{"id":"n-db0f0bf32d2c2b43","revision":1,"first_index":0,"fingerprint":"f6-a30fbdfa808edb9b","name":"basis_map_A013610 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*12^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"],["23","36"],["529","1656","1296"],["12167","57132","89424","46656"],["279841","1752048","4113504","4292352","1679616"],["6436343","50371380","157684320","246810240","193155840","60466176"],["148035889","1390250088","5440109040","11353271040","13327752960","8344332288","2176782336"],["3404825447","37305044028","175171511088","456969159360","715256075520","671718749184","350461956096","78364164096"],["78310985281","980589728736","5371926340032","16816465064448","32901779473920","41198749949952","32242499960832","14419006193664","2821109907456"],["1801152661463","25372759231044","158855536055232","580168044723456","1362133670220288","2132035309910016","2224732497297408","1492367141044224","583969750843392","101559956668416"],["41426511213649","648414958126680","4567096661587920","19062664326627840","52215124025111040","98073624255860736","127922118594600960","114414814146723840","67156521346990080","23358790033735680","3656158440062976"],["952809757913927","16404898440605004","128386161709082640","602856759329605440","1887203768336156160","4135437822788794368","6472859200886808576","7236736994780282880","5663533300262830080","2954886939267563520","925008085335932928","131621703842267136"],["21914624432020321","411613815418816464","3543458063170680864","18487607286107900160","65108530007597387520","163054405584243892224","297751523240793194496","399467882111871614976","390783797718135275520","271849598412615843840","127651115776358744064","36327590260465729536","4738381338321616896"],["504036361936467383","10256044234185510228","96317632808003052576","552779457854626214784","2163050052474624318720","6094158408711115471872","12718243635571023593472","19906816125241602146304","23368871103544489476096","20320757481343034327040","12722561205710421491712","5430974743939626565632","1416776020158163451904","170581728179578208256"],["11592836324538749809","254034326415979561032","2584523147014748577456","16181362311744512832768","69650211689682903062784","218035445289442131326976","511909306331733699637248","915713541761113698729984","1254129415890220935217152","1308656781798491410661376","1024166177059688930082816","582924622516186584711168","228100939245464315756544","54927316473824183058432","6140942214464815497216"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","59","3481","205379","12117361","714924299","42180533641","2488651484819","146830437604321","8662995818654939","511116753300641401","30155888444737842659","1779197418239532716881","104972647676132430295979","6193386212891813387462761"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013610","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["23","36"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":167552,"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-ffff6b452715bdc2"]},{"id":167553,"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-ffff6b452715bdc2"]},{"id":167554,"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-ffff6b452715bdc2"]}],"region_exact":{},"incoming_edges":[{"id":"e-ffff6b452715bdc2","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-db0f0bf32d2c2b43","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":"7f65de160eb887301a3d4451a5d1b21578b9ccf37b2434b89aa738cdced2098d","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-9e472291b944d56e","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*12^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":[],"oeis":[],"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:             1;\n  1:            23,             36;\n  2:           529,           1656,            1296;\n  3:         12167,          57132,           89424,           46656;\n  4:        279841,        1752048,         4113504,         4292352,          1679616;\n  5:       6436343,       50371380,       157684320,       246810240,        193155840,         60466176;\n  6:     148035889,     1390250088,      5440109040,     11353271040,      13327752960,       8344332288,       2176782336;\n  7:    3404825447,    37305044028,    175171511088,    456969159360,     715256075520,     671718749184,     350461956096,      78364164096;\n  8:   78310985281,   980589728736,   5371926340032,  16816465064448,   32901779473920,   41198749949952,   32242499960832,   14419006193664,   2821109907456;\n  9: 1801152661463, 25372759231044, 158855536055232, 580168044723456, 1362133670220288, 2132035309910016, 2224732497297408, 1492367141044224, 583969750843392, 101559956668416;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (23+36*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[(23 + 36*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] = (23 + 36*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-9e472291b944d56e","steps":["basis A013610"],"edges":["e-ffff6b452715bdc2"],"text":"A038326 \u2192 basis A013610"},"headline":"A038326 \u2192 basis A013610","chain":null}