{"id":"n-15c5aa72a34be0a6","revision":0,"first_index":1,"fingerprint":"f6-5db7e795cb165ce2","name":"mobius_x_over_x_plus_one of reversal of Table a(d,m) of number of points of L1 norm m in cubic lattice Z^d, read by antidiagonals (d >= 1, m >= 0).","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"],["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":1,"initial_polynomials":[["1"],["2","3"]],"recurrence":{"denominator":{"coeffs":[["-2","-8","-12","-8","-2"],["1","4","6","4","1"]]},"terms":[{"coeff":{"coeffs":[["-2","2","84","324","566","522","248","48"],["1","-7","-82","-278","-467","-427","-204","-40"],["0","3","20","58","92","83","40","8"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2","39","212","546","774","623","268","48"],["0","-1","-14","-70","-176","-249","-202","-88","-16"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","1","13","58","130","165","121","48","8"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":172449,"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-f1e547f1147ee075"]},{"id":172450,"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-f1e547f1147ee075"]},{"id":172451,"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-f1e547f1147ee075"]}],"region_exact":{},"incoming_edges":[{"id":"e-f1e547f1147ee075","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-15c5aa72a34be0a6","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":"f4b275b8e543cb859e690c4c1507689166b16fde2f29bbecca020f3c96128190","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-63c9448cec96afa7","role":"sequence","name":"reversal of Table a(d,m) of number of points of L1 norm m in cubic lattice Z^d, read by antidiagonals (d >= 1, m >= 0).","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-19fdb928412e1356","kind":"fingerprint_related","node_a_id":"n-7e0e00e3b41ff4d0","node_b_id":"n-15c5aa72a34be0a6","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-7e0e00e3b41ff4d0","other":{"id":"n-7e0e00e3b41ff4d0","name":"mobius_x_over_x_plus_one of A Riordan array of coordination sequences.","first_index":0,"fingerprint":"f6-5db7e795cb165ce2","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","3"],["2","8","7"]],"card_rows":[["1"],["2","3"],["2","8","7"],["2","14","28","17"]],"row_sums_preview":["1","5","17","61","217","773","2753","9805"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"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}},{"id":"l-25f68cee326f2a73","kind":"fingerprint_related","node_a_id":"n-40ba0d1f2a40a97e","node_b_id":"n-15c5aa72a34be0a6","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-2f56cd409899ed83","kind":"fingerprint_related","node_a_id":"n-3a1022cbfb5e5fa3","node_b_id":"n-15c5aa72a34be0a6","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-3a1022cbfb5e5fa3","other":{"id":"n-3a1022cbfb5e5fa3","name":"shift_x_plus_one of reversal 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":"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-b0fc620949fb5990","steps":["reverse","x \u21a6 x+1"],"edges":["e-0819b37fda6a96d6","e-021a83bf68dcbf48"],"text":"A122542 \u2192 reverse \u2192 x \u21a6 x+1"},"headline":"A122542 \u2192 reverse \u2192 x \u21a6 x+1","chain":null}},{"id":"l-3e2986228325966b","kind":"fingerprint_related","node_a_id":"n-4a65bf6e6040840d","node_b_id":"n-15c5aa72a34be0a6","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-2078cb2a58c0f143"],"text":"A122542 \u2192 x \u21a6 x/(x+1)"},"headline":"A122542 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-4e5d80acb8ec1760","kind":"fingerprint_related","node_a_id":"n-9ac33c0e4aa39b7d","node_b_id":"n-15c5aa72a34be0a6","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-6277a5ab10569984","kind":"fingerprint_related","node_a_id":"n-15c5aa72a34be0a6","node_b_id":"n-0c3cd83a294504fa","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-0c3cd83a294504fa","other":{"id":"n-0c3cd83a294504fa","name":"basis_map_A007318 of reversal 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":"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-b0fc620949fb5990","steps":["reverse","basis A007318"],"edges":["e-0819b37fda6a96d6","e-559576b5e75fd8c6"],"text":"A122542 \u2192 reverse \u2192 basis A007318"},"headline":"A122542 \u2192 reverse \u2192 basis A007318","chain":null}},{"id":"l-838e4a19d49d86d2","kind":"fingerprint_related","node_a_id":"n-195811b74d9080d1","node_b_id":"n-15c5aa72a34be0a6","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-195811b74d9080d1","other":{"id":"n-195811b74d9080d1","name":"shift_x_plus_one of reversal of A Riordan array of coordination sequences.","first_index":0,"fingerprint":"f6-5db7e795cb165ce2","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","2"],["7","8","2"]],"card_rows":[["1"],["3","2"],["7","8","2"],["17","28","14","2"]],"row_sums_preview":["1","5","17","61","217","773","2753","9805"],"regions":["(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ec35d1145e9a1211","steps":["reverse","x \u21a6 x+1"],"edges":["e-9850f850f603c588","e-a704b8dfcd47093a"],"text":"A113413 \u2192 reverse \u2192 x \u21a6 x+1"},"headline":"A113413 \u2192 reverse \u2192 x \u21a6 x+1","chain":null}}],"oeis":[],"bfiles":[],"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: 2,  3;\n   3: 2,  8,   7;\n   4: 2, 14,  28,   17;\n   5: 2, 20,  66,   88,    41;\n   6: 2, 26, 122,  262,   262,    99;\n   7: 2, 32, 196,  592,   940,   752,   239;\n   8: 2, 38, 288, 1132,  2508,  3156,  2104,   577;\n   9: 2, 44, 398, 1936,  5560,  9704, 10108,  5776,  1393;\n  10: 2, 50, 526, 3058, 10852, 24464, 35180, 31252, 15626, 3363;\n","recurrence_sentence":"The row polynomials satisfy (n-2+(4*n-8)*x+(6*n-12)*x^2+(4*n-8)*x^3+(n-2)*x^4)*P(n,x) = (n-2+(3*n^2-7*n+2)*x+(20*n^2-82*n+84)*x^2+(58*n^2-278*n+324)*x^3+(92*n^2-467*n+566)*x^4+(83*n^2-427*n+522)*x^5+(40*n^2-204*n+248)*x^6+(8*n^2-40*n+48)*x^7)*P(n-1,x) - ((n-2)*x+(14*n-39)*x^2+(70*n-212)*x^3+(176*n-546)*x^4+(249*n-774)*x^5+(202*n-623)*x^6+(88*n-268)*x^7+(16*n-48)*x^8)*d/dx P(n-1,x) + (x^2+13*x^3+58*x^4+130*x^5+165*x^6+121*x^7+48*x^8+8*x^9)*d^2/dx^2 P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + 3*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 2 + 3*x;\nP[n_, x_] := P[n, x] = Expand[Together[((n - 2 + (3*n^2 - 7*n + 2)*x + (20*n^2 - 82*n + 84)*x^2 + (58*n^2 - 278*n + 324)*x^3 + (92*n^2 - 467*n + 566)*x^4 + (83*n^2 - 427*n + 522)*x^5 + (40*n^2 - 204*n + 248)*x^6 + (8*n^2 - 40*n + 48)*x^7)*P[n - 1, x] - ((n - 2)*x + (14*n - 39)*x^2 + (70*n - 212)*x^3 + (176*n - 546)*x^4 + (249*n - 774)*x^5 + (202*n - 623)*x^6 + (88*n - 268)*x^7 + (16*n - 48)*x^8)*D[P[n - 1, x], x] + (x^2 + 13*x^3 + 58*x^4 + 130*x^5 + 165*x^6 + 121*x^7 + 48*x^8 + 8*x^9)*D[P[n - 1, x], {x, 2}])/(n - 2 + (4*n - 8)*x + (6*n - 12)*x^2 + (4*n - 8)*x^3 + (n - 2)*x^4)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 3*x; for(n = 3, N, my(i = n); v[i] = ((n - 2 + (3*n^2 - 7*n + 2)*x + (20*n^2 - 82*n + 84)*x^2 + (58*n^2 - 278*n + 324)*x^3 + (92*n^2 - 467*n + 566)*x^4 + (83*n^2 - 427*n + 522)*x^5 + (40*n^2 - 204*n + 248)*x^6 + (8*n^2 - 40*n + 48)*x^7)*v[i - 1] - ((n - 2)*x + (14*n - 39)*x^2 + (70*n - 212)*x^3 + (176*n - 546)*x^4 + (249*n - 774)*x^5 + (202*n - 623)*x^6 + (88*n - 268)*x^7 + (16*n - 48)*x^8)*deriv(v[i - 1], x) + (x^2 + 13*x^3 + 58*x^4 + 130*x^5 + 165*x^6 + 121*x^7 + 48*x^8 + 8*x^9)*deriv(deriv(v[i - 1], x), x))/(n - 2 + (4*n - 8)*x + (6*n - 12)*x^2 + (4*n - 8)*x^3 + (n - 2)*x^4)); 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-e073e5e0b021f36a","steps":["reverse","x \u21a6 x/(x+1)"],"edges":["e-735e2173a8b41e53","e-f1e547f1147ee075"],"text":"A035607 \u2192 reverse \u2192 x \u21a6 x/(x+1)"},"headline":"A035607 \u2192 reverse \u2192 x \u21a6 x/(x+1)","chain":null}
