{"id":"n-3a81c23f7a892ffd","revision":0,"first_index":0,"fingerprint":"f6-5cd009bfc7bc0bb1","name":"mobius_x_over_x_plus_one of reversal of Triangle read by rows: T(n,k) = number of vertices of degree k in an origami flip graph OFG(A2n).","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":[["8"],["18","30"],["32","128","112"],["50","350","700","420"],["72","768","2592","3456","1584"],["98","1470","7350","16170","16170","6006"],["128","2560","17600","56320","91520","73216","22880"],["162","4158","37422","162162","378378","486486","324324","87516"],["200","6400","72800","407680","1274000","2329600","2475200","1414400","335920"],["242","9438","132132","924924","3699696","8984976","13477464","12193896","6096948","1293292"],["288","13440","226800","1935360","9596160","29611008","58605120","74419200","58605120","26046720","4992288"],["338","18590","371800","3792360","22754160","86465808","216164520","360274200","396301620","276210220","110484088","19315400"],["392","25088","586432","7037184","50139936","229211136","701959104","1470771456","2114233968","2050166272","1281353920","465946880","74884320"],["450","33150","895050","12471030","103925250","561196350","2057719950","5258617650","9465511770","11951403750","10357883250","5867052750","1955684250","290845350"],["512","43008","1327872","21245952","204492288","1285380096","5543201664","16893566976","36954677760","58231613440","65510565120","51308974080","26570718720","8175605760","1131445440"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["8","48","272","1520","8472","47264","264224","1480608","8316200","46813008","264051504","1492153104","8446316088","47883240000","271832858688"],"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":[["8"]],"recurrence":{"denominator":{"coeffs":[["3","6","3"],["7","14","7"],["5","10","5"],["1","2","1"]]},"terms":[{"coeff":{"coeffs":[["12","42","48","18"],["16","59","70","27"],["7","27","33","13"],["1","4","5","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","6","18","18","6"],["0","7","21","21","7"],["0","2","6","6","2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":177941,"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-14c2f28924cda6a6"]},{"id":177942,"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-14c2f28924cda6a6"]},{"id":177943,"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-14c2f28924cda6a6"]}],"region_exact":{},"incoming_edges":[{"id":"e-14c2f28924cda6a6","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-3a81c23f7a892ffd","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":"1504201fc4159b9e00d98a5ceef0c6c2dc79a008524a49f88aeef616f1a1aa2d","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-4d976aa636e25300","role":"sequence","name":"reversal of Triangle read by rows: T(n,k) = number of vertices of degree k in an origami flip graph OFG(A2n).","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-0dca9696e98a68cd","kind":"fingerprint_related","node_a_id":"n-8cac8e8480894b25","node_b_id":"n-3a81c23f7a892ffd","fingerprint":"f6-5cd009bfc7bc0bb1","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-5cd009bfc7bc0bb1"},"other_id":"n-8cac8e8480894b25","other":{"id":"n-8cac8e8480894b25","name":"shift_x_plus_one of reversal of Triangle of Dyck paths counted by number of long interior inclines.","first_index":0,"fingerprint":"f6-5cd009bfc7bc0bb1","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["2"],["5","3"],["14","16","4"]],"card_rows":[["2"],["5","3"],["14","16","4"],["42","70","35","5"]],"row_sums_preview":["2","8","34","152","706","3376","16514","82256"],"regions":["(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c963abfc0266d66c","steps":["reverse","x \u21a6 x+1"],"edges":["e-7ac651410315272d","e-ca102fc79ed966ef"],"text":"A108838 \u2192 reverse \u2192 x \u21a6 x+1"},"headline":"A108838 \u2192 reverse \u2192 x \u21a6 x+1","chain":null}},{"id":"l-0ef08017187eb388","kind":"fingerprint_related","node_a_id":"n-fabe4ac480a936a7","node_b_id":"n-3a81c23f7a892ffd","fingerprint":"f6-5cd009bfc7bc0bb1","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-5cd009bfc7bc0bb1"},"other_id":"n-fabe4ac480a936a7","other":{"id":"n-fabe4ac480a936a7","name":"shift_x_plus_one of derivative of Triangle read by rows, T(n, k) = binomial(n, k)*binomial(n + 2, k).","first_index":0,"fingerprint":"f6-5cd009bfc7bc0bb1","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["3"],["20","12"]],"card_rows":[["3"],["20","12"],["105","120","30"],["504","840","420","60"]],"row_sums_preview":["0","3","32","255","1824","12355","81024","520191"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-5aa52e3acf29a627","steps":["D"],"edges":["e-e2cb16f67cf9b10f"],"text":"A345013 \u2192 D"},"headline":"A345013 \u2192 D","chain":null}},{"id":"l-39ba359445f066c7","kind":"fingerprint_related","node_a_id":"n-d54c958727b939eb","node_b_id":"n-3a81c23f7a892ffd","fingerprint":"f6-5cd009bfc7bc0bb1","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-5cd009bfc7bc0bb1"},"other_id":"n-d54c958727b939eb","other":{"id":"n-d54c958727b939eb","name":"shift_x_plus_one of Triangle read by rows: T(n,k) = number of vertices of degree k in an origami flip graph OFG(A2n).","first_index":0,"fingerprint":"f6-5cd009bfc7bc0bb1","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["8"],["30","18"],["112","128","32"]],"card_rows":[["8"],["30","18"],["112","128","32"],["420","700","350","50"]],"row_sums_preview":["8","48","272","1520","8472","47264","264224","1480608"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7a4827a398e28fe4","steps":["x \u21a6 x+1"],"edges":["e-5e1fa9e41a7375d8"],"text":"A352880 \u2192 x \u21a6 x+1"},"headline":"A352880 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-6080a58b9088c8d4","kind":"fingerprint_related","node_a_id":"n-f7cfaa6a3770b14b","node_b_id":"n-3a81c23f7a892ffd","fingerprint":"f6-5cd009bfc7bc0bb1","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-5cd009bfc7bc0bb1"},"other_id":"n-f7cfaa6a3770b14b","other":{"id":"n-f7cfaa6a3770b14b","name":"mobius_x_over_x_plus_one of Triangle of Dyck paths counted by number of long interior inclines.","first_index":0,"fingerprint":"f6-5cd009bfc7bc0bb1","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["2"],["3","5"],["4","16","14"]],"card_rows":[["2"],["3","5"],["4","16","14"],["5","35","70","42"]],"row_sums_preview":["2","8","34","152","706","3376","16514","82256"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c963abfc0266d66c","steps":["x \u21a6 x/(x+1)"],"edges":["e-c9da786527cfa7e3"],"text":"A108838 \u2192 x \u21a6 x/(x+1)"},"headline":"A108838 \u2192 x \u21a6 x/(x+1)","chain":null}}],"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:   8;\n  1:  18,   30;\n  2:  32,  128,    112;\n  3:  50,  350,    700,    420;\n  4:  72,  768,   2592,   3456,    1584;\n  5:  98, 1470,   7350,  16170,   16170,    6006;\n  6: 128, 2560,  17600,  56320,   91520,   73216,    22880;\n  7: 162, 4158,  37422, 162162,  378378,  486486,   324324,    87516;\n  8: 200, 6400,  72800, 407680, 1274000, 2329600,  2475200,  1414400,  335920;\n  9: 242, 9438, 132132, 924924, 3699696, 8984976, 13477464, 12193896, 6096948, 1293292;\n","recurrence_sentence":"The row polynomials satisfy (n^3+5*n^2+7*n+3+(2*n^3+10*n^2+14*n+6)*x+(n^3+5*n^2+7*n+3)*x^2)*P(n,x) = (n^3+7*n^2+16*n+12+(4*n^3+27*n^2+59*n+42)*x+(5*n^3+33*n^2+70*n+48)*x^2+(2*n^3+13*n^2+27*n+18)*x^3)*P(n-1,x) + ((2*n^2+7*n+6)*x+(6*n^2+21*n+18)*x^2+(6*n^2+21*n+18)*x^3+(2*n^2+7*n+6)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 8.","mathematica":"P[0, x_] := 8;\nP[n_, x_] := P[n, x] = Expand[Together[((n^3 + 7*n^2 + 16*n + 12 + (4*n^3 + 27*n^2 + 59*n + 42)*x + (5*n^3 + 33*n^2 + 70*n + 48)*x^2 + (2*n^3 + 13*n^2 + 27*n + 18)*x^3)*P[n - 1, x] + ((2*n^2 + 7*n + 6)*x + (6*n^2 + 21*n + 18)*x^2 + (6*n^2 + 21*n + 18)*x^3 + (2*n^2 + 7*n + 6)*x^4)*D[P[n - 1, x], x])/(n^3 + 5*n^2 + 7*n + 3 + (2*n^3 + 10*n^2 + 14*n + 6)*x + (n^3 + 5*n^2 + 7*n + 3)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 8; for(n = 1, N, my(i = n + 1); v[i] = ((n^3 + 7*n^2 + 16*n + 12 + (4*n^3 + 27*n^2 + 59*n + 42)*x + (5*n^3 + 33*n^2 + 70*n + 48)*x^2 + (2*n^3 + 13*n^2 + 27*n + 18)*x^3)*v[i - 1] + ((2*n^2 + 7*n + 6)*x + (6*n^2 + 21*n + 18)*x^2 + (6*n^2 + 21*n + 18)*x^3 + (2*n^2 + 7*n + 6)*x^4)*deriv(v[i - 1], x))/(n^3 + 5*n^2 + 7*n + 3 + (2*n^3 + 10*n^2 + 14*n + 6)*x + (n^3 + 5*n^2 + 7*n + 3)*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-7a4827a398e28fe4","steps":["reverse","x \u21a6 x/(x+1)"],"edges":["e-8b10d4d15ed56acc","e-14c2f28924cda6a6"],"text":"A352880 \u2192 reverse \u2192 x \u21a6 x/(x+1)"},"headline":"A352880 \u2192 reverse \u2192 x \u21a6 x/(x+1)","chain":null}