{"id":"n-7e0e00e3b41ff4d0","revision":1,"first_index":0,"fingerprint":"f6-5db7e795cb165ce2","name":"mobius_x_over_x_plus_one of A Riordan array of coordination sequences.","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"],["2","3"],["2","8","7"],["2","14","28","17"],["2","20","66","88","41"],["2","26","122","262","262","99"],["2","32","196","592","940","752","239"],["2","38","288","1132","2508","3156","2104","577"],["2","44","398","1936","5560","9704","10108","5776","1393"],["2","50","526","3058","10852","24464","35180","31252","15626","3363"],["2","56","672","4552","19302","53664","99372","121424","94014","41784","8119"],["2","62","836","6472","31990","106178","242016","379812","403294","276690","110676","19601"],["2","68","1018","8872","50158","194012","527338","1016880","1383714","1298716","799854","290856","47321"],["2","74","1218","11806","75210","332790","1053530","2419750","4039302","4849250","4077270","2277930","759310","114243"],["2","80","1436","15328","108712","542240","1963280","5248160","10423020","15328448","16458200","12531040","6405880","1971040","275807"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","17","61","217","773","2753","9805","34921","124373","442961","1577629","5618809","20011685","71272673"],"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","3","3","1"],["1","3","3","1"]]},"terms":[{"coeff":{"coeffs":[["11","73","197","279","220","92","16"],["-10","-75","-225","-349","-297","-132","-24"],["3","20","58","92","83","40","8"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["2","37","188","454","606","461","188","32"],["-1","-14","-70","-176","-249","-202","-88","-16"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1","13","58","130","165","121","48","8"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":163380,"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-73b4fdfae3fbc1a0"]},{"id":163381,"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-73b4fdfae3fbc1a0"]},{"id":163382,"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-73b4fdfae3fbc1a0"]}],"region_exact":{},"incoming_edges":[{"id":"e-73b4fdfae3fbc1a0","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-7e0e00e3b41ff4d0","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":"fbfd195ceaa02090db7912c97b685850e5a266a5c164de19330ca34b1b4124d8","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-ec35d1145e9a1211","role":"sequence","name":"A Riordan array of coordination sequences.","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-b744ad8eed383d6c","kind":"fingerprint_related","node_a_id":"n-4a65bf6e6040840d","node_b_id":"n-7e0e00e3b41ff4d0","fingerprint":"f6-5db7e795cb165ce2","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-5db7e795cb165ce2"},"other_id":"n-4a65bf6e6040840d","other":{"id":"n-4a65bf6e6040840d","name":"mobius_x_over_x_plus_one of Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.","first_index":1,"fingerprint":"f6-5db7e795cb165ce2","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["0","1"],["0","2","3"]],"card_rows":[["1"],["0","1"],["0","2","3"],["0","2","8","7"]],"row_sums_preview":["1","1","5","17","61","217","773","2753"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-b0fc620949fb5990","steps":["x \u21a6 x/(x+1)"],"edges":["e-217dc49de8a7d573"],"text":"A122542 \u2192 x \u21a6 x/(x+1)"},"headline":"A122542 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-f0e6f6d4552f18c3","kind":"fingerprint_related","node_a_id":"n-9ac33c0e4aa39b7d","node_b_id":"n-7e0e00e3b41ff4d0","fingerprint":"f6-5db7e795cb165ce2","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-5db7e795cb165ce2"},"other_id":"n-9ac33c0e4aa39b7d","other":{"id":"n-9ac33c0e4aa39b7d","name":"basis_map_A007318 of Square array A(n,r), the number of neighbors at a sharp Manhattan distance r in a finite n-hypercube lattice, read by upwards antidiagonals; A(n,r) = Sum_{k=0..min(n,r)} binomial(r-1,k-1)*binomial(n,k)* 2^k.","first_index":0,"fingerprint":"f6-5db7e795cb165ce2","polytool_family":null,"degree_pattern":"irregular","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1"],["3","2"]],"card_rows":[["1"],["1"],["3","2"],["7","8","2"]],"row_sums_preview":["1","1","5","17","61","217","773","2753"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-e073e5e0b021f36a","steps":["basis A007318"],"edges":["e-017120e86183432c"],"text":"A035607 \u2192 basis A007318"},"headline":"A035607 \u2192 basis A007318","chain":null}},{"id":"l-fabdc6742d15f374","kind":"fingerprint_related","node_a_id":"n-40ba0d1f2a40a97e","node_b_id":"n-7e0e00e3b41ff4d0","fingerprint":"f6-5db7e795cb165ce2","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-5db7e795cb165ce2"},"other_id":"n-40ba0d1f2a40a97e","other":{"id":"n-40ba0d1f2a40a97e","name":"shift_x_plus_one of Square array A(n,r), the number of neighbors at a sharp Manhattan distance r in a finite n-hypercube lattice, read by upwards antidiagonals; A(n,r) = Sum_{k=0..min(n,r)} binomial(r-1,k-1)*binomial(n,k)* 2^k.","first_index":0,"fingerprint":"f6-5db7e795cb165ce2","polytool_family":null,"degree_pattern":"irregular","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1"],["3","2"]],"card_rows":[["1"],["1"],["3","2"],["7","8","2"]],"row_sums_preview":["1","1","5","17","61","217","773","2753"],"regions":["(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-e073e5e0b021f36a","steps":["x \u21a6 x+1"],"edges":["e-15636530e2574f08"],"text":"A035607 \u2192 x \u21a6 x+1"},"headline":"A035607 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-5d954f111935db31","kind":"oeis_match","node_a_id":"n-7e0e00e3b41ff4d0","node_b_id":null,"fingerprint":null,"oeis_id":"A007483","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A007483","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A007483","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-7e0e00e3b41ff4d0","oeis_id":"A007483","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":27,"oeis_terms_available":27,"term_count":200,"raw_size":11993}],"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: 2,  3;\n  2: 2,  8,   7;\n  3: 2, 14,  28,   17;\n  4: 2, 20,  66,   88,    41;\n  5: 2, 26, 122,  262,   262,    99;\n  6: 2, 32, 196,  592,   940,   752,   239;\n  7: 2, 38, 288, 1132,  2508,  3156,  2104,   577;\n  8: 2, 44, 398, 1936,  5560,  9704, 10108,  5776,  1393;\n  9: 2, 50, 526, 3058, 10852, 24464, 35180, 31252, 15626, 3363;\n","recurrence_sentence":"The row polynomials satisfy (n+1+(3*n+3)*x+(3*n+3)*x^2+(n+1)*x^3)*P(n,x) = (3*n^2-10*n+11+(20*n^2-75*n+73)*x+(58*n^2-225*n+197)*x^2+(92*n^2-349*n+279)*x^3+(83*n^2-297*n+220)*x^4+(40*n^2-132*n+92)*x^5+(8*n^2-24*n+16)*x^6)*P(n-1,x) - (n-2+(14*n-37)*x+(70*n-188)*x^2+(176*n-454)*x^3+(249*n-606)*x^4+(202*n-461)*x^5+(88*n-188)*x^6+(16*n-32)*x^7)*d/dx P(n-1,x) + (x+13*x^2+58*x^3+130*x^4+165*x^5+121*x^6+48*x^7+8*x^8)*d^2/dx^2 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[((3*n^2 - 10*n + 11 + (20*n^2 - 75*n + 73)*x + (58*n^2 - 225*n + 197)*x^2 + (92*n^2 - 349*n + 279)*x^3 + (83*n^2 - 297*n + 220)*x^4 + (40*n^2 - 132*n + 92)*x^5 + (8*n^2 - 24*n + 16)*x^6)*P[n - 1, x] - (n - 2 + (14*n - 37)*x + (70*n - 188)*x^2 + (176*n - 454)*x^3 + (249*n - 606)*x^4 + (202*n - 461)*x^5 + (88*n - 188)*x^6 + (16*n - 32)*x^7)*D[P[n - 1, x], x] + (x + 13*x^2 + 58*x^3 + 130*x^4 + 165*x^5 + 121*x^6 + 48*x^7 + 8*x^8)*D[P[n - 1, x], {x, 2}])/(n + 1 + (3*n + 3)*x + (3*n + 3)*x^2 + (n + 1)*x^3)]];\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] = ((3*n^2 - 10*n + 11 + (20*n^2 - 75*n + 73)*x + (58*n^2 - 225*n + 197)*x^2 + (92*n^2 - 349*n + 279)*x^3 + (83*n^2 - 297*n + 220)*x^4 + (40*n^2 - 132*n + 92)*x^5 + (8*n^2 - 24*n + 16)*x^6)*v[i - 1] - (n - 2 + (14*n - 37)*x + (70*n - 188)*x^2 + (176*n - 454)*x^3 + (249*n - 606)*x^4 + (202*n - 461)*x^5 + (88*n - 188)*x^6 + (16*n - 32)*x^7)*deriv(v[i - 1], x) + (x + 13*x^2 + 58*x^3 + 130*x^4 + 165*x^5 + 121*x^6 + 48*x^7 + 8*x^8)*deriv(deriv(v[i - 1], x), x))/(n + 1 + (3*n + 3)*x + (3*n + 3)*x^2 + (n + 1)*x^3)); 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-ec35d1145e9a1211","steps":["x \u21a6 x/(x+1)"],"edges":["e-73b4fdfae3fbc1a0"],"text":"A113413 \u2192 x \u21a6 x/(x+1)"},"headline":"A113413 \u2192 x \u21a6 x/(x+1)","chain":null}
