{"id":"n-573c4ed7a2758dc8","revision":0,"first_index":0,"fingerprint":"f6-ad8665a1ff262206","name":"mobius_x_over_x_plus_one of theta_plus_one of A Jacobsthal triangle.","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"],["0","2"],["0","4","7"],["0","0","12","16"],["0","0","12","48","41"],["0","0","0","48","136","94"],["0","0","0","32","216","396","219"],["0","0","0","0","160","720","1044","492"],["0","0","0","0","80","800","2340","2712","1101"],["0","0","0","0","0","480","3040","6912","6768","2426"],["0","0","0","0","0","192","2640","10880","19728","16596","5311"],["0","0","0","0","0","0","1344","11200","35392","53856","39836","11528"],["0","0","0","0","0","0","448","8064","43680","109760","143100","94272","24881"],["0","0","0","0","0","0","0","3584","37632","154560","324160","370320","220056","53398"],["0","0","0","0","0","0","0","1024","23296","158592","516320","925280","939336","508124","114083"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","2","11","28","101","278","863","2416","7033","19626","55347","153156","424205","1163710","3186055"],"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":[["0","0","0"],["1","2","1"]]},"terms":[{"coeff":{"coeffs":[["4","22","41","32","9"],["-4","-20","-37","-30","-9"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","8","33","51","35","9"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":172550,"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-2ad9825d2a2c8673"]},{"id":172551,"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-2ad9825d2a2c8673"]},{"id":172552,"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-2ad9825d2a2c8673"]}],"region_exact":{},"incoming_edges":[{"id":"e-2ad9825d2a2c8673","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-573c4ed7a2758dc8","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":"eff4e8ddf967147aa54b37bdc6e0556f96103a5755b0e0f314d16192e553f0b0","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-f8bb6517bc8332de","role":"sequence","name":"theta_plus_one of A Jacobsthal triangle.","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-e0353f0e5f6c11f0","kind":"fingerprint_related","node_a_id":"n-4021924eb530c03c","node_b_id":"n-573c4ed7a2758dc8","fingerprint":"f6-ad8665a1ff262206","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-ad8665a1ff262206"},"other_id":"n-4021924eb530c03c","other":{"id":"n-4021924eb530c03c","name":"shift_x_plus_one of derivative of Triangle read by rows: T(n,k) is the number of ways to tile a 3 X n rectangle with k pieces of 2 X 2 tiles and 3n-4k pieces of 1 X 1 tiles (0 <= k <= floor(n/2)).","first_index":0,"fingerprint":"f6-ad8665a1ff262206","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],[],["2"]],"card_rows":[["2"],["4"],["14","8"],["32","24"]],"row_sums_preview":["0","0","2","4","22","56","202","556"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-e3d901814302c059","steps":["x \u21a6 x+1","D"],"edges":["e-31754865d296b758","e-6d6c1b443c41d0da"],"text":"A128099 \u2192 x \u21a6 x+1 \u2192 D"},"headline":"A128099 \u2192 x \u21a6 x+1 \u2192 D","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: 1;\n  1: 0, 2;\n  2: 0, 4,  7;\n  3: 0, 0, 12, 16;\n  4: 0, 0, 12, 48,  41;\n  5: 0, 0,  0, 48, 136,  94;\n  6: 0, 0,  0, 32, 216, 396,  219;\n  7: 0, 0,  0,  0, 160, 720, 1044,  492;\n  8: 0, 0,  0,  0,  80, 800, 2340, 2712, 1101;\n  9: 0, 0,  0,  0,   0, 480, 3040, 6912, 6768, 2426;\n","recurrence_sentence":"The row polynomials satisfy (n+2*n*x+n*x^2)*P(n,x) = -(4*n-4+(20*n-22)*x+(37*n-41)*x^2+(30*n-32)*x^3+(9*n-9)*x^4)*P(n-1,x) + (8*x+33*x^2+51*x^3+35*x^4+9*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 - 4 + (20*n - 22)*x + (37*n - 41)*x^2 + (30*n - 32)*x^3 + (9*n - 9)*x^4)*P[n - 1, x] + (8*x + 33*x^2 + 51*x^3 + 35*x^4 + 9*x^5)*D[P[n - 1, x], x])/(n + 2*n*x + n*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 - 4 + (20*n - 22)*x + (37*n - 41)*x^2 + (30*n - 32)*x^3 + (9*n - 9)*x^4)*v[i - 1] + (8*x + 33*x^2 + 51*x^3 + 35*x^4 + 9*x^5)*deriv(v[i - 1], x))/(n + 2*n*x + n*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-ba196dcd01800741","steps":["\u03b8+1","x \u21a6 x/(x+1)"],"edges":["e-ecb19626d7d3a316","e-2ad9825d2a2c8673"],"text":"A113953 \u2192 \u03b8+1 \u2192 x \u21a6 x/(x+1)"},"headline":"A113953 \u2192 \u03b8+1 \u2192 x \u21a6 x/(x+1)","chain":null}