{"id":"n-27bb5540b6efd4ae","revision":1,"first_index":0,"fingerprint":"f6-98172a07d2808c64","name":"basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":false,"rows":[["1"],["0","1"],["-1","0","1"],["1","-2","0","1"],["0","2","-3","0","1"],["-1","1","3","-4","0","1"],["1","-4","3","4","-5","0","1"],["0","3","-9","6","5","-6","0","1"],["-1","2","5","-16","10","6","-7","0","1"],["1","-6","9","6","-25","15","7","-8","0","1"],["0","4","-18","24","5","-36","21","8","-9","0","1"],["-1","3","7","-39","50","1","-49","28","9","-10","0","1"],["1","-8","18","4","-70","90","-7","-64","36","10","-11","0","1"],["0","5","-30","60","-15","-111","147","-20","-81","45","11","-12","0","1"],["-1","4","9","-76","149","-64","-161","224","-39","-100","55","12","-13","0","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","1","0","0","0","0","0","0","0","0","0","0","0","0","0"],"nonn":false,"definitions":[{"position":0,"source":"derived:basis_map_coefficients_of:oeis_catalog:A030528","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["0","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-1","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-1","1"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":160012,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-3, 1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-3"},"upper":{"closed":true,"value":"1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-3, 1]","region_exact":null,"edges":["e-51281bad32c8f5cd"]},{"id":160013,"position":1,"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-51281bad32c8f5cd"]}],"region_exact":{},"incoming_edges":[{"id":"e-51281bad32c8f5cd","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"-1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-27bb5540b6efd4ae","premise_count":1,"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"22655a469341f2e8874ffb79d69a896ca01ec1cd8fcce4b96cf6b350cdb87856","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-3"},"upper":{"closed":true,"value":"1"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-21854f4257effb01","role":"sequence","name":"Triangle read by rows: T(n,k) = binomial(k,n-k).","uses":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-4"},"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4, 0]"}]}],"used_by":[{"id":"e-ba7f4c6c74dd05c0","operation":"basis_map","operation_detail":"A007318","conclusion_id":"n-21854f4257effb01","conclusion_name":"Triangle read by rows: T(n,k) = binomial(k,n-k)."}],"used_by_total":1,"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:  0,  1;\n  2: -1,  0,  1;\n  3:  1, -2,  0,   1;\n  4:  0,  2, -3,   0,   1;\n  5: -1,  1,  3,  -4,   0,  1;\n  6:  1, -4,  3,   4,  -5,  0,  1;\n  7:  0,  3, -9,   6,   5, -6,  0,  1;\n  8: -1,  2,  5, -16,  10,  6, -7,  0, 1;\n  9:  1, -6,  9,   6, -25, 15,  7, -8, 0, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(1-x)*P(n-1,x) - (1-x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := x;\nP[n_, x_] := P[n, x] = Expand[-(1 - x)*P[n - 1, x] - (1 - x)*P[n - 2, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = x; for(n = 2, N, my(i = n + 1); v[i] = -(1 - x)*v[i - 1] - (1 - x)*v[i - 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-21854f4257effb01","steps":["x \u21a6 x\u22121"],"edges":["e-51281bad32c8f5cd"],"text":"A030528 \u2192 x \u21a6 x\u22121"},"headline":"A030528 \u2192 x \u21a6 x\u22121","chain":null}
