{"id":"n-03750a627f7ed2dc","revision":0,"first_index":0,"fingerprint":"f6-80fa4861659be6fe","name":"basis_map_A013611 of Triangular array of generalized Narayana numbers T(n,k) = 2*binomial(n+1,k)* binomial(n-2,k-1)/(n+1) for n >= 1 and 0 <= k <= n-1, read by rows.","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","8"],["5","32","48"],["14","120","320","256"],["42","448","1680","2560","1280"],["132","1680","8064","17920","17920","6144"],["429","6336","36960","107520","161280","114688","28672"],["1430","24024","164736","591360","1182720","1290240","688128","131072"],["4862","91520","720720","3075072","7687680","11354112","9461760","3932160","589824"],["16796","350064","3111680","15375360","46126080","86102016","98402304","64880640","21626880","2621440"],["58786","1343680","13302432","74680320","261381120","590413824","861020160","787218432","421724160","115343360","11534336"],["208012","5173168","56434560","354731520","1418926080","3763888128","6691356672","7872184320","5904138240","2624061440","599785472","50331648"],["742900","19969152","237965728","1655413760","7449361920","22702817280","47675916288","68825382912","66913566720","41984983040","15744368640","3053453312","218103808"],["2674440","77261600","998457600","7614903296","38074516480","131108769792","317839441920","544867614720","653841137664","535308533760","285497884672","91603599360","15267266560","939524096"],["9694845","299537280","4172126400","34613196800","190372582400","731030716416","2010334470144","3995695841280","5721109954560","5811921223680","4068344856576","1868713426944","519087063040","75161927680","4026531840"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","10","85","710","6010","51860","455885","4073710","36917710","338613260","3138020610","29341219260","276482045460","2623041585960","25034893149885"],"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","8"]],"recurrence":{"denominator":{"coeffs":[["-2"],["3"],["2"]]},"terms":[{"coeff":{"coeffs":[["0","0","32"],["0","0","48"],["0","0","-32"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-2","8"],["0","0"],["8","16"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":204649,"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-eeceffe8f22b66c0"]},{"id":204650,"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-eeceffe8f22b66c0"]},{"id":204651,"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-eeceffe8f22b66c0"]},{"id":204652,"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-eeceffe8f22b66c0"]}],"region_exact":{},"incoming_edges":[{"id":"e-eeceffe8f22b66c0","operation":{"map":"A013611","operation":"basis_map"},"operation_detail":"A013611","conclusion_id":"n-03750a627f7ed2dc","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":"e904dee011561832dbdb224df954e461c2465e21a890ca21e33a2a3ff0da95d5","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-8a65e165240c40e0","role":"sequence","name":"Triangular array of generalized Narayana numbers T(n,k) = 2*binomial(n+1,k)* binomial(n-2,k-1)/(n+1) for n >= 1 and 0 <= k <= n-1, read by rows.","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,      8;\n  2:     5,     32,      48;\n  3:    14,    120,     320,      256;\n  4:    42,    448,    1680,     2560,     1280;\n  5:   132,   1680,    8064,    17920,    17920,     6144;\n  6:   429,   6336,   36960,   107520,   161280,   114688,    28672;\n  7:  1430,  24024,  164736,   591360,  1182720,  1290240,   688128,   131072;\n  8:  4862,  91520,  720720,  3075072,  7687680, 11354112,  9461760,  3932160,   589824;\n  9: 16796, 350064, 3111680, 15375360, 46126080, 86102016, 98402304, 64880640, 21626880, 2621440;\n","recurrence_sentence":"The row polynomials satisfy (2*n^2+3*n-2)*P(n,x) = -(32*n^2-48*n-32)*x^2*P(n-2,x) + (8*n^2-2+(16*n^2+8)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 8*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 8*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(32*n^2 - 48*n - 32)*x^2*P[n - 2, x] + (8*n^2 - 2 + (16*n^2 + 8)*x)*P[n - 1, x])/(2*n^2 + 3*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 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = (-(32*n^2 - 48*n - 32)*x^2*v[i - 2] + (8*n^2 - 2 + (16*n^2 + 8)*x)*v[i - 1])/(2*n^2 + 3*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-8a65e165240c40e0","steps":["basis A013611"],"edges":["e-eeceffe8f22b66c0"],"text":"A281260 \u2192 basis A013611"},"headline":"A281260 \u2192 basis A013611","chain":null}
