{"id":"n-f263ca9517b1fbce","revision":1,"first_index":1,"fingerprint":"f6-261a981b55e1ac6d","name":"mobius_x_over_x_plus_one of Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.","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","2"],["2","7","6"],["3","16","28","16"],["5","35","91","104","44"],["8","70","243","418","356","120"],["13","136","589","1351","1730","1172","328"],["21","256","1330","3816","6528","6656","3744","896"],["34","473","2865","9866","21121","28776","24360","11712","2448"],["55","860","5951","23914","61489","104890","118680","85872","36048","6688"],["89","1545","12023","55223","165766","339739","481390","465576","294096","109552","18272"],["144","2748","23754","122756","421345","1008456","1717196","2080048","1756272","984320","329536","49920"],["233","4848","46086","264646","1022338","2798629","5566247","8103664","8569972","6419856","3233296","982912","136384"],["377","8494","88068","556332","2388751","7361308","16748759","28484706","36208108","33973032","22868048","10456672","2911040","372608"],["610","14795","166152","1145068","5409863","18533745","47478185","92381887","137190978","154726348","130446920","79715280","33376096","8569664","1017984"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","15","63","279","1215","5319","23247","101655","444447","1943271","8496495","37149111","162426303","710173575"],"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","-2","-1"],["1","2","1"]]},"terms":[{"coeff":{"coeffs":[["15","68","115","86","24"],["-7","-32","-55","-42","-12"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["5","31","75","89","52","12"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":163618,"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-5274886585caff6b"]},{"id":163619,"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-5274886585caff6b"]},{"id":163620,"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-5274886585caff6b"]}],"region_exact":{},"incoming_edges":[{"id":"e-5274886585caff6b","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-f263ca9517b1fbce","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":"d2f75287e947ae502303fe526e4303058d7c8108dbae3aa70b4ff3edfe7bd130","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-c72653d9ab0ee8be","role":"sequence","name":"Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.","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-24417ee93acb771b","kind":"oeis_match","node_a_id":"n-f263ca9517b1fbce","node_b_id":null,"fingerprint":null,"oeis_id":"A083858","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":0,"oeis_term_position":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A083858","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A083858","relation":"match_row_sums"}],"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:  1,   2;\n   3:  2,   7,    6;\n   4:  3,  16,   28,    16;\n   5:  5,  35,   91,   104,    44;\n   6:  8,  70,  243,   418,   356,    120;\n   7: 13, 136,  589,  1351,  1730,   1172,    328;\n   8: 21, 256, 1330,  3816,  6528,   6656,   3744,   896;\n   9: 34, 473, 2865,  9866, 21121,  28776,  24360, 11712,  2448;\n  10: 55, 860, 5951, 23914, 61489, 104890, 118680, 85872, 36048, 6688;\n","recurrence_sentence":"The row polynomials satisfy (n-1+(2*n-2)*x+(n-1)*x^2)*P(n,x) = -(7*n-15+(32*n-68)*x+(55*n-115)*x^2+(42*n-86)*x^3+(12*n-24)*x^4)*P(n-1,x) + (5+31*x+75*x^2+89*x^3+52*x^4+12*x^5)*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[(-(7*n - 15 + (32*n - 68)*x + (55*n - 115)*x^2 + (42*n - 86)*x^3 + (12*n - 24)*x^4)*P[n - 1, x] + (5 + 31*x + 75*x^2 + 89*x^3 + 52*x^4 + 12*x^5)*D[P[n - 1, x], x])/(n - 1 + (2*n - 2)*x + (n - 1)*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] = (-(7*n - 15 + (32*n - 68)*x + (55*n - 115)*x^2 + (42*n - 86)*x^3 + (12*n - 24)*x^4)*v[i - 1] + (5 + 31*x + 75*x^2 + 89*x^3 + 52*x^4 + 12*x^5)*deriv(v[i - 1], x))/(n - 1 + (2*n - 2)*x + (n - 1)*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}}],"derivation":{"root":"n-c72653d9ab0ee8be","steps":["x \u21a6 x/(x+1)"],"edges":["e-5274886585caff6b"],"text":"A063967 \u2192 x \u21a6 x/(x+1)"},"headline":"A063967 \u2192 x \u21a6 x/(x+1)","chain":null}
