{"id":"n-45b8d33479b2056d","revision":1,"first_index":1,"fingerprint":"f6-c933affcacfcef22","name":"basis_map_A013609 of Triangle read by rows: T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["4","2"],["11","14","4"],["29","62","40","8"],["76","230","240","104","16"],["199","780","1140","784","256","32"],["521","2508","4740","4528","2320","608","64"],["1364","7786","18096","22280","15760","6432","1408","128"],["3571","23578","65120","98504","89520","50208","17024","3200","256"],["9349","70090","224420","403472","449808","323232","150080","43520","7168","512"],["24476","205390","748320","1560752","2066848","1819104","1079680","427520","108288","15872","1024"],["64079","595032","2431320","5775424","8872240","9267776","6727168","3398400","1172736","263680","34816","2048"],["167761","1707864","7735704","20628160","36100720","43728704","37637376","23222016","10206720","3120640","630784","75776","4096"],["439204","4864082","24191520","71580464","140686240","194119776","193642368","141542400","75891456","29511680","8098816","1486848","163840","8192"],["1149851","13762790","74567020","242496272","529118928","820003104","931529280","788689920","500552448","237197312","82689024","20582400","3461120","352256","16384"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","6","29","139","666","3191","15289","73254","350981","1681651","8057274","38604719","184966321","886226886","4246168109"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013609","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["4","2"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["3","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":166413,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-0d48af5cb3d93d03"]},{"id":166414,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-0d48af5cb3d93d03"]},{"id":166415,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-0d48af5cb3d93d03"]},{"id":166416,"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-0d48af5cb3d93d03"]}],"region_exact":{},"incoming_edges":[{"id":"e-0d48af5cb3d93d03","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-45b8d33479b2056d","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":"bc78eeabc1302ced2c51c0ed78a30e41683446efbb16c5c2c1c92ec69c028ba5","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-318700a428d7b97b","role":"sequence","name":"Triangle read by rows: T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-c07727207acaa18e","kind":"oeis_match","node_a_id":"n-45b8d33479b2056d","node_b_id":null,"fingerprint":null,"oeis_id":"A030221","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A030221","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A030221","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-45b8d33479b2056d","oeis_id":"A030221","target":"row_sums","oeis_offset":0,"first_row":1,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":24,"oeis_terms_available":24,"term_count":200,"raw_size":14552}],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:    1;\n   2:    4,     2;\n   3:   11,    14,      4;\n   4:   29,    62,     40,      8;\n   5:   76,   230,    240,    104,     16;\n   6:  199,   780,   1140,    784,    256,     32;\n   7:  521,  2508,   4740,   4528,   2320,    608,     64;\n   8: 1364,  7786,  18096,  22280,  15760,   6432,   1408,   128;\n   9: 3571, 23578,  65120,  98504,  89520,  50208,  17024,  3200,  256;\n  10: 9349, 70090, 224420, 403472, 449808, 323232, 150080, 43520, 7168, 512;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -P(n-2,x) + (3+2*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 4 + 2*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 4 + 2*x;\nP[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (3 + 2*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 4 + 2*x; for(n = 3, N, my(i = n); v[i] = -v[i - 2] + (3 + 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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-318700a428d7b97b","steps":["basis A013609"],"edges":["e-0d48af5cb3d93d03"],"text":"A111125 \u2192 basis A013609"},"headline":"A111125 \u2192 basis A013609","chain":null}