_index":0,"fingerprint":"f6-dca555f310728b4d","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["10","18"],["100","360","324"]],"card_rows":[["1"],["10","18"],["100","360","324"],["1000","5400","9720","5832"]],"row_sums_preview":["1","28","784","21952","614656","17210368","481890304","13492928512"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d28ad9841fc73317","steps":["basis A013610","basis A013610"],"edges":["e-1a1a6d136d357a71","e-0529bc80e9e68c55"],"text":"A038208 \u2192 basis A013610 \u2192 basis A013610"},"headline":"A038208 \u2192 basis A013610 \u2192 basis A013610","chain":null}},{"id":"l-865ee8a35ca0e560","kind":"fingerprint_related","node_a_id":"n-51c3e1916ec6b28c","node_b_id":"n-bd16f85c006e2508","fingerprint":"f6-dca555f310728b4d","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-dca555f310728b4d"},"other_id":"n-51c3e1916ec6b28c","other":{"id":"n-51c3e1916ec6b28c","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*9^j.","first_index":0,"fingerprint":"f6-dca555f310728b4d","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["20","36"],["400","1440","1296"]],"card_rows":[["1"],["20","36"],["400","1440","1296"],["8000","43200","77760","46656"]],"row_sums_preview":["1","56","3136","175616","9834496","550731776","30840979456","1727094849536"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4e096b5929fd3f2e","steps":["basis A013611"],"edges":["e-2829c9269d688cfe"],"text":"A038323 \u2192 basis A013611"},"headline":"A038323 \u2192 basis A013611","chain":null}},{"id":"l-995549b623231535","kind":"fingerprint_related","node_a_id":"n-352735e5f7822499","node_b_id":"n-bd16f85c006e2508","fingerprint":"f6-dca555f310728b4d","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-dca555f310728b4d"},"other_id":"n-352735e5f7822499","other":{"id":"n-352735e5f7822499","name":"mobius_x_over_x_plus_one of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^j.","first_index":0,"fingerprint":"f6-dca555f310728b4d","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["400","1440","1296"],["8000","43200","77760","46656"],["160000","1152000","3110400","3732480","1679616"]],"card_rows":[["400","1440","1296"],["8000","43200","77760","46656"],["160000","1152000","3110400","3732480","1679616"],["3200000","28800000","103680000","186624000","167961600","60466176"]],"row_sums_preview":["3136","175616","9834496","550731776","30840979456","1727094849536","96717311574016","5416169448144896"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-6874f381e9c7e0a2","steps":["basis A013609","x \u21a6 x/(x+1)"],"edges":["e-4eca5e8e987e069c","e-bff01706d994a3a7"],"text":"A038334 \u2192 basis A013609 \u2192 x \u21a6 x/(x+1)"},"headline":"A038334 \u2192 basis A013609 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-bac397c966bdc65c","kind":"fingerprint_related","node_a_id":"n-950319b6c870b4e9","node_b_id":"n-bd16f85c006e2508","fingerprint":"f6-dca555f310728b4d","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-dca555f310728b4d"},"other_id":"n-950319b6c870b4e9","other":{"id":"n-950319b6c870b4e9","name":"mobius_x_over_x_plus_one of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.","first_index":0,"fingerprint":"f6-dca555f310728b4d","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["225","810","729"],["3375","18225","32805","19683"],["50625","364500","984150","1180980","531441"]],"card_rows":[["225","810","729"],["3375","18225","32805","19683"],["50625","364500","984150","1180980","531441"],["759375","6834375","24603750","44286750","39858075","14348907"]],"row_sums_preview":["1764","74088","3111696","130691232","5489031744","230539333248","9682651996416","406671383849472"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-615df6ea54d3bbef","steps":["basis A013611","x \u21a6 x/(x+1)"],"edges":["e-a5d7492b5780a0c3","e-0f4ceb54ded14de9"],"text":"A038329 \u2192 basis A013611 \u2192 x \u21a6 x/(x+1)"},"headline":"A038329 \u2192 basis A013611 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-bed76ad9e52bd605","kind":"fingerprint_related","node_a_id":"n-41ebfdeeaf308611","node_b_id":"n-bd16f85c006e2508","fingerprint":"f6-dca555f310728b4d","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-dca555f310728b4d"},"other_id":"n-41ebfdeeaf308611","other":{"id":"n-41ebfdeeaf308611","name":"basis_map_A013610 of basis_map_A013610 of Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).","first_index":0,"fingerprint":"f6-dca555f310728b4d","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["5","9"],["50","180","162"]],"card_rows":[["1"],["5","9"],["50","180","162"],["750","4050","7290","4374"]],"row_sums_preview":["1","14","392","16464","921984","64538880","5421265920","531284060160"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cb5d49e71e46f2ed","steps":["basis A013610","basis A013610"],"edges":["e-542183e09dec6678","e-d02e0ef9bae6d51b"],"text":"A196347 \u2192 basis A013610 \u2192 basis A013610"},"headline":"A196347 \u2192 basis A013610 \u2192 basis A013610","chain":null}},{"id":"l-c646930cc5dee652","kind":"fingerprint_related","node_a_id":"n-d8e81e9a78e7e2d2","node_b_id":"n-bd16f85c006e2508","fingerprint":"f6-dca555f310728b4d","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-dca555f310728b4d"},"other_id":"n-d8e81e9a78e7e2d2","other":{"id":"n-d8e81e9a78e7e2d2","name":"basis_map_A013610 of basis_map_A013610 of Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A000108).","first_index":1,"fingerprint":"f6-dca555f310728b4d","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["5","9"],["50","180","162"]],"card_rows":[["1"],["5","9"],["50","180","162"],["625","3375","6075","3645"]],"row_sums_preview":["1","14","392","13720","537824","22588608","993898752","45222393216"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a98fae1bf9eeb913","steps":["basis A013610","basis A013610"],"edges":["e-13e67b7b83b67a66","e-aa07e95cf62ebea1"],"text":"A085880 \u2192 basis A013610 \u2192 basis A013610"},"headline":"A085880 \u2192 basis A013610 \u2192 basis A013610","chain":null}},{"id":"l-dab993ab481a602e","kind":"fingerprint_related","node_a_id":"n-ee54987254c3cd08","node_b_id":"n-bd16f85c006e2508","fingerprint":"f6-dca555f310728b4d","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-dca555f310728b4d"},"other_id":"n-ee54987254c3cd08","other":{"id":"n-ee54987254c3cd08","name":"basis_map_A013610 of basis_map_A013610 of Triangle read by rows: T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-dca555f310728b4d","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["30","54"],["525","1890","1701"]],"card_rows":[["1"],["30","54"],["525","1890","1701"],["7000","37800","68040","40824"]],"row_sums_preview":["1","84","4116","153664","4840416","135531648","3478645632","83487495168"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-97228a88c81561dc","steps":["basis A013610","basis A013610"],"edges":["e-3fa86ece9d0bedd1","e-560ac885b0acd0ec"],"text":"A178822 \u2192 basis A013610 \u2192 basis A013610"},"headline":"A178822 \u2192 basis A013610 \u2192 basis A013610","chain":null}},{"id":"l-dd432781a0320036","kind":"fingerprint_related","node_a_id":"n-ab0ac67f032e783e","node_b_id":"n-bd16f85c006e2508","fingerprint":"f6-dca555f310728b4d","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-dca555f310728b4d"},"other_id":"n-ab0ac67f032e783e","other":{"id":"n-ab0ac67f032e783e","name":"basis_map_A013610 of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*3^j.","first_index":0,"fingerprint":"f6-dca555f310728b4d","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["15","27"],["225","810","729"]],"card_rows":[["1"],["15","27"],["225","810","729"],["3375","18225","32805","19683"]],"row_sums_preview":["1","42","1764","74088","3111696","130691232","5489031744","230539333248"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8d88ef9de74b305f","steps":["basis A013610","basis A013610"],"edges":["e-b88e4ed4246b1dea","e-f03ec0615713698c"],"text":"A038221 \u2192 basis A013610 \u2192 basis A013610"},"headline":"A038221 \u2192 basis A013610 \u2192 basis A013610","chain":null}},{"id":"l-df2ffef0797d84b2","kind":"fingerprint_related","node_a_id":"n-a13c2de84911142b","node_b_id":"n-bd16f85c006e2508","fingerprint":"f6-dca555f310728b4d","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-dca555f310728b4d"},"other_id":"n-a13c2de84911142b","other":{"id":"n-a13c2de84911142b","name":"basis_map_A013610 of Triangle of coefficients in expansion of (2+3x)^n.","first_index":0,"fingerprint":"f6-dca555f310728b4d","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["5","9"],["25","90","81"]],"card_rows":[["1"],["5","9"],["25","90","81"],["125","675","1215","729"]],"row_sums_preview":["1","14","196","2744","38416","537824","7529536","105413504"],"regions":["(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-5733d79f45a14084","steps":["basis A013610"],"edges":["e-4eef4a156684ec8f"],"text":"A013620 \u2192 basis A013610"},"headline":"A013620 \u2192 basis A013610","chain":null}},{"id":"l-e906447d58614ac2","kind":"fingerprint_related","node_a_id":"n-9960d61ffc0b2713","node_b_id":"n-bd16f85c006e2508","fingerprint":"f6-dca555f310728b4d","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-dca555f310728b4d"},"other_id":"n-9960d61ffc0b2713","other":{"id":"n-9960d61ffc0b2713","name":"basis_map_A013610 of basis_map_A013610 of Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-dca555f310728b4d","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["20","36"],["250","900","810"]],"card_rows":[["1"],["20","36"],["250","900","810"],["2500","13500","24300","14580"]],"row_sums_preview":["1","56","1960","54880","1344560","30118144","632481024","12649620480"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a820c1859f68e71c","steps":["basis A013610","basis A013610"],"edges":["e-24fdbe8219e7135e","e-17e3f9c6f72cd3fc"],"text":"A178820 \u2192 basis A013610 \u2192 basis A013610"},"headline":"A178820 \u2192 basis A013610 \u2192 basis A013610","chain":null}},{"id":"l-ffdc0d5e0dcd8bd8","kind":"fingerprint_related","node_a_id":"n-dff5aa7e3bad489c","node_b_id":"n-bd16f85c006e2508","fingerprint":"f6-dca555f310728b4d","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-dca555f310728b4d"},"other_id":"n-dff5aa7e3bad489c","other":{"id":"n-dff5aa7e3bad489c","name":"derivative of basis_map_A013610 of Triangle of coefficients in expansion of (2+3x)^n.","first_index":0,"fingerprint":"f6-dca555f310728b4d","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["9"],["90","162"]],"card_rows":[["9"],["90","162"],["675","2430","2187"],["4500","24300","43740","26244"]],"row_sums_preview":["0","9","252","5292","98784","1728720","29042496","474360768"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-5733d79f45a14084","steps":["basis A013610","D"],"edges":["e-4eef4a156684ec8f","e-66dc6a9f85976a5b"],"text":"A013620 \u2192 basis A013610 \u2192 D"},"headline":"A013620 \u2192 basis A013610 \u2192 D","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:                 3600,                 12960,                  11664;\n  1:               216000,               1166400,                2099520,                 1259712;\n  2:             12960000,              93312000,              251942400,               302330880,                136048896;\n  3:            777600000,            6998400000,            25194240000,             45349632000,              40814668800,              14693280768;\n  4:          46656000000,          503884800000,          2267481600000,           5441955840000,            7346640384000,            5289581076480,            1586874322944;\n  5:        2799360000000,        35271936000000,        190468454400000,         571405363200000,         1028529653760000,         1110812026060800,          666487215636480,          171382426877952;\n  6:      167961600000000,      2418647040000000,      15237476352000000,       54854914867200000,       123423558451200000,       177729924169728000,       159956931752755200,        82263564901416960,        18509302102818816;\n  7:    10077696000000000,    163258675200000000,    1175462461440000000,     4936942338048000000,     13329744312729600000,     23993539762913280000,     28792247715495936000,     22211162523382579200,      9995023135522160640,      1999004627104432128;\n  8:   604661760000000000,  10883911680000000000,   88159684608000000000,   423166486118400000000,   1332974431272960000000,   2879224771549593600000,   4318837157324390400000,   4442232504676515840000,   2998506940656648192000,   1199402776262659276800,    215892499727278669824;\n  9: 36279705600000000000, 718338170880000000000, 6465043537920000000000, 34911235104768000000000, 125680446377164800000000, 316714724870455296000000, 570086504766819532800000, 732968363271625113600000, 659671526944462602240000, 395802916166677561344000, 142489049820003922083840, 23316389970546096340992;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (60+108*x)*P(n-1,x) for n >= 1, with P(0,x) = 3600 + 12960*x + 11664*x^2.","mathematica":"P[0, x_] := 3600 + 12960*x + 11664*x^2;\nP[n_, x_] := P[n, x] = Expand[(60 + 108*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] = 3600 + 12960*x + 11664*x^2; for(n = 1, N, my(i = n + 1); v[i] = (60 + 108*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-c1532953a6736702","steps":["basis A013610","basis A013610"],"edges":["e-85e8843cd1dca083","e-b3602fe9ead223e6"],"text":"A038338 \u2192 basis A013610 \u2192 basis A013610"},"headline":"A038338 \u2192 basis A013610 \u2192 basis A013610","chain":null}
