{"id":"n-b7cf68162d40a956","revision":1,"first_index":0,"fingerprint":"f6-428cb055b0457fc7","name":"Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*11^j.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["1"],["7","11"],["49","154","121"],["343","1617","2541","1331"],["2401","15092","35574","37268","14641"],["16807","132055","415030","652190","512435","161051"],["117649","1109262","4357815","9130660","10761135","6764142","1771561"],["823543","9058973","42706587","111850585","175765205","165721479","86806489","19487171"],["5764801","72471784","398594812","1252726552","2460712870","3093467608","2430581692","1091281576","214358881"],["40353607","570715299","3587353308","13153628796","31004982162","48722114826","51042215532","34375369644","13504609503","2357947691"],["282475249","4438896770","31389341445","131536287960","361724791890","682109607564","893238771810","802091958360","472661332605","165056338370","25937424601"],["1977326743","34179505129","268553254585","1266036771615","3978972710790","8753739963738","13755877085874","15440270198430","12131640870195","6354669027245","1997181694277","285311670611"],["13841287201","261007130076","2255847338514","11816343201740","41779213463295","105044879564856","192582279202236","259396539333624","254764458274095","177930732762860","83881631159634","23966180331324","3138428376721"],["96889010407","1979304069743","18662009800434","107528723135834","422434269462205","1194885505050237","2503569629629068","3934180846559964","4636713140588529","4047924170355065","2544409478508898","1090461205075242","285596982281611","34522712143931"],["678223072849","14920907602678","152406413370211","957983169755612","4139855840729609","13010975499435914","30668727962956083","55078531851839496","75732981296279307","79339313738959274","62338032223468001","35621732699124572","13994252131798939","3383225790105238","379749833583241"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","18","324","5832","104976","1889568","34012224","612220032","11019960576","198359290368","3570467226624","64268410079232","1156831381426176","20822964865671168","374813367582081024"],"nonn":true,"definitions":[{"position":0,"source":"oeis_catalog:A038277","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["7","11"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":158495,"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-f9e3d73ad0eb8fa6"]},{"id":158496,"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-399c4cc80f25ee4f","e-f9e3d73ad0eb8fa6"]},{"id":158497,"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-399c4cc80f25ee4f","e-f9e3d73ad0eb8fa6"]},{"id":158498,"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-f9e3d73ad0eb8fa6"]},{"id":158499,"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-399c4cc80f25ee4f"]},{"id":158500,"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-dc6cd425a1850984"]},{"id":158501,"position":6,"claim":"basis_map_preserver","from_index":null,"region_key":null,"provenance":"edges","claim_data":{"claim":"basis_map_preserver","input_interval":"(-inf, 7]","output_interval":"(-inf, 0]","statement":"an input root r <= 7 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, 7] to roots in (-inf, 0]: an input root r <= 7 gives the root (r - a)/d <= 0; a larger root gives a positive root","region_exact":null,"edges":["e-dc6cd425a1850984"]}],"region_exact":{},"incoming_edges":[{"id":"e-dc6cd425a1850984","operation":{"operation":"basis_map_theorem"},"operation_detail":null,"conclusion_id":"n-b7cf68162d40a956","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":"883096c41b3f207b9ebe0696bb08049cc29892dbbe7e68f51c3a7e1e341a9959","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, 7]","output_interval":"(-inf, 0]","statement":"an input root r <= 7 gives the root (r - a)/d <= 0; a larger root gives a positive root"}],"premises":[]},{"id":"e-399c4cc80f25ee4f","operation":{"operation":"parent_provers","route":"periodic_product"},"operation_detail":"periodic_product","conclusion_id":"n-b7cf68162d40a956","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":"6881a416154e4c63cd6947811af67b977a3364b169469ee3003e96397eb60130","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[]},{"id":"e-f9e3d73ad0eb8fa6","operation":{"operation":"theorem_check","theorem":"liu_wang"},"operation_detail":"liu_wang","conclusion_id":"n-b7cf68162d40a956","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":"87f69a17650624a8e42ef08569823b4004829fe55a152a4e9973858dd45b28c2","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-59f75239fe41171d","operation":"basis_map","operation_detail":"A013609","conclusion_id":"n-b2ae2bce0e1523fc","conclusion_name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*11^j."},{"id":"e-77c9a747c1207e17","operation":"basis_map","operation_detail":"A013610","conclusion_id":"n-196d480a741a9e86","conclusion_name":"basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*11^j."},{"id":"e-8cea3676de7ce3cf","operation":"basis_map","operation_detail":"A008277","conclusion_id":"n-c6d0c3e6cdaf94a0","conclusion_name":"basis_map_A008277 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*11^j."},{"id":"e-c73faa2b910a081c","operation":"basis_map","operation_detail":"A007318","conclusion_id":"n-77900e9176a9461a","conclusion_name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*11^j."},{"id":"e-f0378e56b9f5156d","operation":"basis_map","operation_detail":"A013611","conclusion_id":"n-16b272df8b69180c","conclusion_name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*11^j."},{"id":"e-3667cd139473d050","operation":"obreschkoff_combination","operation_detail":null,"conclusion_id":"n-d3ee9b6e124c7b6d","conclusion_name":"obreschkoff_combination of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*11^j."},{"id":"e-3667cd139473d050","operation":"obreschkoff_combination","operation_detail":null,"conclusion_id":"n-d3ee9b6e124c7b6d","conclusion_name":"obreschkoff_combination of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*11^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-7dcf11eddf2b9a8a","other":{"id":"n-7dcf11eddf2b9a8a","name":"Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^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":[["121","154","49"],["1331","2541","1617","343"],["14641","37268","35574","15092","2401"]],"card_rows":[["121","154","49"],["1331","2541","1617","343"],["14641","37268","35574","15092","2401"],["161051","512435","652190","415030","132055","16807"]],"row_sums_preview":["324","5832","104976","1889568","34012224","612220032","11019960576","198359290368"],"regions":["(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"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}},{"id":"l-618187d5ae367dbd","kind":"oeis_match","node_a_id":"n-b7cf68162d40a956","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":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A001027","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-f01a243c60c89667","kind":"oeis_match","node_a_id":"n-b7cf68162d40a956","node_b_id":null,"fingerprint":null,"oeis_id":"A038277","in_catalog":1,"target":"triangle","row_alignment":"tabl","rows_matched":7,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":7,"target":"triangle","terms_compared":32},"in_catalog":true,"oeis_id":"A038277","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-f21d42c5d1a67ce7","kind":"oeis_match","node_a_id":"n-b7cf68162d40a956","node_b_id":null,"fingerprint":null,"oeis_id":"A038321","in_catalog":1,"target":"reversed_triangle","row_alignment":"tabl","rows_matched":8,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":8,"target":"reversed_triangle","terms_compared":36},"in_catalog":true,"oeis_id":"A038321","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A038277","relation":"identification"},{"oeis_id":"A038321","relation":"match_reversed_triangle"},{"oeis_id":"A001027","relation":"match_row_sums"},{"oeis_id":"A038277","relation":"match_triangle"}],"bfiles":[{"node_id":"n-b7cf68162d40a956","oeis_id":"A001027","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":21,"oeis_terms_available":21,"term_count":200,"raw_size":25970},{"node_id":"n-b7cf68162d40a956","oeis_id":"A038277","target":"triangle","oeis_offset":0,"first_row":0,"rows_used":44,"terms_matched":32,"oeis_terms_agreeing":32,"oeis_terms_available":32,"term_count":990,"raw_size":37402},{"node_id":"n-b7cf68162d40a956","oeis_id":"A038321","target":"reversed_triangle","oeis_offset":0,"first_row":0,"rows_used":44,"terms_matched":36,"oeis_terms_agreeing":36,"oeis_terms_available":36,"term_count":990,"raw_size":37402}],"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:        1;\n  1:        7,        11;\n  2:       49,       154,        121;\n  3:      343,      1617,       2541,        1331;\n  4:     2401,     15092,      35574,       37268,       14641;\n  5:    16807,    132055,     415030,      652190,      512435,      161051;\n  6:   117649,   1109262,    4357815,     9130660,    10761135,     6764142,     1771561;\n  7:   823543,   9058973,   42706587,   111850585,   175765205,   165721479,    86806489,    19487171;\n  8:  5764801,  72471784,  398594812,  1252726552,  2460712870,  3093467608,  2430581692,  1091281576,   214358881;\n  9: 40353607, 570715299, 3587353308, 13153628796, 31004982162, 48722114826, 51042215532, 34375369644, 13504609503, 2357947691;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (7+11*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(7 + 11*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] = 1; for(n = 1, N, my(i = n + 1); v[i] = (7 + 11*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}},{"oeis_id":"A038277","target":"triangle","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:        1;\n  1:        7,        11;\n  2:       49,       154,        121;\n  3:      343,      1617,       2541,        1331;\n  4:     2401,     15092,      35574,       37268,       14641;\n  5:    16807,    132055,     415030,      652190,      512435,      161051;\n  6:   117649,   1109262,    4357815,     9130660,    10761135,     6764142,     1771561;\n  7:   823543,   9058973,   42706587,   111850585,   175765205,   165721479,    86806489,    19487171;\n  8:  5764801,  72471784,  398594812,  1252726552,  2460712870,  3093467608,  2430581692,  1091281576,   214358881;\n  9: 40353607, 570715299, 3587353308, 13153628796, 31004982162, 48722114826, 51042215532, 34375369644, 13504609503, 2357947691;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (7+11*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(7 + 11*x)*P[n - 1, x]];\nTable[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (7 + 11*x)*v[i - 1]); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); 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":true}},{"oeis_id":"A038321","target":"reversed_triangle","first_row":0,"oeis_offset":0,"reversed":true,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:          1;\n  1:         11,           7;\n  2:        121,         154,          49;\n  3:       1331,        2541,        1617,         343;\n  4:      14641,       37268,       35574,       15092,        2401;\n  5:     161051,      512435,      652190,      415030,      132055,       16807;\n  6:    1771561,     6764142,    10761135,     9130660,     4357815,     1109262,      117649;\n  7:   19487171,    86806489,   165721479,   175765205,   111850585,    42706587,     9058973,     823543;\n  8:  214358881,  1091281576,  2430581692,  3093467608,  2460712870,  1252726552,   398594812,   72471784,   5764801;\n  9: 2357947691, 13504609503, 34375369644, 51042215532, 48722114826, 31004982162, 13153628796, 3587353308, 570715299, 40353607;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (7+11*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(7 + 11*x)*P[n - 1, x]];\nTable[Reverse[PadRight[CoefficientList[P[n, x], x], n + 1]], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (7 + 11*x)*v[i - 1]); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c = Vecrev(c); 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":true,"trim_zeros":"none","regular_triangle":true}}],"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}
