{"id":"n-cffba19abf0eaf93","revision":1,"first_index":0,"fingerprint":"f6-b4b323c4cf375c4a","name":"basis_map_A013611 of A Riordan array of coordination sequences.","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"],["3","4"],["7","24","16"],["17","92","144","64"],["41","304","768","768","256"],["99","932","3264","5248","3840","1024"],["239","2728","12240","27392","32000","18432","4096"],["577","7740","42384","122048","198400","181248","86016","16384"],["1393","21472","138880","489984","1026560","1302528","974848","393216","65536"],["3363","58564","436992","1827072","4699648","7686144","7970816","5046272","1769472","262144"],["8119","157624","1333008","6447616","19694080","39579648","52871168","46268416","25362432","7864320","1048576"],["19601","419740","3967760","21802304","77186560","184420352","302776320","340951040","257753088","124518400","34603008","4194304"],["47321","1107984","11577984","71255296","287066880","795942912","1554423808","2150760448","2089746432","1389363200","599785472","150994944","16777216"],["114243","2903396","33234624","226495872","1023550720","3233319936","7333076992","12071272448","14404091904","12293242880","7289700352","2843738112","654311424","67108864"],["275807","7561032","94092752","703497472","3525172992","12505053184","32337629184","61843308544","87786061824","91951202304","69909610496","37396414464","13304332288","2818572288","268435456"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","7","47","317","2137","14407","97127","654797","4414417","29760487","200635007","1352612477","9118849897","61476161767","414451220087"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013611","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["3","4"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","4"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["2","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":167750,"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-14cc72d2ba3c3aa2"]},{"id":167751,"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-14cc72d2ba3c3aa2"]},{"id":167752,"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-14cc72d2ba3c3aa2"]},{"id":167753,"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-14cc72d2ba3c3aa2"]}],"region_exact":{},"incoming_edges":[{"id":"e-14cc72d2ba3c3aa2","operation":{"map":"A013611","operation":"basis_map"},"operation_detail":"A013611","conclusion_id":"n-cffba19abf0eaf93","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":"90f92ea3a66272cb48ad5d40326b0ca4a7a11b2ccf3cf115ec67e1644ff47b56","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-ec35d1145e9a1211","role":"sequence","name":"A Riordan array of coordination sequences.","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-f0f24b2ce4bbb3d4","kind":"oeis_match","node_a_id":"n-cffba19abf0eaf93","node_b_id":null,"fingerprint":null,"oeis_id":"A126528","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":"A126528","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A126528","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-cffba19abf0eaf93","oeis_id":"A126528","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":22,"oeis_terms_available":22,"term_count":200,"raw_size":17485}],"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,     4;\n  2:    7,    24,     16;\n  3:   17,    92,    144,      64;\n  4:   41,   304,    768,     768,     256;\n  5:   99,   932,   3264,    5248,    3840,    1024;\n  6:  239,  2728,  12240,   27392,   32000,   18432,    4096;\n  7:  577,  7740,  42384,  122048,  198400,  181248,   86016,   16384;\n  8: 1393, 21472, 138880,  489984, 1026560, 1302528,  974848,  393216,   65536;\n  9: 3363, 58564, 436992, 1827072, 4699648, 7686144, 7970816, 5046272, 1769472, 262144;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+4*x)*P(n-2,x) + (2+4*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + 4*x;\nP[n_, x_] := P[n, x] = Expand[(1 + 4*x)*P[n - 2, x] + (2 + 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] = 3 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (1 + 4*x)*v[i - 2] + (2 + 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-ec35d1145e9a1211","steps":["basis A013611"],"edges":["e-14cc72d2ba3c3aa2"],"text":"A113413 \u2192 basis A013611"},"headline":"A113413 \u2192 basis A013611","chain":null}
