{"id":"n-ff3f26cc45b3d80a","revision":1,"first_index":0,"fingerprint":"f6-ca6d936c14642db2","name":"mobius_x_over_x_plus_one of Triangle read by rows: T(n,k) is the number of Schroeder paths of semilength n containing exactly k peaks but no peaks at level one (n >= 1; 0 <= k <= n-1).","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","3"],["5","15","11"],["14","63","93","45"],["42","252","560","546","197"],["132","990","2940","4320","3140","903"],["429","3861","14355","28215","30915","17901","4279"],["1430","15015","67067","165165","242165","211365","101675","20793"],["4862","58344","304304","900900","1655654","1933932","1402016","576708","103049"],["16796","226746","1352520","4678128","10339056","15140034","14688492","9103536","3270438","518859"],["58786","881790","5920590","23430420","60518640","106594488","129653160","107526510","58188585","18553275","2646723"],["208012","3432198","25617130","114157890","337463940","694796256","1016590344","1056991320","765316860","367494435","105324681","13648869"],["742900","13372200","109830336","544248540","1812136590","4270848120","7305252240","9137235888","8293857660","5327906760","2299160160","598407750","71039373"],["2674440","52151580","467432680","2549632800","9442140136","25069264506","49091516760","71786574288","78385689096","63121702644","36435419592","14275737216","3402883848","372693519"],["9694845","203591745","1977414075","11773219185","48002149195","141775518885","312798513027","523691390079","668552103219","647614352385","468563894799","245538217197","88091707431","19367988633","1968801519"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","31","215","1597","12425","99955","824675","6939769","59334605","513972967","4501041935","39784038517","354455513105","3179928556219"],"nonn":true,"definitions":[{"position":0,"source":"derived:mobius_x_over_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["2","4","2"],["1","2","1"]]},"terms":[{"coeff":{"coeffs":[["2","4","-2","-8","-4"],["4","17","26","17","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-2","-10","-18","-14","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":164908,"position":0,"claim":"real_rooted","from_index":0,"region_key":"[-1, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in [-1, 0]","region_exact":null,"edges":["e-38a5f21f7e3f9e41"]},{"id":164909,"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-38a5f21f7e3f9e41"]},{"id":164910,"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-38a5f21f7e3f9e41"]}],"region_exact":{},"incoming_edges":[{"id":"e-38a5f21f7e3f9e41","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"mobius\",\"a\":\"1\",\"b\":\"0\",\"c\":\"1\",\"e\":\"1\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":1}}"},"operation_detail":"mobius, shift 0","conclusion_id":"n-ff3f26cc45b3d80a","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":"fTransform","lean_declarations":["RealRooted.splits_fPolynomial"],"certificate_hash":"84a414d35594a54646a55dd2d96113fe6b7fadda0fe4a402e61932fca813f655","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-57b4b14604f4c0d7","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of Schroeder paths of semilength n containing exactly k peaks but no peaks at level one (n >= 1; 0 <= k <= n-1).","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-ed5441e8f99f3414","kind":"fingerprint_related","node_a_id":"n-07af2e6e3c9f0c77","node_b_id":"n-ff3f26cc45b3d80a","fingerprint":"f6-ca6d936c14642db2","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-ca6d936c14642db2"},"other_id":"n-07af2e6e3c9f0c77","other":{"id":"n-07af2e6e3c9f0c77","name":"basis_map_A007318 of Triangle read by rows: T(n, k) is the number of diagonal dissections of a convex n-gon into k+1 regions.","first_index":0,"fingerprint":"f6-ca6d936c14642db2","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","2"],["11","15","5"]],"card_rows":[["1"],["3","2"],["11","15","5"],["45","93","63","14"]],"row_sums_preview":["1","5","31","215","1597","12425","99955","824675"],"regions":["(-inf, 0]","(-2, -1)"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-66e2c3448245b0b7","steps":["x \u21a6 x+1"],"edges":["e-190c283dd8b47413"],"text":"A033282 \u2192 x \u21a6 x+1"},"headline":"A033282 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-4db7680e465d8389","kind":"oeis_match","node_a_id":"n-ff3f26cc45b3d80a","node_b_id":null,"fingerprint":null,"oeis_id":"A269730","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":1,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A269730","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A269730","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-ff3f26cc45b3d80a","oeis_id":"A269730","target":"row_sums","oeis_offset":1,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":23,"oeis_terms_available":23,"term_count":200,"raw_size":20321}],"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,      3;\n  2:     5,     15,      11;\n  3:    14,     63,      93,      45;\n  4:    42,    252,     560,     546,      197;\n  5:   132,    990,    2940,    4320,     3140,      903;\n  6:   429,   3861,   14355,   28215,    30915,    17901,     4279;\n  7:  1430,  15015,   67067,  165165,   242165,   211365,   101675,   20793;\n  8:  4862,  58344,  304304,  900900,  1655654,  1933932,  1402016,  576708,  103049;\n  9: 16796, 226746, 1352520, 4678128, 10339056, 15140034, 14688492, 9103536, 3270438, 518859;\n","recurrence_sentence":"The row polynomials satisfy (n+2+(2*n+4)*x+(n+2)*x^2)*P(n,x) = (4*n+2+(17*n+4)*x+(26*n-2)*x^2+(17*n-8)*x^3+(4*n-4)*x^4)*P(n-1,x) - (2*x+10*x^2+18*x^3+14*x^4+4*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((4*n + 2 + (17*n + 4)*x + (26*n - 2)*x^2 + (17*n - 8)*x^3 + (4*n - 4)*x^4)*P[n - 1, x] - (2*x + 10*x^2 + 18*x^3 + 14*x^4 + 4*x^5)*D[P[n - 1, x], x])/(n + 2 + (2*n + 4)*x + (n + 2)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((4*n + 2 + (17*n + 4)*x + (26*n - 2)*x^2 + (17*n - 8)*x^3 + (4*n - 4)*x^4)*v[i - 1] - (2*x + 10*x^2 + 18*x^3 + 14*x^4 + 4*x^5)*deriv(v[i - 1], x))/(n + 2 + (2*n + 4)*x + (n + 2)*x^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-57b4b14604f4c0d7","steps":["x \u21a6 x/(x+1)"],"edges":["e-38a5f21f7e3f9e41"],"text":"A126216 \u2192 x \u21a6 x/(x+1)"},"headline":"A126216 \u2192 x \u21a6 x/(x+1)","chain":null}