{"id":"n-4324dc8d94b46a38","revision":0,"first_index":0,"fingerprint":"f6-db9ee1b7b985cf8f","name":"mobius_x_over_x_plus_one of Triangle read by rows: T(n,k) is the number of ternary strings of length n+1 with k+1 indispensable digits and a nonzero leading digit, with 0 <= k <= n.","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":[["2"],["3","6"],["4","18","18"],["5","37","81","54"],["6","64","226","324","162"],["7","100","501","1144","1215","486"],["8","146","966","3098","5194","4374","1458"],["9","203","1694","7113","16630","21961","15309","4374"],["10","272","2772","14568","44326","81192","88300","52488","13122"],["11","354","4302","27420","103873","246498","370286","341992","177147","39366"],["12","450","6402","48348","221080","650614","1261710","1604164","1286776","590490","118098"],["13","561","9207","80916","436513","1545171","3704477","6062824","6676227","4731301","1948617","354294"],["14","688","12870","129756","811360","3377496","9712846","19602420","27719352","26904200","17072542","6377292","1062882"],["15","832","17563","200772","1434862","6902974","23286147","56244456","97870254","121737354","105590683","60650704","20726199","3188646"],["16","994","23478","301366","2433574","13344552","51904776","146760054","305053182","466086208","517187584","405356622","212642830","66961566","9565938"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["2","9","40","177","782","3453","15244","67293","297050","1311249","5788144","25550121","112783718","497851461","2197622740"],"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":[["2"],["3","6"]],"recurrence":{"denominator":{"coeffs":[["1","2","1"]]},"terms":[{"coeff":{"coeffs":[["-1","-5","-10","-9","-3"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["2","8","10","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":174775,"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-ad71aabf00d312b7"]},{"id":174776,"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-ad71aabf00d312b7"]},{"id":174777,"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-ad71aabf00d312b7"]}],"region_exact":{},"incoming_edges":[{"id":"e-ad71aabf00d312b7","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-4324dc8d94b46a38","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":"73ebf46005143b43953c73f6858f97cf6f239ecae368d030cf38471f7ee6e997","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-539ff84504d2bf3a","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of ternary strings of length n+1 with k+1 indispensable digits and a nonzero leading digit, with 0 <= k <= n.","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-8f64283302ea4782","kind":"fingerprint_related","node_a_id":"n-94470298ec1d4aa5","node_b_id":"n-4324dc8d94b46a38","fingerprint":"f6-db9ee1b7b985cf8f","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-db9ee1b7b985cf8f"},"other_id":"n-94470298ec1d4aa5","other":{"id":"n-94470298ec1d4aa5","name":"shift_x_plus_one of Triangle read by rows: T(n,k) is the number of ternary strings of length n+1 with k+1 indispensable digits and a nonzero leading digit, with 0 <= k <= n.","first_index":0,"fingerprint":"f6-db9ee1b7b985cf8f","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["2"],["6","3"],["18","18","4"]],"card_rows":[["2"],["6","3"],["18","18","4"],["54","81","37","5"]],"row_sums_preview":["2","9","40","177","782","3453","15244","67293"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-539ff84504d2bf3a","steps":["x \u21a6 x+1"],"edges":["e-06150a129b04f752"],"text":"A330510 \u2192 x \u21a6 x+1"},"headline":"A330510 \u2192 x \u21a6 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:  2;\n  1:  3,   6;\n  2:  4,  18,   18;\n  3:  5,  37,   81,    54;\n  4:  6,  64,  226,   324,    162;\n  5:  7, 100,  501,  1144,   1215,    486;\n  6:  8, 146,  966,  3098,   5194,   4374,   1458;\n  7:  9, 203, 1694,  7113,  16630,  21961,  15309,   4374;\n  8: 10, 272, 2772, 14568,  44326,  81192,  88300,  52488,  13122;\n  9: 11, 354, 4302, 27420, 103873, 246498, 370286, 341992, 177147, 39366;\n","recurrence_sentence":"The row polynomials satisfy (1+2*x+x^2)*P(n,x) = -(1+5*x+10*x^2+9*x^3+3*x^4)*P(n-2,x) + (2+8*x+10*x^2+4*x^3)*P(n-1,x) for n >= 2, with P(0,x) = 2, P(1,x) = 3 + 6*x.","mathematica":"P[0, x_] := 2;\nP[1, x_] := 3 + 6*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(1 + 5*x + 10*x^2 + 9*x^3 + 3*x^4)*P[n - 2, x] + (2 + 8*x + 10*x^2 + 4*x^3)*P[n - 1, x])/(1 + 2*x + x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 2; v[2] = 3 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = (-(1 + 5*x + 10*x^2 + 9*x^3 + 3*x^4)*v[i - 2] + (2 + 8*x + 10*x^2 + 4*x^3)*v[i - 1])/(1 + 2*x + 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-539ff84504d2bf3a","steps":["x \u21a6 x/(x+1)"],"edges":["e-ad71aabf00d312b7"],"text":"A330510 \u2192 x \u21a6 x/(x+1)"},"headline":"A330510 \u2192 x \u21a6 x/(x+1)","chain":null}