{"id":"n-1690f654ed66c83b","revision":0,"first_index":0,"fingerprint":"f6-9d2ce69001c67321","name":"basis_map_A013610 of A triangular array related to A077028 and distributing the values of A007582.","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"],["3","6"],["10","39","36"],["36","204","369","216"],["136","996","2646","3051","1296"],["528","4704","16344","27864","23409","7776"],["2080","21744","92880","208440","260010","171315","46656"],["8256","98880","499824","1386720","2285820","2242404","1213785","279936"],["32896","443712","2586528","8530704","17440920","22664124","18299358","8400267","1679616"],["131328","1969152","12992256","49593600","120857184","195193152","209100528","143362224","57113505","10077696"],["524800","8657664","63742464","276182784","780663744","1505425824","2007071136","1827737136","1088509266","382972131","60466176"],["2098176","37764096","306812160","1486771200","4778663040","10703726208","17058390048","19351275840","15319278900","8062550460","2539638441","362797056"],["8390656","163611648","1453579776","7786817280","28031857920","71479698048","132445139904","179743896288","177376110120","124161544980","58532557446","16688841723","2176782336"],["33558528","704716800","6795380736","39865706496","158838071040","454085305344","958615637760","1513590997248","1787915855088","1560868198560","979083910728","417946460040","108839648241","13060694016"],["134225920","3020058624","31408386048","200227203072","874516870656","2769289585920","6558814259712","11805817630464","16232939195616","16970742751248","13281409730160","7546234034664","2943023535738","705279071187","78364164096"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","9","85","825","8125","80625","803125","8015625","80078125","800390625","8001953125","80009765625","800048828125","8000244140625","80001220703125"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013610","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["3","6"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-8","-24","-18"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["6","9"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":202140,"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-3ebfbb0957769697"]},{"id":202141,"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-3ebfbb0957769697"]},{"id":202142,"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-3ebfbb0957769697"]},{"id":202143,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-3ebfbb0957769697"]}],"region_exact":{},"incoming_edges":[{"id":"e-3ebfbb0957769697","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-1690f654ed66c83b","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":"aa92df9e39100081e54666790c86a675e749a4668c7a6008e506638fed80bfea","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-cf9f646a76b9c283","role":"sequence","name":"A triangular array related to A077028 and distributing the values of A007582.","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:      3,       6;\n  2:     10,      39,       36;\n  3:     36,     204,      369,      216;\n  4:    136,     996,     2646,     3051,      1296;\n  5:    528,    4704,    16344,    27864,     23409,      7776;\n  6:   2080,   21744,    92880,   208440,    260010,    171315,     46656;\n  7:   8256,   98880,   499824,  1386720,   2285820,   2242404,   1213785,    279936;\n  8:  32896,  443712,  2586528,  8530704,  17440920,  22664124,  18299358,   8400267,  1679616;\n  9: 131328, 1969152, 12992256, 49593600, 120857184, 195193152, 209100528, 143362224, 57113505, 10077696;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(8+24*x+18*x^2)*P(n-2,x) + (6+9*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 6*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + 6*x;\nP[n_, x_] := P[n, x] = Expand[-(8 + 24*x + 18*x^2)*P[n - 2, x] + (6 + 9*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; v[2] = 3 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = -(8 + 24*x + 18*x^2)*v[i - 2] + (6 + 9*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-cf9f646a76b9c283","steps":["basis A013610"],"edges":["e-3ebfbb0957769697"],"text":"A110552 \u2192 basis A013610"},"headline":"A110552 \u2192 basis A013610","chain":null}