{"id":"n-35ed341964ed7074","revision":0,"first_index":1,"fingerprint":"f6-c569cb6e352750ae","name":"mobius_x_over_x_plus_one of basis_map_A013609 of Square array of Delannoy numbers D(i,j) (i >= 0, j >= 0) read by antidiagonals.","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","4"],["5","20","19"],["12","72","140","88"],["29","232","682","872","409"],["70","700","2756","5336","5078","1900"],["169","2028","10009","25992","37443","28364","8827"],["408","5712","33896","110480","213544","244720","153944","41008"],["985","15760","109268","428656","1040446","1599728","1521380","818192","190513"],["2378","42804","339544","1557584","4552668","8791576","11215128","9112848","4279658","885076"],["5741","114820","1025567","5386352","18418546","42841304","68638190","74789104","53038129","22105412","4111843"],["13860","304920","3028308","17919144","70184136","191031792","368681032","504479056","479609108","301705592","113024612","19102600"],["33461","803064","8778366","57784600","255085503","795474672","1796748276","2961548016","3535243555","2980472856","1684519470","573067256","88745929"],["80782","2100332","25058556","181629800","892255042","3136817124","8118755848","15662387696","22520247602","23835499124","18049604668","9260530216","2885245118","412291516"],["195025","5460700","70609445","558801080","3023588285","11831824900","34528683665","76334763920","128461726435","163758079780","155653347391","106972042808","50247184463","14439748988","1915403851"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","6","44","312","2224","15840","112832","803712","5724928","40779264","290475008","2069084160","14738305024","104982503424","747801460736"],"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","-3","-3","-1"],["1","3","3","1"]]},"terms":[{"coeff":{"coeffs":[["52","376","1112","1726","1486","674","126"],["-41","-295","-877","-1376","-1201","-553","-105"],["8","56","165","260","230","108","21"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["26","247","969","2052","2548","1863","745","126"],["-8","-76","-300","-642","-808","-600","-244","-42"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["2","24","121","336","565","592","379","136","21"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":176971,"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-c51adcc32940cfeb"]},{"id":176972,"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-c51adcc32940cfeb"]},{"id":176973,"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-c51adcc32940cfeb"]}],"region_exact":{},"incoming_edges":[{"id":"e-c51adcc32940cfeb","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-35ed341964ed7074","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":"10dde530b5a4b5f4e14cca957267ffff55cdf59eb785175a49eacda97669d310","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-4b9af074c19fbdaf","role":"sequence","name":"basis_map_A013609 of Square array of Delannoy numbers D(i,j) (i >= 0, j >= 0) read by antidiagonals.","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-39f21e4aa485d7b1","kind":"fingerprint_related","node_a_id":"n-35ed341964ed7074","node_b_id":"n-266498645a09d3d0","fingerprint":"f6-c569cb6e352750ae","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-c569cb6e352750ae"},"other_id":"n-266498645a09d3d0","other":{"id":"n-266498645a09d3d0","name":"mobius_x_over_x_plus_one of basis_map_A013609 of delannoy","first_index":0,"fingerprint":"f6-c569cb6e352750ae","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","4"],["5","20","19"]],"card_rows":[["1"],["2","4"],["5","20","19"],["12","72","140","88"]],"row_sums_preview":["1","6","44","312","2224","15840","112832","803712"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-e00a1cb428776eaa","steps":["basis A013609","x \u21a6 x/(x+1)"],"edges":["e-301914f41f1bfd45","e-044e8f894125881a"],"text":"delannoy \u2192 basis A013609 \u2192 x \u21a6 x/(x+1)"},"headline":"delannoy \u2192 basis A013609 \u2192 x \u21a6 x/(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,     4;\n   3:    5,    20,     19;\n   4:   12,    72,    140,      88;\n   5:   29,   232,    682,     872,     409;\n   6:   70,   700,   2756,    5336,    5078,    1900;\n   7:  169,  2028,  10009,   25992,   37443,   28364,     8827;\n   8:  408,  5712,  33896,  110480,  213544,  244720,   153944,   41008;\n   9:  985, 15760, 109268,  428656, 1040446, 1599728,  1521380,  818192,  190513;\n  10: 2378, 42804, 339544, 1557584, 4552668, 8791576, 11215128, 9112848, 4279658, 885076;\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) = (8*n^2-41*n+52+(56*n^2-295*n+376)*x+(165*n^2-877*n+1112)*x^2+(260*n^2-1376*n+1726)*x^3+(230*n^2-1201*n+1486)*x^4+(108*n^2-553*n+674)*x^5+(21*n^2-105*n+126)*x^6)*P(n-1,x) - (8*n-26+(76*n-247)*x+(300*n-969)*x^2+(642*n-2052)*x^3+(808*n-2548)*x^4+(600*n-1863)*x^5+(244*n-745)*x^6+(42*n-126)*x^7)*d/dx P(n-1,x) + (2+24*x+121*x^2+336*x^3+565*x^4+592*x^5+379*x^6+136*x^7+21*x^8)*d^2/dx^2 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[((8*n^2 - 41*n + 52 + (56*n^2 - 295*n + 376)*x + (165*n^2 - 877*n + 1112)*x^2 + (260*n^2 - 1376*n + 1726)*x^3 + (230*n^2 - 1201*n + 1486)*x^4 + (108*n^2 - 553*n + 674)*x^5 + (21*n^2 - 105*n + 126)*x^6)*P[n - 1, x] - (8*n - 26 + (76*n - 247)*x + (300*n - 969)*x^2 + (642*n - 2052)*x^3 + (808*n - 2548)*x^4 + (600*n - 1863)*x^5 + (244*n - 745)*x^6 + (42*n - 126)*x^7)*D[P[n - 1, x], x] + (2 + 24*x + 121*x^2 + 336*x^3 + 565*x^4 + 592*x^5 + 379*x^6 + 136*x^7 + 21*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, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((8*n^2 - 41*n + 52 + (56*n^2 - 295*n + 376)*x + (165*n^2 - 877*n + 1112)*x^2 + (260*n^2 - 1376*n + 1726)*x^3 + (230*n^2 - 1201*n + 1486)*x^4 + (108*n^2 - 553*n + 674)*x^5 + (21*n^2 - 105*n + 126)*x^6)*v[i - 1] - (8*n - 26 + (76*n - 247)*x + (300*n - 969)*x^2 + (642*n - 2052)*x^3 + (808*n - 2548)*x^4 + (600*n - 1863)*x^5 + (244*n - 745)*x^6 + (42*n - 126)*x^7)*deriv(v[i - 1], x) + (2 + 24*x + 121*x^2 + 336*x^3 + 565*x^4 + 592*x^5 + 379*x^6 + 136*x^7 + 21*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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-811c78acf96cbc19","steps":["basis A013609","x \u21a6 x/(x+1)"],"edges":["e-14fad2b62d6c6dcf","e-c51adcc32940cfeb"],"text":"A008288 \u2192 basis A013609 \u2192 x \u21a6 x/(x+1)"},"headline":"A008288 \u2192 basis A013609 \u2192 x \u21a6 x/(x+1)","chain":null}
