{"id":"n-f475a0646d719a92","revision":1,"first_index":0,"fingerprint":"f6-c03785f2011a6138","name":"basis_map_A013609 of Triangular array binomial(2*n-k, k), k=0..n, n >= 0.","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","2"],["5","10","4"],["13","40","36","8"],["34","142","204","112","16"],["89","470","932","840","320","32"],["233","1488","3768","4752","3040","864","64"],["610","4570","14088","22880","20720","10080","2240","128"],["1597","13730","49892","99216","116560","81088","31360","5632","256"],["4181","40568","169788","399704","577952","525024","292992","92928","13824","512"],["10946","118310","560436","1524560","2619680","2942688","2157120","995328","264960","33280","1024"],["28657","341422","1805764","5574296","11101360","14871136","13537216","8249472","3218688","732160","78848","2048"],["75025","976800","5705520","19706400","44641600","69499200","75689600","57502720","29779200","9996800","1971200","184320","4096"],["196418","2774450","17737680","67778240","172128800","305490240","387638400","353383680","228983040","102502400","30018560","5191680","425984","8192"],["514229","7832122","54398020","227849760","641210080","1278094720","1852623360","1972446720","1537935360","864314880","338990080","87592960","13418496","974848","16384"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","19","97","508","2683","14209","75316","399331","2117473","11228332","59541067","315732481","1674257764","8878212019"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013609","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["2","2"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-1","-4","-4"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["3","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":166347,"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-dfc51938f5f39b6d"]},{"id":166348,"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-dfc51938f5f39b6d"]},{"id":166349,"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-dfc51938f5f39b6d"]},{"id":166350,"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-dfc51938f5f39b6d"]}],"region_exact":{},"incoming_edges":[{"id":"e-dfc51938f5f39b6d","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-f475a0646d719a92","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":"58db653d0430c31b7c88c66a7a6609605d4fc07ab65b814a195f29c75eec22d0","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-6758fe7729845be8","role":"sequence","name":"Triangular array binomial(2*n-k, k), k=0..n, n >= 0.","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,     2;\n  2:    5,    10,      4;\n  3:   13,    40,     36,      8;\n  4:   34,   142,    204,    112,     16;\n  5:   89,   470,    932,    840,    320,     32;\n  6:  233,  1488,   3768,   4752,   3040,    864,     64;\n  7:  610,  4570,  14088,  22880,  20720,  10080,   2240,   128;\n  8: 1597, 13730,  49892,  99216, 116560,  81088,  31360,  5632,   256;\n  9: 4181, 40568, 169788, 399704, 577952, 525024, 292992, 92928, 13824, 512;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(1+4*x+4*x^2)*P(n-2,x) + (3+4*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 2*x;\nP[n_, x_] := P[n, x] = Expand[-(1 + 4*x + 4*x^2)*P[n - 2, x] + (3 + 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] = 2 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(1 + 4*x + 4*x^2)*v[i - 2] + (3 + 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-6758fe7729845be8","steps":["basis A013609"],"edges":["e-dfc51938f5f39b6d"],"text":"A054142 \u2192 basis A013609"},"headline":"A054142 \u2192 basis A013609","chain":null}
