{"id":"n-0c674e242339be8f","revision":0,"first_index":0,"fingerprint":"f6-747352d83647270b","name":"mobius_x_over_x_plus_one of Triangular array A066667 or A008297 unsigned and transposed.","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"],["1","3"],["1","8","13"],["1","15","63","73"],["1","24","186","544","501"],["1","35","430","2290","5225","4051"],["1","48","855","7160","29415","55656","37633"],["1","63","1533","18515","118755","400701","653023","394353"],["1","80","2548","41888","385630","2018576","5817700","8379008","4596553"],["1","99","3996","85764","1071126","8012466","35620284","90064836","116780049","58941091"],["1","120","5985","162480","2642850","26736192","169175370","656418960","1484548605","1757211400","824073141"],["1","143","8635","289245","5940330","78087702","666283926","3666081210","12663953445","25994697355","28394129951","12470162233"],["1","168","12078","489280","12381435","205148592","2269620276","16794002016","81984834855","255946230760","482301147966","490371506208","202976401213"],["1","195","16458","793078","24244935","494325117","6881940780","65892067956","431482596831","1897628103085","5417272641066","9457329713910","9013522796473","3535017524403"],["1","224","21931","1239784","45046001","1108379272","18963948003","228131306832","1929281677323","11354115044272","45527247686921","119976022306504","195497275570051","175679564492264","65573803186921"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","22","152","1256","12032","130768","1586944","21241984","310579712","4920975104","83939634176","1532861774848","29825546764288","615785559906304"],"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":[["1","2","1"]]},"terms":[{"coeff":{"coeffs":[["1","3","2","-1","-1"],["0","2","5","4","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-1","-3","-3","-1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":181906,"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-31bab05fa4fccafe"]},{"id":181907,"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-31bab05fa4fccafe"]},{"id":181908,"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-31bab05fa4fccafe"]}],"region_exact":{},"incoming_edges":[{"id":"e-31bab05fa4fccafe","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-0c674e242339be8f","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":"5d1126f1fd10c093cce4940f285082e45e15e133a04cb8bbef0d84d3a89bfe57","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-d55979c30b97975a","role":"sequence","name":"Triangular array A066667 or A008297 unsigned and transposed.","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-d2857da4e3860846","kind":"fingerprint_related","node_a_id":"n-674e6855d1cd3f21","node_b_id":"n-0c674e242339be8f","fingerprint":"f6-747352d83647270b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-747352d83647270b"},"other_id":"n-674e6855d1cd3f21","other":{"id":"n-674e6855d1cd3f21","name":"shift_x_plus_one of basis_map_A008277 of Triangle T(n,k), 0 <= k <= n, read by rows, giving coefficients of the polynomial (x+1)(x+2)...(x+n), expanded in increasing powers of x. T(n,k) is also the unsigned Stirling number |s(n+1, k+1)|, denoting the number of permutations on n+1 elements that contain exactly k+1 cycles.","first_index":0,"fingerprint":"f6-747352d83647270b","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","1"],["13","8","1"]],"card_rows":[["1"],["3","1"],["13","8","1"],["73","63","15","1"]],"row_sums_preview":["1","4","22","152","1256","12032","130768","1586944"],"regions":["(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-efe23a76953cf69d","steps":["basis A008277","x \u21a6 x+1"],"edges":["e-31e8b29857f291e5","e-a41d774ed65701cc"],"text":"A130534 \u2192 basis A008277 \u2192 x \u21a6 x+1"},"headline":"A130534 \u2192 basis A008277 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-d3bb8aec784b2819","kind":"fingerprint_related","node_a_id":"n-54d11b5bf400eddf","node_b_id":"n-0c674e242339be8f","fingerprint":"f6-747352d83647270b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-747352d83647270b"},"other_id":"n-54d11b5bf400eddf","other":{"id":"n-54d11b5bf400eddf","name":"shift_x_plus_one of Triangle read by rows: T(n,k) = binomial(n,k)*(n-1)!/(k-1)!.","first_index":1,"fingerprint":"f6-747352d83647270b","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","1"],["13","8","1"]],"card_rows":[["1"],["3","1"],["13","8","1"],["73","63","15","1"]],"row_sums_preview":["1","4","22","152","1256","12032","130768","1586944"],"regions":["(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4a0fd05ab0621175","steps":["x \u21a6 x+1"],"edges":["e-60b9c88fd5e752c7"],"text":"A105278 \u2192 x \u21a6 x+1"},"headline":"A105278 \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: 1;\n  1: 1,  3;\n  2: 1,  8,   13;\n  3: 1, 15,   63,    73;\n  4: 1, 24,  186,   544,     501;\n  5: 1, 35,  430,  2290,    5225,    4051;\n  6: 1, 48,  855,  7160,   29415,   55656,    37633;\n  7: 1, 63, 1533, 18515,  118755,  400701,   653023,   394353;\n  8: 1, 80, 2548, 41888,  385630, 2018576,  5817700,  8379008,   4596553;\n  9: 1, 99, 3996, 85764, 1071126, 8012466, 35620284, 90064836, 116780049, 58941091;\n","recurrence_sentence":"The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (1+(2*n+3)*x+(5*n+2)*x^2+(4*n-1)*x^3+(n-1)*x^4)*P(n-1,x) - (x^2+3*x^3+3*x^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[((1 + (2*n + 3)*x + (5*n + 2)*x^2 + (4*n - 1)*x^3 + (n - 1)*x^4)*P[n - 1, x] - (x^2 + 3*x^3 + 3*x^4 + x^5)*D[P[n - 1, x], 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] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((1 + (2*n + 3)*x + (5*n + 2)*x^2 + (4*n - 1)*x^3 + (n - 1)*x^4)*v[i - 1] - (x^2 + 3*x^3 + 3*x^4 + 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 - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-d55979c30b97975a","steps":["x \u21a6 x/(x+1)"],"edges":["e-31bab05fa4fccafe"],"text":"A089231 \u2192 x \u21a6 x/(x+1)"},"headline":"A089231 \u2192 x \u21a6 x/(x+1)","chain":null}