{"id":"n-4397e742863fdbe2","revision":1,"first_index":0,"fingerprint":"f6-1024c0779b1dc97d","name":"Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^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":[["100","60","9"],["1000","900","270","27"],["10000","12000","5400","1080","81"],["100000","150000","90000","27000","4050","243"],["1000000","1800000","1350000","540000","121500","14580","729"],["10000000","21000000","18900000","9450000","2835000","510300","51030","2187"],["100000000","240000000","252000000","151200000","56700000","13608000","2041200","174960","6561"],["1000000000","2700000000","3240000000","2268000000","1020600000","306180000","61236000","7873200","590490","19683"],["10000000000","30000000000","40500000000","32400000000","17010000000","6123600000","1530900000","262440000","29524500","1968300","59049"],["100000000000","330000000000","495000000000","445500000000","267300000000","112266000000","33679800000","7217100000","1082565000","108256500","6495390","177147"],["1000000000000","3600000000000","5940000000000","5940000000000","4009500000000","1924560000000","673596000000","173210400000","32476950000","4330260000","389723400","21257640","531441"],["10000000000000","39000000000000","70200000000000","77220000000000","57915000000000","31274100000000","12509640000000","3752892000000","844400700000","140733450000","16888014000","1381746600","69087330","1594323"],["100000000000000","420000000000000","819000000000000","982800000000000","810810000000000","486486000000000","218918700000000","75057840000000","19702683000000","3940536600000","591080490000","64481508000","4836113100","223205220","4782969"],["1000000000000000","4500000000000000","9450000000000000","12285000000000000","11056500000000000","7297290000000000","3648645000000000","1407334500000000","422200350000000","98513415000000","17732414700000","2418056550000","241805655000","16740391500","717445350","14348907"],["10000000000000000","48000000000000000","108000000000000000","151200000000000000","147420000000000000","106142400000000000","58378320000000000","25019280000000000","8444007000000000","2251735200000000","472864392000000","77377809600000","9672226200000","892820880000","57395628000","2295825120","43046721"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["169","2197","28561","371293","4826809","62748517","815730721","10604499373","137858491849","1792160394037","23298085122481","302875106592253","3937376385699289","51185893014090757","665416609183179841"],"nonn":true,"definitions":[{"position":0,"source":"oeis_catalog:A038305","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["100","60","9"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["10","3"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":158544,"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":false,"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-e186153c671b61f0"]},{"id":158545,"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-e186153c671b61f0","e-f733aee5869de636"]},{"id":158546,"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-e186153c671b61f0","e-f733aee5869de636"]},{"id":158547,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"previous_interlaces_next"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 1","region_exact":null,"edges":["e-e186153c671b61f0"]},{"id":158548,"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-f733aee5869de636"]},{"id":158549,"position":5,"claim":"basis_map_preserver","from_index":null,"region_key":null,"provenance":"edges","claim_data":{"claim":"basis_map_preserver","input_interval":"(-inf, +inf)","output_interval":"(-inf, +inf)","statement":"every real-rooted input has a real-rooted image (an affine substitution times g)"},"label":"Basis map preserves real-rootedness","description":"Basis map u^k -> P_k sends roots in (-inf, +inf) to roots in (-inf, +inf): every real-rooted input has a real-rooted image (an affine substitution times g)","region_exact":null,"edges":["e-7687994fc4fefbbe"]},{"id":158550,"position":6,"claim":"basis_map_preserver","from_index":null,"region_key":null,"provenance":"edges","claim_data":{"claim":"basis_map_preserver","input_interval":"(-inf, 10]","output_interval":"(-inf, 0]","statement":"an input root r <= 10 gives the root (r - a)/d <= 0; a larger root gives a positive root"},"label":"Basis map preserves real-rootedness","description":"Basis map u^k -> P_k sends roots in (-inf, 10] to roots in (-inf, 0]: an input root r <= 10 gives the root (r - a)/d <= 0; a larger root gives a positive root","region_exact":null,"edges":["e-7687994fc4fefbbe"]}],"region_exact":{},"incoming_edges":[{"id":"e-7687994fc4fefbbe","operation":{"operation":"basis_map_theorem"},"operation_detail":null,"conclusion_id":"n-4397e742863fdbe2","premise_count":0,"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":"24bfc1953aba885473e414dff72d91acfa400bd2d9c4e39cb8eca97200c22a3a","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"basis_map_preserver","input_interval":"(-inf, +inf)","output_interval":"(-inf, +inf)","statement":"every real-rooted input has a real-rooted image (an affine substitution times g)"},{"claim":"basis_map_preserver","input_interval":"(-inf, 10]","output_interval":"(-inf, 0]","statement":"an input root r <= 10 gives the root (r - a)/d <= 0; a larger root gives a positive root"}],"premises":[]},{"id":"e-f733aee5869de636","operation":{"operation":"parent_provers","route":"periodic_product"},"operation_detail":"periodic_product","conclusion_id":"n-4397e742863fdbe2","premise_count":0,"theorem":"polytool parent provers, route periodic_product: every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign","lean_route":null,"lean_declarations":[],"certificate_hash":"52cc471cf8d1a8879ea21bfeb67b460bc6ac00143803b7839d8e65dc04b28330","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[]},{"id":"e-e186153c671b61f0","operation":{"operation":"theorem_check","theorem":"liu_wang"},"operation_detail":"liu_wang","conclusion_id":"n-4397e742863fdbe2","premise_count":0,"theorem":"L. L. Liu and Y. Wang, A unified approach to polynomial sequences with only real zeros, Adv. in Appl. Math. 38 (2007) 542-560, Theorem 2.1 with g = f' (as in their Corollary 2.4), with nonnegativity of the coefficients replaced by an exact induction keeping every root in (-inf, 0)","lean_route":null,"lean_declarations":[],"certificate_hash":"fd2f62c696cc5bcced67801414d39d699f5a9114c6f0c86de61ba43d959535a3","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":false,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"previous_interlaces_next"}],"premises":[]}],"used_by":[{"id":"e-1eab37781c60e76a","operation":"basis_map","operation_detail":"A008277","conclusion_id":"n-8540506b111a5858","conclusion_name":"basis_map_A008277 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^j."},{"id":"e-54bb49968ded72d0","operation":"basis_map","operation_detail":"A013609","conclusion_id":"n-1abe254a355aa2af","conclusion_name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^j."},{"id":"e-7b93be005ec5b750","operation":"basis_map","operation_detail":"A013611","conclusion_id":"n-b8a3f88eab200094","conclusion_name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^j."},{"id":"e-c80be26a429cc3fc","operation":"basis_map","operation_detail":"A013610","conclusion_id":"n-7e380db7ce29966b","conclusion_name":"basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^j."},{"id":"e-ebc4517445c15609","operation":"basis_map","operation_detail":"A007318","conclusion_id":"n-4ba4cfdaaf5fda7d","conclusion_name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^j."},{"id":"e-e4e37e9713aaf48a","operation":"obreschkoff_combination","operation_detail":null,"conclusion_id":"n-91e781c52666cb4c","conclusion_name":"obreschkoff_combination of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^j."},{"id":"e-e4e37e9713aaf48a","operation":"obreschkoff_combination","operation_detail":null,"conclusion_id":"n-91e781c52666cb4c","conclusion_name":"obreschkoff_combination of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^j."}],"used_by_total":7,"links":[{"id":"l-9797774950d34a19","kind":"fingerprint_related","node_a_id":"n-4397e742863fdbe2","node_b_id":"n-6ca7f61b6ad83377","fingerprint":"f6-1024c0779b1dc97d","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1024c0779b1dc97d"},"other_id":"n-6ca7f61b6ad83377","other":{"id":"n-6ca7f61b6ad83377","name":"basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*1^j.","first_index":0,"fingerprint":"f6-1024c0779b1dc97d","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["10","3"],["100","60","9"]],"card_rows":[["1"],["10","3"],["100","60","9"],["1000","900","270","27"]],"row_sums_preview":["1","13","169","2197","28561","371293","4826809","62748517"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3dd184aba99388f7","steps":["basis A013610"],"edges":["e-8d10edd099560dc3"],"text":"A038291 \u2192 basis A013610"},"headline":"A038291 \u2192 basis A013610","chain":null}},{"id":"l-a0c4eb789ea1b957","kind":"fingerprint_related","node_a_id":"n-4397e742863fdbe2","node_b_id":"n-a2769223501d377d","fingerprint":"f6-1024c0779b1dc97d","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1024c0779b1dc97d"},"other_id":"n-a2769223501d377d","other":{"id":"n-a2769223501d377d","name":"basis_map_A013611 of Triangle of coefficients in expansion of (1+5x)^n.","first_index":0,"fingerprint":"f6-1024c0779b1dc97d","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["36","240","400"],["216","2160","7200","8000"],["1296","17280","86400","192000","160000"]],"card_rows":[["36","240","400"],["216","2160","7200","8000"],["1296","17280","86400","192000","160000"],["7776","129600","864000","2880000","4800000","3200000"]],"row_sums_preview":["676","17576","456976","11881376","308915776","8031810176","208827064576","5429503678976"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-bbfc5a54ff13a785","steps":["basis A013611"],"edges":["e-db3287eb465d1cf9"],"text":"A013612 \u2192 basis A013611"},"headline":"A013612 \u2192 basis A013611","chain":null}},{"id":"l-1b89dd5f2c430bb5","kind":"oeis_match","node_a_id":"n-4397e742863fdbe2","node_b_id":null,"fingerprint":null,"oeis_id":"A001022","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":2,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A001022","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A001022","relation":"match_row_sums"},{"oeis_id":"A038305","relation":"tail"}],"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:          100,           60,            9;\n  1:         1000,          900,          270,           27;\n  2:        10000,        12000,         5400,         1080,           81;\n  3:       100000,       150000,        90000,        27000,         4050,          243;\n  4:      1000000,      1800000,      1350000,       540000,       121500,        14580,         729;\n  5:     10000000,     21000000,     18900000,      9450000,      2835000,       510300,       51030,       2187;\n  6:    100000000,    240000000,    252000000,    151200000,     56700000,     13608000,     2041200,     174960,       6561;\n  7:   1000000000,   2700000000,   3240000000,   2268000000,   1020600000,    306180000,    61236000,    7873200,     590490,     19683;\n  8:  10000000000,  30000000000,  40500000000,  32400000000,  17010000000,   6123600000,  1530900000,  262440000,   29524500,   1968300,   59049;\n  9: 100000000000, 330000000000, 495000000000, 445500000000, 267300000000, 112266000000, 33679800000, 7217100000, 1082565000, 108256500, 6495390, 177147;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (10+3*x)*P(n-1,x) for n >= 1, with P(0,x) = 100 + 60*x + 9*x^2.","mathematica":"P[0, x_] := 100 + 60*x + 9*x^2;\nP[n_, x_] := P[n, x] = Expand[(10 + 3*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] = 100 + 60*x + 9*x^2; for(n = 1, N, my(i = n + 1); v[i] = (10 + 3*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-4397e742863fdbe2","steps":[],"edges":[],"text":"A038305"},"headline":"Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^j.","chain":null}
