{"id":"n-22be1063864c8769","revision":0,"first_index":0,"fingerprint":"f6-c64000922872a417","name":"basis_map_A013611 of Triangle read by rows: T(n,k) is the number of k-matchings in the n-sunlet graph (0 <= k <= n).","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"],["2","4"],["6","24","16"],["14","84","144","64"],["34","272","736","768","256"],["82","820","3040","5120","3840","1024"],["198","2376","11184","26112","31488","18432","4096"],["478","6692","38192","114240","191744","179200","86016","16384"],["1154","18464","123840","452096","976384","1269760","966656","393216","65536"],["2786","50148","386496","1666560","4409856","7391232","7815168","5013504","1769472","262144"],["6726","134520","1171280","5826560","18270720","37597184","51240960","45547520","25231360","7864320","1048576"],["16238","357236","3467728","19550784","70923776","173307904","290250752","332333056","254476288","123994112","34603008","4194304"],["39202","940848","10073760","63484160","261627648","740990976","1475362816","2076180480","2045706240","1374683136","597688320","150994944","16777216"],["94642","2460692","28807584","200685056","926318848","2985495552","6898171904","11547148288","13979942912","12074090496","7224688640","2835349504","654311424","67108864"],["228486","6397608","81295088","620379648","3170942208","11463768064","30177652736","58664615936","84498907136","89625460736","68842160128","37111201792","13270777856","2818572288","268435456"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","6","46","306","2066","13926","93886","632946","4267106","28767366","193939726","1307475186","8814549746","59424674406","400620795166"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013611","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["2","4"],["6","24","16"]],"recurrence":{"denominator":{"coeffs":[["-2"],["1"]]},"terms":[{"coeff":{"coeffs":[["-2","-8"],["1","4"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-4","-8"],["2","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":204509,"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-51c207fadbfbd40d"]},{"id":204510,"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-51c207fadbfbd40d"]},{"id":204511,"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-51c207fadbfbd40d"]},{"id":204512,"position":3,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2","region_exact":null,"edges":["e-51c207fadbfbd40d"]}],"region_exact":{},"incoming_edges":[{"id":"e-51c207fadbfbd40d","operation":{"map":"A013611","operation":"basis_map"},"operation_detail":"A013611","conclusion_id":"n-22be1063864c8769","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":"565781bd2aff108be97d182fd012b537ba9439968a7c21021200144670a29b08","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-9d2e609ab19d361f","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of k-matchings in the n-sunlet graph (0 <= k <= n).","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:    2,     4;\n  2:    6,    24,     16;\n  3:   14,    84,    144,      64;\n  4:   34,   272,    736,     768,     256;\n  5:   82,   820,   3040,    5120,    3840,    1024;\n  6:  198,  2376,  11184,   26112,   31488,   18432,    4096;\n  7:  478,  6692,  38192,  114240,  191744,  179200,   86016,   16384;\n  8: 1154, 18464, 123840,  452096,  976384, 1269760,  966656,  393216,   65536;\n  9: 2786, 50148, 386496, 1666560, 4409856, 7391232, 7815168, 5013504, 1769472, 262144;\n","recurrence_sentence":"The row polynomials satisfy (n-2)*P(n,x) = (n-2+(4*n-8)*x)*P(n-2,x) + (2*n-4+(4*n-8)*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 2 + 4*x, P(2,x) = 6 + 24*x + 16*x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 4*x;\nP[2, x_] := 6 + 24*x + 16*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((n - 2 + (4*n - 8)*x)*P[n - 2, x] + (2*n - 4 + (4*n - 8)*x)*P[n - 1, x])/(n - 2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 4*x; v[3] = 6 + 24*x + 16*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((n - 2 + (4*n - 8)*x)*v[i - 2] + (2*n - 4 + (4*n - 8)*x)*v[i - 1])/(n - 2)); 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-9d2e609ab19d361f","steps":["basis A013611"],"edges":["e-51c207fadbfbd40d"],"text":"A102413 \u2192 basis A013611"},"headline":"A102413 \u2192 basis A013611","chain":null}
