{"id":"n-00566dc174c09dbc","revision":1,"first_index":1,"fingerprint":"f6-bfe4d29ca3d34feb","name":"mobius_x_over_x_plus_one of Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.","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"],["1","4"],["1","12","16"],["1","24","80","64"],["1","40","240","448","256"],["1","60","560","1792","2304","1024"],["1","84","1120","5376","11520","11264","4096"],["1","112","2016","13440","42240","67584","53248","16384"],["1","144","3360","29568","126720","292864","372736","245760","65536"],["1","180","5280","59136","329472","1025024","1863680","1966080","1114112","262144"],["1","220","7920","109824","768768","3075072","7454720","11141120","10027008","4980736","1048576"],["1","264","11440","192192","1647360","8200192","25346048","50135040","63504384","49807360","22020096","4194304"],["1","312","16016","320320","3294720","19914752","76038144","190513152","317521920","348651520","242221056","96468992","16777216"],["1","364","21840","512512","6223360","44808192","206389248","635043840","1333592064","1917583360","1857028096","1157627904","419430400","67108864"],["1","420","29120","792064","11202048","94595072","515973120","1905131520","4889837568","8820883456","11142168576","9646899200","5452595200","1811939328","268435456"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","29","169","985","5741","33461","195025","1136689","6625109","38613965","225058681","1311738121","7645370045","44560482149"],"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":[["-3","-6","-3"],["2","4","2"]]},"terms":[{"coeff":{"coeffs":[["-3","-10","-11","-4"],["2","8","10","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","4","12","12","4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":164967,"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-f1f6e022074eb849"]},{"id":164968,"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-f1f6e022074eb849"]},{"id":164969,"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-f1f6e022074eb849"]}],"region_exact":{},"incoming_edges":[{"id":"e-f1f6e022074eb849","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-00566dc174c09dbc","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":"dc98e1cd0402c9fd35afd88eae6c9e5f1e72610ec012bbbf42a534bd72d0a104","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-ba8de4e02acfb6ca","role":"sequence","name":"Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.","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-275bf8a7822db299","kind":"fingerprint_related","node_a_id":"n-87eef110d487e3a4","node_b_id":"n-00566dc174c09dbc","fingerprint":"f6-bfe4d29ca3d34feb","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-bfe4d29ca3d34feb"},"other_id":"n-87eef110d487e3a4","other":{"id":"n-87eef110d487e3a4","name":"basis_map_A007318 of Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).","first_index":0,"fingerprint":"f6-bfe4d29ca3d34feb","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["4","1"],["16","12","1"]],"card_rows":[["1"],["4","1"],["16","12","1"],["64","80","24","1"]],"row_sums_preview":["1","5","29","169","985","5741","33461","195025"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c30cb06ad75c5e21","steps":["basis A007318"],"edges":["e-6f2b6c095b5c4221"],"text":"A103327 \u2192 basis A007318"},"headline":"A103327 \u2192 basis A007318","chain":null}},{"id":"l-33d0cac4231dfdf5","kind":"oeis_match","node_a_id":"n-00566dc174c09dbc","node_b_id":null,"fingerprint":null,"oeis_id":"A001653","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":1,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A001653","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-d4396998332be3bb","kind":"oeis_match","node_a_id":"n-00566dc174c09dbc","node_b_id":null,"fingerprint":null,"oeis_id":"A373547","in_catalog":0,"target":"triangle","row_alignment":"tabl","rows_matched":10,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":10,"target":"triangle","terms_compared":55},"in_catalog":false,"oeis_id":"A373547","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A001653","relation":"match_row_sums"},{"oeis_id":"A373547","relation":"match_triangle"}],"bfiles":[{"node_id":"n-00566dc174c09dbc","oeis_id":"A001653","target":"row_sums","oeis_offset":1,"first_row":1,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":24,"oeis_terms_available":24,"term_count":200,"raw_size":16213},{"node_id":"n-00566dc174c09dbc","oeis_id":"A373547","target":"triangle","oeis_offset":0,"first_row":1,"rows_used":44,"terms_matched":55,"oeis_terms_agreeing":55,"oeis_terms_available":55,"term_count":990,"raw_size":21435}],"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: 1,   4;\n   3: 1,  12,   16;\n   4: 1,  24,   80,    64;\n   5: 1,  40,  240,   448,    256;\n   6: 1,  60,  560,  1792,   2304,    1024;\n   7: 1,  84, 1120,  5376,  11520,   11264,    4096;\n   8: 1, 112, 2016, 13440,  42240,   67584,   53248,   16384;\n   9: 1, 144, 3360, 29568, 126720,  292864,  372736,  245760,   65536;\n  10: 1, 180, 5280, 59136, 329472, 1025024, 1863680, 1966080, 1114112, 262144;\n","recurrence_sentence":"The row polynomials satisfy (2*n-3+(4*n-6)*x+(2*n-3)*x^2)*P(n,x) = (2*n-3+(8*n-10)*x+(10*n-11)*x^2+(4*n-4)*x^3)*P(n-1,x) + (4*x+12*x^2+12*x^3+4*x^4)*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[((2*n - 3 + (8*n - 10)*x + (10*n - 11)*x^2 + (4*n - 4)*x^3)*P[n - 1, x] + (4*x + 12*x^2 + 12*x^3 + 4*x^4)*D[P[n - 1, x], x])/(2*n - 3 + (4*n - 6)*x + (2*n - 3)*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] = ((2*n - 3 + (8*n - 10)*x + (10*n - 11)*x^2 + (4*n - 4)*x^3)*v[i - 1] + (4*x + 12*x^2 + 12*x^3 + 4*x^4)*deriv(v[i - 1], x))/(2*n - 3 + (4*n - 6)*x + (2*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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}},{"oeis_id":"A373547","target":"triangle","first_row":1,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0: 1;\n  1: 1,   4;\n  2: 1,  12,   16;\n  3: 1,  24,   80,    64;\n  4: 1,  40,  240,   448,    256;\n  5: 1,  60,  560,  1792,   2304,    1024;\n  6: 1,  84, 1120,  5376,  11520,   11264,    4096;\n  7: 1, 112, 2016, 13440,  42240,   67584,   53248,   16384;\n  8: 1, 144, 3360, 29568, 126720,  292864,  372736,  245760,   65536;\n  9: 1, 180, 5280, 59136, 329472, 1025024, 1863680, 1966080, 1114112, 262144;\n","recurrence_sentence":"The row polynomials satisfy (2*n-1+(4*n-2)*x+(2*n-1)*x^2)*P(n,x) = (2*n-1+(8*n-2)*x+(10*n-1)*x^2+4*n*x^3)*P(n-1,x) + (4*x+12*x^2+12*x^3+4*x^4)*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[((2*n - 1 + (8*n - 2)*x + (10*n - 1)*x^2 + 4*n*x^3)*P[n - 1, x] + (4*x + 12*x^2 + 12*x^3 + 4*x^4)*D[P[n - 1, x], x])/(2*n - 1 + (4*n - 2)*x + (2*n - 1)*x^2)]];\nTable[PadRight[CoefficientList[P[n, x], x], n + 1], {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] = ((2*n - 1 + (8*n - 2)*x + (10*n - 1)*x^2 + 4*n*x^3)*v[i - 1] + (4*x + 12*x^2 + 12*x^3 + 4*x^4)*deriv(v[i - 1], x))/(2*n - 1 + (4*n - 2)*x + (2*n - 1)*x^2)); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":1,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-ba8de4e02acfb6ca","steps":["x \u21a6 x/(x+1)"],"edges":["e-f1f6e022074eb849"],"text":"A091042 \u2192 x \u21a6 x/(x+1)"},"headline":"A091042 \u2192 x \u21a6 x/(x+1)","chain":null}
