{"id":"n-6ffee7a877aa82b8","revision":1,"first_index":0,"fingerprint":"f6-dde9c6f2b382eac4","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*6^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":[["289","204","36"],["4913","5202","1836","216"],["83521","117912","62424","14688","1296"],["1419857","2505630","1768680","624240","110160","7776"],["24137569","51114852","45101340","21224160","5618160","793152","46656"],["410338673","1013777898","1073411892","631418760","222853680","47192544","5552064","279936"],["6975757441","19696256304","24330669552","17174590272","7577025120","2139395328","377540352","38071296","1679616"],["118587876497","376690901814","531798920208","437952051936","231856968672","81831871296","19254557952","2912454144","256981248","10077696"],["2015993900449","7115272589820","11300727054420","10635978404160","6569280779040","2782283624064","818318712960","165039068160","21843406080","1713208320","60466176"],["34271896307633","133055597429634","234803995464060","248615995197240","175493643668640","86714506283328","30605119864704","7715576436480","1361572312320","160184977920","11307174912","362797056"],["582622237229761","2467576534149576","4790001507466824","5635295891137440","4475087913550320","2527108468828416","1040574075399936","314795518608384","69440187928320","10892578498560","1153331841024","74010599424","2176782336"],["9904578032905937","45444534503921358","96235484831833464","124540039194137424","109888269877180080","69811371451384992","32852410094769408","11594968268742144","3069256306431744","601814962045440","84962112288768","8178171236352","481068896256","13060694016"],["168377826559400929","831984554764098708","1908670449164697036","2694593575291336992","2615340823076885904","1846122933936625344","977359200319389888","394228921137232896","121747166821792512","28646392193362944","5055245681181696","648801584750592","57247198654464","3108445175808","78364164096"],["2862423051509815793","15154004390346083610","37439304964384441860","57260113474940911080","60628355444055082320","47076134815383946272","27691844009049380160","12566046861249298560","4435075362793870080","1217471668217925120","257817529740266496","41361101027850240","4866011885629440","396326759915520","19982861844480","470184984576"],["48661191875666868481","274792612944942316128","727392210736612013280","1198057758860302139520","1374242723398581865920","1164064424525857580544","753218157046143140352","379773860695534356480","150792562334991582720","47307470536467947520","11687728014892081152","2250043895915053056","330888808222801920","35933626232340480","2717669210849280","127890315804672","2821109907456"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["529","12167","279841","6436343","148035889","3404825447","78310985281","1801152661463","41426511213649","952809757913927","21914624432020321","504036361936467383","11592836324538749809","266635235464391245607","6132610415680998648961"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A007318","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["289","204","36"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["17","6"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162042,"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":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-a19af80736e38760"]},{"id":162043,"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-a19af80736e38760"]},{"id":162044,"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-a19af80736e38760"]}],"region_exact":{},"incoming_edges":[{"id":"e-a19af80736e38760","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-6ffee7a877aa82b8","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":"ed2693aea67cd1c7749908e3f78c26f57001febff7703ba5dac7a918cb9455b5","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-1f2dec04776afb2b","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*6^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-d44d69569ae30996","kind":"oeis_match","node_a_id":"n-6ffee7a877aa82b8","node_b_id":null,"fingerprint":null,"oeis_id":"A009967","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":"A009967","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A009967","relation":"match_row_sums"}],"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:            289,             204,              36;\n  1:           4913,            5202,            1836,             216;\n  2:          83521,          117912,           62424,           14688,            1296;\n  3:        1419857,         2505630,         1768680,          624240,          110160,           7776;\n  4:       24137569,        51114852,        45101340,        21224160,         5618160,         793152,          46656;\n  5:      410338673,      1013777898,      1073411892,       631418760,       222853680,       47192544,        5552064,        279936;\n  6:     6975757441,     19696256304,     24330669552,     17174590272,      7577025120,     2139395328,      377540352,      38071296,       1679616;\n  7:   118587876497,    376690901814,    531798920208,    437952051936,    231856968672,    81831871296,    19254557952,    2912454144,     256981248,     10077696;\n  8:  2015993900449,   7115272589820,  11300727054420,  10635978404160,   6569280779040,  2782283624064,   818318712960,  165039068160,   21843406080,   1713208320,    60466176;\n  9: 34271896307633, 133055597429634, 234803995464060, 248615995197240, 175493643668640, 86714506283328, 30605119864704, 7715576436480, 1361572312320, 160184977920, 11307174912, 362797056;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (17+6*x)*P(n-1,x) for n >= 1, with P(0,x) = 289 + 204*x + 36*x^2.","mathematica":"P[0, x_] := 289 + 204*x + 36*x^2;\nP[n_, x_] := P[n, x] = Expand[(17 + 6*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] = 289 + 204*x + 36*x^2; for(n = 1, N, my(i = n + 1); v[i] = (17 + 6*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-1f2dec04776afb2b","steps":["basis A007318"],"edges":["e-a19af80736e38760"],"text":"A038320 \u2192 basis A007318"},"headline":"A038320 \u2192 basis A007318","chain":null}