{"id":"n-2090bc852fe1ed0c","revision":0,"first_index":0,"fingerprint":"f6-dca555f310728b4d","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.","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":[["1296","1440","400"],["46656","77760","43200","8000"],["1679616","3732480","3110400","1152000","160000"],["60466176","167961600","186624000","103680000","28800000","3200000"],["2176782336","7255941120","10077696000","7464960000","3110400000","691200000","64000000"],["78364164096","304749527040","507915878400","470292480000","261273600000","87091200000","16128000000","1280000000"],["2821109907456","12538266255360","24379962163200","27088846848000","18811699200000","8360755200000","2322432000000","368640000000","25600000000"],["101559956668416","507799783342080","1128443962982400","1462797729792000","1218998108160000","677221171200000","250822656000000","59719680000000","8294400000000","512000000000"],["3656158440062976","20311991333683200","50779978334208000","75229597532160000","73139886489600000","48759924326400000","22574039040000000","7166361600000000","1492992000000000","184320000000000","10240000000000"],["131621703842267136","804354856813854720","2234319046705152000","3723865077841920000","4137627864268800000","3218155005542400000","1787863891968000000","709469798400000000","197074944000000000","36495360000000000","4055040000000000","204800000000000"],["4738381338321616896","31589208922144112640","96522582817662566400","178745523736412160000","223431904670515200000","198606137484902400000","128726200221696000000","61298190581760000000","21284093952000000000","5255331840000000000","875888640000000000","88473600000000000","4096000000000000"],["170581728179578208256","1231979147963620392960","4106597159878734643200","8365290510864089088000","11618459042866790400000","11618459042866790400000","8606265957679104000000","4781258865377280000000","1992191193907200000000","614873825280000000000","136638627840000000000","20702822400000000000","1916928000000000000","81920000000000000"],["6140942214464815497216","47762883890281898311680","172477080714906855014400","383282401588681900032000","585570335760486236160000","650633706400540262400000","542194755333783552000000","344250638307164160000000","167344060288204800000000","61979281588224000000000","17216467107840000000000","3478074163200000000000","483065856000000000000","41287680000000000000","1638400000000000000"],["221073919720733357899776","1842282664339444649164800","7164432583542284746752000","17247708071490685501440000","28746180119151142502400000","35134220145629174169600000","32531685320027013120000000","23236918085733580800000000","12909398936518656000000000","5578135342940160000000000","1859378447646720000000000","469540012032000000000000","86951854080000000000000","11147673600000000000000","884736000000000000000","32768000000000000000"],["7958661109946400884391936","70743654310634674527928320","294765226294311143866368000","764206142244510372986880000","1379816645719254840115200000","1839755527625673120153600000","1873825074433555955712000000","1487162757486949171200000000","929476723429343232000000000","459000851076218880000000000","178500330974085120000000000","54091009386086400000000000","12521066987520000000000000","2140353331200000000000000","254803968000000000000000","18874368000000000000000","655360000000000000000"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["3136","175616","9834496","550731776","30840979456","1727094849536","96717311574016","5416169448144896","303305489096114176","16985107389382393856","951166013805414055936","53265296773103187132416","2982856619293778479415296","167039970680451594847256576","9354238358105289311446368256"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1296","1440","400"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["36","20"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":194719,"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-cef54db26bf72db9"]},{"id":194720,"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-cef54db26bf72db9","e-d6c434ca60102d19"]},{"id":194721,"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-cef54db26bf72db9","e-d6c434ca60102d19"]},{"id":194722,"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-cef54db26bf72db9"]},{"id":194723,"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-d6c434ca60102d19"]}],"region_exact":{},"incoming_edges":[{"id":"e-d6c434ca60102d19","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-2090bc852fe1ed0c","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":"bd76ed1c0120d91e494e98c73b6ef63b24533c6d19213f643b9a5ebc9e3db5c2","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-93016690ff7736f4","role":"sequence","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*5^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]"}]},{"id":"e-cef54db26bf72db9","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-2090bc852fe1ed0c","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":"f2b4e441bcc69584611c960805d6113c0b2a9f6f1ee8998d957627a225e8b6a2","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-93016690ff7736f4","role":"sequence","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*5^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-02f3c1b1de113b52","kind":"fingerprint_related","node_a_id":"n-51c3e1916ec6b28c","node_b_id":"n-2090bc852fe1ed0c","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-0456df9a9e8033a2","kind":"fingerprint_related","node_a_id":"n-2090bc852fe1ed0c","node_b_id":"n-8fb4a561853504a4","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-071e9f4bbb939e0e","kind":"fingerprint_related","node_a_id":"n-950319b6c870b4e9","node_b_id":"n-2090bc852fe1ed0c","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-279d32cf33df8b47","kind":"fingerprint_related","node_a_id":"n-352735e5f7822499","node_b_id":"n-2090bc852fe1ed0c","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-4973481bd1b57883","kind":"fingerprint_related","node_a_id":"n-6d9baf9da524d2a0","node_b_id":"n-2090bc852fe1ed0c","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-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-bd16f85c006e2508","other":{"id":"n-bd16f85c006e2508","name":"basis_map_A013610 of basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^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":[["3600","12960","11664"],["216000","1166400","2099520","1259712"],["12960000","93312000","251942400","302330880","136048896"]],"card_rows":[["3600","12960","11664"],["216000","1166400","2099520","1259712"],["12960000","93312000","251942400","302330880","136048896"],["777600000","6998400000","25194240000","45349632000","40814668800","14693280768"]],"row_sums_preview":["28224","4741632","796594176","133827821568","22483074023424","3777156435935232","634562281237118976","106606463247835987968"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"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}},{"id":"l-5ff80f509695bee4","kind":"fingerprint_related","node_a_id":"n-2090bc852fe1ed0c","node_b_id":"n-ab0ac67f032e783e","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-698687815ad10c29","kind":"fingerprint_related","node_a_id":"n-2090bc852fe1ed0c","node_b_id":"n-41ebfdeeaf308611","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-7127104dc7244de1","kind":"fingerprint_related","node_a_id":"n-2090bc852fe1ed0c","node_b_id":"n-4547a8b17f528145","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-8838a0ff6f1bbbac","kind":"fingerprint_related","node_a_id":"n-2090bc852fe1ed0c","node_b_id":"n-ee54987254c3cd08","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-93dd2304048713ba","kind":"fingerprint_related","node_a_id":"n-2090bc852fe1ed0c","node_b_id":"n-1273b38f23038cfd","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-994056616ed28d2a","kind":"fingerprint_related","node_a_id":"n-dff5aa7e3bad489c","node_b_id":"n-2090bc852fe1ed0c","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}},{"id":"l-9c5faeb682733542","kind":"fingerprint_related","node_a_id":"n-3d8880ae4f67526d","node_b_id":"n-2090bc852fe1ed0c","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-9df8ba6c29ec3ce5","kind":"fingerprint_related","node_a_id":"n-2090bc852fe1ed0c","node_b_id":"n-411e78f82a532dd1","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-a84cbd63b55c32fc","kind":"fingerprint_related","node_a_id":"n-2090bc852fe1ed0c","node_b_id":"n-d8e81e9a78e7e2d2","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-b3fcb479133b6140","kind":"fingerprint_related","node_a_id":"n-2090bc852fe1ed0c","node_b_id":"n-9960d61ffc0b2713","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-dff337320923d52f","kind":"fingerprint_related","node_a_id":"n-38edd6a4bb3bb62b","node_b_id":"n-2090bc852fe1ed0c","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-f41b9f03850ea674","kind":"fingerprint_related","node_a_id":"n-a13c2de84911142b","node_b_id":"n-2090bc852fe1ed0c","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}}],"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:               1296,               1440,                 400;\n  1:              46656,              77760,               43200,                8000;\n  2:            1679616,            3732480,             3110400,             1152000,              160000;\n  3:           60466176,          167961600,           186624000,           103680000,            28800000,             3200000;\n  4:         2176782336,         7255941120,         10077696000,          7464960000,          3110400000,           691200000,            64000000;\n  5:        78364164096,       304749527040,        507915878400,        470292480000,        261273600000,         87091200000,         16128000000,         1280000000;\n  6:      2821109907456,     12538266255360,      24379962163200,      27088846848000,      18811699200000,       8360755200000,       2322432000000,       368640000000,        25600000000;\n  7:    101559956668416,    507799783342080,    1128443962982400,    1462797729792000,    1218998108160000,     677221171200000,     250822656000000,     59719680000000,      8294400000000,      512000000000;\n  8:   3656158440062976,  20311991333683200,   50779978334208000,   75229597532160000,   73139886489600000,   48759924326400000,   22574039040000000,   7166361600000000,   1492992000000000,   184320000000000,   10240000000000;\n  9: 131621703842267136, 804354856813854720, 2234319046705152000, 3723865077841920000, 4137627864268800000, 3218155005542400000, 1787863891968000000, 709469798400000000, 197074944000000000, 36495360000000000, 4055040000000000, 204800000000000;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (36+20*x)*P(n-1,x) for n >= 1, with P(0,x) = 1296 + 1440*x + 400*x^2.","mathematica":"P[0, x_] := 1296 + 1440*x + 400*x^2;\nP[n_, x_] := P[n, x] = Expand[(36 + 20*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] = 1296 + 1440*x + 400*x^2; for(n = 1, N, my(i = n + 1); v[i] = (36 + 20*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-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}
