{"id":"n-32899adace0e98a1","revision":1,"first_index":0,"fingerprint":"f6-74aba9cdd78cf583","name":"basis_map_A013611 of A Jacobsthal triangle.","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":false,"rows":[["1"],["1","4"],["3","16","16"],["5","44","112","64"],["11","120","448","640","256"],["21","292","1504","3456","3328","1024"],["43","704","4528","14336","22784","16384","4096"],["85","1628","12688","51392","114432","136192","77824","16384"],["171","3720","33888","167040","480256","808960","761856","360448","65536"],["341","8340","87168","506880","1788416","3917824","5242880","4063232","1638400","262144"],["683","18512","218064","1460736","6112768","16531456","28909568","31850496","20905984","7340032","1048576"],["1365","40652","533168","4044096","19587584","63125504","136863744","197558272","184090624","104595456","32505856","4194304"],["2731","88600","1280000","10842752","59675392","223440896","579436544","1039990784","1270939648","1022885888","511705088","142606336","16777216"],["5461","191748","3025952","28316288","174574336","745094144","2251931648","4847763456","7379550208","7788560384","5505024000","2457862144","620756992","67108864"],["10923","412640","7061744","72345600","493932288","2367676416","8178708480","20570963968","37632409600","49520050176","45865762816","28856811520","11626610688","2684354560","268435456"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","35","225","1475","9625","62875","410625","2681875","17515625","114396875","747140625","4879671875","31869765625","208145546875"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013611","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["1","4"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","8"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":167690,"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-4cbbf4b06abf62f6"]},{"id":167691,"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-4cbbf4b06abf62f6"]},{"id":167692,"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-4cbbf4b06abf62f6"]}],"region_exact":{},"incoming_edges":[{"id":"e-4cbbf4b06abf62f6","operation":{"map":"A013611","operation":"basis_map"},"operation_detail":"A013611","conclusion_id":"n-32899adace0e98a1","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":"97b93762b0d803510387a6e5592ac4d0d0ca45fe15b86f6b91065966c91009cf","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-ba196dcd01800741","role":"sequence","name":"A Jacobsthal triangle.","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-6c5b53b015ae5a0c","kind":"oeis_match","node_a_id":"n-32899adace0e98a1","node_b_id":null,"fingerprint":null,"oeis_id":"A180250","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":1,"oeis_term_position":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A180250","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A180250","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:   1;\n  1:   1,    4;\n  2:   3,   16,    16;\n  3:   5,   44,   112,     64;\n  4:  11,  120,   448,    640,     256;\n  5:  21,  292,  1504,   3456,    3328,    1024;\n  6:  43,  704,  4528,  14336,   22784,   16384,    4096;\n  7:  85, 1628, 12688,  51392,  114432,  136192,   77824,   16384;\n  8: 171, 3720, 33888, 167040,  480256,  808960,  761856,  360448,   65536;\n  9: 341, 8340, 87168, 506880, 1788416, 3917824, 5242880, 4063232, 1638400, 262144;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2+8*x)*P(n-2,x) + (1+4*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 4*x;\nP[n_, x_] := P[n, x] = Expand[(2 + 8*x)*P[n - 2, x] + (1 + 4*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] = 1 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (2 + 8*x)*v[i - 2] + (1 + 4*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-ba196dcd01800741","steps":["basis A013611"],"edges":["e-4cbbf4b06abf62f6"],"text":"A113953 \u2192 basis A013611"},"headline":"A113953 \u2192 basis A013611","chain":null}