{"id":"n-39736a4ef132a31d","revision":1,"first_index":0,"fingerprint":"f6-611b245e25712c49","name":"basis_map_A013609 of Triangle of coefficients of Chebyshev's S(n,x+5) polynomials (exponents of x in increasing order).","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"],["6","2"],["35","24","4"],["204","212","72","8"],["1189","1656","852","192","16"],["6930","12102","8352","2848","480","32"],["40391","84816","73464","33600","8560","1152","64"],["235416","577576","602064","345680","118080","24000","2688","128"],["1372105","3851472","4694072","3244608","1391280","379008","64064","6144","256"],["7997214","25275466","35265312","28510112","14718816","5032608","1139712","164864","13824","512"],["46611179","163795752","257448732","238346688","143941840","59254272","16839424","3262464","412416","30720","1024"],["271669860","1050721404","1837018584","1916467480","1325625600","638376704","218405376","53088768","8985600","1008640","67584","2048"],["1583407981","6683872392","12866105580","14934495360","11642746720","6422257152","2570346240","752080896","159678720","23992320","2421760","147456","4096"],["9228778026","42219328910","88727359680","113422715840","98399845440","61180659648","28048186368","9600089088","2453248512","462302720","62447616","5726208","319488","8192"],["53789260175","265089657120","603936710320","843056519040","805601757600","557461391616","288080091264","112944826368","33759990528","7656321024","1296869376","159105024","13365248","688128","16384"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","8","63","496","3905","30744","242047","1905632","15003009","118118440","929944511","7321437648","57641556673","453811015736","3572846569215"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013609","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["6","2"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["6","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":166687,"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-63a398c0ccd31fef"]},{"id":166688,"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-63a398c0ccd31fef"]},{"id":166689,"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-63a398c0ccd31fef"]},{"id":166690,"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-63a398c0ccd31fef"]}],"region_exact":{},"incoming_edges":[{"id":"e-63a398c0ccd31fef","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-39736a4ef132a31d","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":"618f672b402e17c689cc0f599f7e40247ef64ba6cac763a17fcf89ef7167f42c","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-c2b7c74f8491f01b","role":"sequence","name":"Triangle of coefficients of Chebyshev's S(n,x+5) polynomials (exponents of x in increasing order).","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-3d4bc8bee9518ac7","kind":"oeis_match","node_a_id":"n-39736a4ef132a31d","node_b_id":null,"fingerprint":null,"oeis_id":"A001090","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":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A001090","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A001090","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:       6,        2;\n  2:      35,       24,        4;\n  3:     204,      212,       72,        8;\n  4:    1189,     1656,      852,      192,       16;\n  5:    6930,    12102,     8352,     2848,      480,      32;\n  6:   40391,    84816,    73464,    33600,     8560,    1152,      64;\n  7:  235416,   577576,   602064,   345680,   118080,   24000,    2688,    128;\n  8: 1372105,  3851472,  4694072,  3244608,  1391280,  379008,   64064,   6144,   256;\n  9: 7997214, 25275466, 35265312, 28510112, 14718816, 5032608, 1139712, 164864, 13824, 512;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -P(n-2,x) + (6+2*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 6 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 6 + 2*x;\nP[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (6 + 2*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] = 6 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -v[i - 2] + (6 + 2*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-c2b7c74f8491f01b","steps":["basis A013609"],"edges":["e-63a398c0ccd31fef"],"text":"A207824 \u2192 basis A013609"},"headline":"A207824 \u2192 basis A013609","chain":null}