{"id":"n-3f3274366bbb610a","revision":1,"first_index":1,"fingerprint":"f6-6c43fbf0495b302d","name":"mobius_x_over_x_plus_one of Triangle of numbers T(n,k) = k!*Stirling2(n,k) = A000142(k)*A048993(n,k) read by rows, T(n, k) for 0 <= k <= n.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["0","1"],["0","1","3"],["0","1","8","13"],["0","1","17","67","75"],["0","1","34","246","634","541"],["0","1","67","798","3562","6793","4683"],["0","1","132","2451","16904","54111","81780","47293"],["0","1","261","7341","73649","352371","876255","1095967","545835"],["0","1","518","21748","306146","2056690","7487306","15196348","16205198","7087261"],["0","1","1031","64192","1239080","11247058","56259206","164793976","282270272","262283125","102247563"],["0","1","2056","189465","4939824","59053842","390168624","1547420106","3785616240","5605838349","4614490792","1622632573"],["0","1","4105","560151","19532823","302117418","2567123418","13244643630","43404864846","91071811653","118730462413","87715420747","28091567595"],["0","1","8202","1659906","76928890","1520037195","16298685012","106452725484","448879874052","1251463111575","2297214691810","2674317544146","1791916217922","526858348381"],["0","1","16395","4930482","302555398","7566987855","100967501637","818640593340","4320931931604","15372293786991","37255294851205","60755652545586","63880374891270","39157877582497","10641342970443"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","1","4","22","160","1456","15904","202672","2951680","48361216","880405504","17630351872","385148108800","9114999832576","232311251144704"],"nonn":true,"definitions":[{"position":0,"source":"derived:mobius_x_over_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["1","2","1"]]},"terms":[{"coeff":{"coeffs":[["0","3","10","11","4"],["0","-1","-4","-5","-2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1","5","9","7","2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":165751,"position":0,"claim":"real_rooted","from_index":1,"region_key":"[-1, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in [-1, 0]","region_exact":null,"edges":["e-841cb00766f5fd80"]},{"id":165752,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-841cb00766f5fd80"]},{"id":165753,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-841cb00766f5fd80"]}],"region_exact":{},"incoming_edges":[{"id":"e-841cb00766f5fd80","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"mobius\",\"a\":\"1\",\"b\":\"0\",\"c\":\"1\",\"e\":\"1\",\"schedule\":{\"first_index\":1,\"first_degree\":0,\"step\":1}}"},"operation_detail":"mobius, shift 0","conclusion_id":"n-3f3274366bbb610a","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":"f79c1f05ef93ca0dbbdb701ba710a4da8acbad2ff9ac525314be0acdf9d14304","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-2cabfd060b04f15d","role":"sequence","name":"Triangle of numbers T(n,k) = k!*Stirling2(n,k) = A000142(k)*A048993(n,k) read by rows, T(n, k) for 0 <= k <= n.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-63b7c32b55502ed6","kind":"oeis_match","node_a_id":"n-3f3274366bbb610a","node_b_id":null,"fingerprint":null,"oeis_id":"A122704","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A122704","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A122704","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-3f3274366bbb610a","oeis_id":"A122704","target":"row_sums","oeis_offset":0,"first_row":1,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":21,"oeis_terms_available":21,"term_count":200,"raw_size":39198}],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 1;\n   2: 0, 1;\n   3: 0, 1,   3;\n   4: 0, 1,   8,    13;\n   5: 0, 1,  17,    67,     75;\n   6: 0, 1,  34,   246,    634,     541;\n   7: 0, 1,  67,   798,   3562,    6793,    4683;\n   8: 0, 1, 132,  2451,  16904,   54111,   81780,    47293;\n   9: 0, 1, 261,  7341,  73649,  352371,  876255,  1095967,   545835;\n  10: 0, 1, 518, 21748, 306146, 2056690, 7487306, 15196348, 16205198, 7087261;\n","recurrence_sentence":"The row polynomials satisfy (1+2*x+x^2)*P(n,x) = -((n-3)*x+(4*n-10)*x^2+(5*n-11)*x^3+(2*n-4)*x^4)*P(n-1,x) + (x+5*x^2+9*x^3+7*x^4+2*x^5)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-((n - 3)*x + (4*n - 10)*x^2 + (5*n - 11)*x^3 + (2*n - 4)*x^4)*P[n - 1, x] + (x + 5*x^2 + 9*x^3 + 7*x^4 + 2*x^5)*D[P[n - 1, x], x])/(1 + 2*x + x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-((n - 3)*x + (4*n - 10)*x^2 + (5*n - 11)*x^3 + (2*n - 4)*x^4)*v[i - 1] + (x + 5*x^2 + 9*x^3 + 7*x^4 + 2*x^5)*deriv(v[i - 1], x))/(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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-2cabfd060b04f15d","steps":["x \u21a6 x/(x+1)"],"edges":["e-841cb00766f5fd80"],"text":"A131689 \u2192 x \u21a6 x/(x+1)"},"headline":"A131689 \u2192 x \u21a6 x/(x+1)","chain":null}