{"id":"n-b58ba5723c3a3a3f","revision":1,"first_index":1,"fingerprint":"f6-52b402d0a3aa4f28","name":"Triangle, read by rows, T(n,k) = t(n-k, k) where t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(x) = x + 3.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":1,"palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":false,"rows":[["9","24","9"],["27","141","141","27"],["81","726","1410","726","81"],["243","3471","11406","11406","3471","243"],["729","15828","81327","136872","81327","15828","729"],["2187","69873","533259","1390521","1390521","533259","69873","2187"],["6561","301362","3295152","12609198","19467294","12609198","3295152","301362","6561"],["19683","1277619","19489380","105311556","237144642","237144642","105311556","19489380","1277619","19683"],["59049","5346672","111500709","826763136","2607816498","3794314272","2607816498","826763136","111500709","5346672","59049"],["177147","22154325","621663609","6187086615","26522346846","53824862658","53824862658","26522346846","6187086615","621663609","22154325","177147"],["531441","91097358","3396324270","44582482998","253714380687","695822369724","968847527844","695822369724","253714380687","44582482998","3396324270","91097358","531441"],["1594323","372361047","18256984362","311647113498","2310990460785","8357437145349","15677851447836","15677851447836","8357437145349","2310990460785","311647113498","18256984362","372361047","1594323"],["4782969","1514953356","96870337515","2125480462056","20228345700969","94591382692212","233032471629363","313557028956720","233032471629363","94591382692212","20228345700969","2125480462056","96870337515","1514953356","4782969"],["14348907","6141123897","508590941271","14205937835061","171355146375567","1019699555650293","3232388836970811","5698927477490193","5698927477490193","3232388836970811","1019699555650293","171355146375567","14205937835061","508590941271","6141123897","14348907"],["43046721","24822775914","2647353812604","93373082070702","1412575092154884","10556568494460282","42347593756191108","95777940818551662","125376404504784246","95777940818551662","42347593756191108","10556568494460282","1412575092154884","93373082070702","2647353812604","24822775914","43046721"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["42","336","3024","30240","332640","3991680","51891840","726485760","10897286400","174356582400","2964061900800","53353114214400","1013709170073600","20274183401472000","425757851430912000"],"nonn":true,"definitions":[{"position":0,"source":"oeis_catalog:A257180","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["9","24","9"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["3","3"],["0","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1","-1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":159715,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0)","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":false,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0)","region_exact":null,"edges":["e-27a325e06cca4d67"]},{"id":159716,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-27a325e06cca4d67"]},{"id":159717,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-27a325e06cca4d67"]},{"id":159718,"position":3,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"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 >= 2","region_exact":null,"edges":["e-27a325e06cca4d67"]}],"region_exact":{},"incoming_edges":[{"id":"e-27a325e06cca4d67","operation":{"operation":"theorem_check","theorem":"liu_wang"},"operation_detail":"liu_wang","conclusion_id":"n-b58ba5723c3a3a3f","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":"4cf9d8357ca5d57c4c3ea90bacc1ab16037db26838bf242821e3b5d1d9cc4e59","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":false,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"previous_interlaces_next"}],"premises":[]}],"used_by":[{"id":"e-418ae03de1c699ba","operation":"gamma_polynomial","operation_detail":"scale 1, shift 0","conclusion_id":"n-9fdebfaf3bddb35b","conclusion_name":"gamma_polynomial of Triangle, read by rows, T(n,k) = t(n-k, k) where t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(x) = x + 3."},{"id":"e-52155003e339a993","operation":"obreschkoff_combination","operation_detail":null,"conclusion_id":"n-41f5536fe04f4d13","conclusion_name":"obreschkoff_combination of Triangle, read by rows, T(n,k) = t(n-k, k) where t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(x) = x + 3."},{"id":"e-52155003e339a993","operation":"obreschkoff_combination","operation_detail":null,"conclusion_id":"n-41f5536fe04f4d13","conclusion_name":"obreschkoff_combination of Triangle, read by rows, T(n,k) = t(n-k, k) where t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(x) = x + 3."}],"used_by_total":3,"links":[{"id":"l-40916a7a11d68ac1","kind":"oeis_match","node_a_id":"n-b58ba5723c3a3a3f","node_b_id":null,"fingerprint":null,"oeis_id":"A001725","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":5,"oeis_term_position":2,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A001725","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A001725","relation":"match_row_sums"},{"oeis_id":"A257180","relation":"tail"}],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:      9,       24,         9;\n   2:     27,      141,       141,         27;\n   3:     81,      726,      1410,        726,          81;\n   4:    243,     3471,     11406,      11406,        3471,         243;\n   5:    729,    15828,     81327,     136872,       81327,       15828,         729;\n   6:   2187,    69873,    533259,    1390521,     1390521,      533259,       69873,        2187;\n   7:   6561,   301362,   3295152,   12609198,    19467294,    12609198,     3295152,      301362,       6561;\n   8:  19683,  1277619,  19489380,  105311556,   237144642,   237144642,   105311556,    19489380,    1277619,     19683;\n   9:  59049,  5346672, 111500709,  826763136,  2607816498,  3794314272,  2607816498,   826763136,  111500709,   5346672,    59049;\n  10: 177147, 22154325, 621663609, 6187086615, 26522346846, 53824862658, 53824862658, 26522346846, 6187086615, 621663609, 22154325, 177147;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3+(n+3)*x)*P(n-1,x) + (x-x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 9 + 24*x + 9*x^2.","mathematica":"P[1, x_] := 9 + 24*x + 9*x^2;\nP[n_, x_] := P[n, x] = Expand[(3 + (n + 3)*x)*P[n - 1, x] + (x - x^2)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 9 + 24*x + 9*x^2; for(n = 2, N, my(i = n); v[i] = (3 + (n + 3)*x)*v[i - 1] + (x - x^2)*deriv(v[i - 1], x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-b58ba5723c3a3a3f","steps":[],"edges":[],"text":"A257180"},"headline":"Triangle, read by rows, T(n,k) = t(n-k, k) where t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(x) = x + 3.","chain":null}