{"id":"n-7dcf11eddf2b9a8a","revision":1,"first_index":0,"fingerprint":"f6-428cb055b0457fc7","name":"Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^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":[["121","154","49"],["1331","2541","1617","343"],["14641","37268","35574","15092","2401"],["161051","512435","652190","415030","132055","16807"],["1771561","6764142","10761135","9130660","4357815","1109262","117649"],["19487171","86806489","165721479","175765205","111850585","42706587","9058973","823543"],["214358881","1091281576","2430581692","3093467608","2460712870","1252726552","398594812","72471784","5764801"],["2357947691","13504609503","34375369644","51042215532","48722114826","31004982162","13153628796","3587353308","570715299","40353607"],["25937424601","165056338370","472661332605","802091958360","893238771810","682109607564","361724791890","131536287960","31389341445","4438896770","282475249"],["285311670611","1997181694277","6354669027245","12131640870195","15440270198430","13755877085874","8753739963738","3978972710790","1266036771615","268553254585","34179505129","1977326743"],["3138428376721","23966180331324","83881631159634","177930732762860","254764458274095","259396539333624","192582279202236","105044879564856","41779213463295","11816343201740","2255847338514","261007130076","13841287201"],["34522712143931","285596982281611","1090461205075242","2544409478508898","4047924170355065","4636713140588529","3934180846559964","2503569629629068","1194885505050237","422434269462205","107528723135834","18662009800434","1979304069743","96889010407"],["379749833583241","3383225790105238","13994252131798939","35621732699124572","62338032223468001","79339313738959274","75732981296279307","55078531851839496","30668727962956083","13010975499435914","4139855840729609","957983169755612","152406413370211","14920907602678","678223072849"],["4177248169415651","39873732526240305","177619353980524995","489798824612962865","935070483352020015","1309098676692828021","1388437990431787295","1135994719444189605","722905730555393385","357801826234487635","136615242744077097","39516805752418995","8382352735361605","1230974877220935","111906807020085","4747561509943"],["45949729863572161","467851794974552912","2232929021469457080","6631122548606266480","13714367089162960220","20945578827085248336","24436508631599456392","22215007846908596720","15903926072218654470","8996160202467117680","4007380453826261512","1390991562485148624","368823520355910620","72217192796961520","9847799017767480","835570825749968","33232930569601"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["324","5832","104976","1889568","34012224","612220032","11019960576","198359290368","3570467226624","64268410079232","1156831381426176","20822964865671168","374813367582081024","6746640616477458432","121439531096594251776"],"nonn":true,"definitions":[{"position":0,"source":"oeis_catalog:A038321","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["121","154","49"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["11","7"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":158572,"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-f6f8234ea600f6f8"]},{"id":158573,"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-16634379058ff4dd","e-f6f8234ea600f6f8"]},{"id":158574,"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-16634379058ff4dd","e-f6f8234ea600f6f8"]},{"id":158575,"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-f6f8234ea600f6f8"]},{"id":158576,"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-16634379058ff4dd"]},{"id":158577,"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-78bb8973195ccb8f"]},{"id":158578,"position":6,"claim":"basis_map_preserver","from_index":null,"region_key":null,"provenance":"edges","claim_data":{"claim":"basis_map_preserver","input_interval":"(-inf, 11]","output_interval":"(-inf, 0]","statement":"an input root r <= 11 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, 11] to roots in (-inf, 0]: an input root r <= 11 gives the root (r - a)/d <= 0; a larger root gives a positive root","region_exact":null,"edges":["e-78bb8973195ccb8f"]}],"region_exact":{},"incoming_edges":[{"id":"e-78bb8973195ccb8f","operation":{"operation":"basis_map_theorem"},"operation_detail":null,"conclusion_id":"n-7dcf11eddf2b9a8a","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":"76317001e9ddab934f547d0aaf8b07e3aa3f86fa6f0699cee020b576611544d4","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, 11]","output_interval":"(-inf, 0]","statement":"an input root r <= 11 gives the root (r - a)/d <= 0; a larger root gives a positive root"}],"premises":[]},{"id":"e-16634379058ff4dd","operation":{"operation":"parent_provers","route":"periodic_product"},"operation_detail":"periodic_product","conclusion_id":"n-7dcf11eddf2b9a8a","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":"be5584919a749b2148fc346b8133a80dbe4051f4004f6f5c485155823fac1ee8","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[]},{"id":"e-f6f8234ea600f6f8","operation":{"operation":"theorem_check","theorem":"liu_wang"},"operation_detail":"liu_wang","conclusion_id":"n-7dcf11eddf2b9a8a","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":"b450511b41efb09c69f19ea66498837200d5b57b201b7d7c5651801b1ef48cc2","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-322a2e4c0bbd3231","operation":"basis_map","operation_detail":"A013609","conclusion_id":"n-53dc6d1eeb1907f4","conclusion_name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^j."},{"id":"e-5a47242ff4b92549","operation":"basis_map","operation_detail":"A007318","conclusion_id":"n-32cf9641286d8e9f","conclusion_name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^j."},{"id":"e-632cc4d34c6e419b","operation":"basis_map","operation_detail":"A013610","conclusion_id":"n-63d702ebcae1deea","conclusion_name":"basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^j."},{"id":"e-86a79a465fd2222b","operation":"basis_map","operation_detail":"A008277","conclusion_id":"n-0003d96fddee0ab5","conclusion_name":"basis_map_A008277 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^j."},{"id":"e-e460f89b751823ed","operation":"basis_map","operation_detail":"A013611","conclusion_id":"n-4b2df01c2184142e","conclusion_name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^j."},{"id":"e-57fae4cb2b7c6f75","operation":"obreschkoff_combination","operation_detail":null,"conclusion_id":"n-e22a7f0db816391e","conclusion_name":"obreschkoff_combination of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^j."},{"id":"e-57fae4cb2b7c6f75","operation":"obreschkoff_combination","operation_detail":null,"conclusion_id":"n-e22a7f0db816391e","conclusion_name":"obreschkoff_combination of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^j."}],"used_by_total":7,"links":[{"id":"l-b264edf3722aabff","kind":"fingerprint_related","node_a_id":"n-b7cf68162d40a956","node_b_id":"n-7dcf11eddf2b9a8a","fingerprint":"f6-428cb055b0457fc7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-428cb055b0457fc7"},"other_id":"n-b7cf68162d40a956","other":{"id":"n-b7cf68162d40a956","name":"Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*11^j.","first_index":0,"fingerprint":"f6-428cb055b0457fc7","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["7","11"],["49","154","121"]],"card_rows":[["1"],["7","11"],["49","154","121"],["343","1617","2541","1331"]],"row_sums_preview":["1","18","324","5832","104976","1889568","34012224","612220032"],"regions":["(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":["A038277"],"derivation":{"root":"n-b7cf68162d40a956","steps":[],"edges":[],"text":"A038277"},"headline":"A038277: Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*11^j.","chain":null}},{"id":"l-36b824b315a5ae9d","kind":"oeis_match","node_a_id":"n-7dcf11eddf2b9a8a","node_b_id":null,"fingerprint":null,"oeis_id":"A001027","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":"A001027","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A001027","relation":"match_row_sums"},{"oeis_id":"A038321","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:          121,           154,            49;\n  1:         1331,          2541,          1617,            343;\n  2:        14641,         37268,         35574,          15092,           2401;\n  3:       161051,        512435,        652190,         415030,         132055,          16807;\n  4:      1771561,       6764142,      10761135,        9130660,        4357815,        1109262,        117649;\n  5:     19487171,      86806489,     165721479,      175765205,      111850585,       42706587,       9058973,        823543;\n  6:    214358881,    1091281576,    2430581692,     3093467608,     2460712870,     1252726552,     398594812,      72471784,       5764801;\n  7:   2357947691,   13504609503,   34375369644,    51042215532,    48722114826,    31004982162,   13153628796,    3587353308,     570715299,     40353607;\n  8:  25937424601,  165056338370,  472661332605,   802091958360,   893238771810,   682109607564,  361724791890,  131536287960,   31389341445,   4438896770,   282475249;\n  9: 285311670611, 1997181694277, 6354669027245, 12131640870195, 15440270198430, 13755877085874, 8753739963738, 3978972710790, 1266036771615, 268553254585, 34179505129, 1977326743;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (11+7*x)*P(n-1,x) for n >= 1, with P(0,x) = 121 + 154*x + 49*x^2.","mathematica":"P[0, x_] := 121 + 154*x + 49*x^2;\nP[n_, x_] := P[n, x] = Expand[(11 + 7*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] = 121 + 154*x + 49*x^2; for(n = 1, N, my(i = n + 1); v[i] = (11 + 7*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-7dcf11eddf2b9a8a","steps":[],"edges":[],"text":"A038321"},"headline":"Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^j.","chain":null}