{"id":"n-e924d06af50baffa","revision":0,"first_index":1,"fingerprint":"f6-b98f707c8be2e8fb","name":"mobius_x_over_x_plus_one of Triangle read by rows: coefficients of the partial fraction decomposition of [d^n/dx^n] (x/(1-x))^n/n!.","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"],["0","1"],["0","2","5"],["0","3","18","25"],["0","4","42","132","129"],["0","5","80","420","900","681"],["0","6","135","1040","3630","5910","3653"],["0","7","210","2205","11060","28875","37926","19825"],["0","8","308","4200","28210","104440","217644","239624","108545"],["0","9","432","7392","63504","311220","911232","1579536","1497096","598417"],["0","10","585","12240","130200","807912","3105900","7512960","11145780","9274410","3317445"],["0","11","770","19305","248160","1889580","9130968","28824180","59361720","76961115","57074490","18474633"],["0","12","990","29260","445995","4070880","23981496","94636080","253143990","453712380","522321030","349367436","103274625"],["0","13","1248","42900","763620","8204625","57595824","275699424","917939880","2128846005","3376641840","3495404484","2129223564","579168825"],["0","14","1547","61152","1255254","15645630","128585457","730116816","2937090156","8448982542","17286844575","24586594032","23119890066","12929165886","3256957317"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","1","7","46","307","2086","14374","100108","702979","4968838","35307442","251984932","1804984174","12969532252","93441190444"],"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"],["0","1"]],"recurrence":{"denominator":{"coeffs":[["5","5"],["-7","-7"],["2","2"]]},"terms":[{"coeff":{"coeffs":[["-9","-27","-27","-9"],["9","27","27","9"],["-2","-6","-6","-2"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["16","62","46"],["-16","-64","-48"],["4","16","12"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":177338,"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-1d83a42fbcaa03fe"]},{"id":177339,"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-1d83a42fbcaa03fe"]},{"id":177340,"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-1d83a42fbcaa03fe"]}],"region_exact":{},"incoming_edges":[{"id":"e-1d83a42fbcaa03fe","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-e924d06af50baffa","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":"11671c7f5741e12a4cb798eeed1148ec2c541a7aa3cbc2ae4ba91faf3222f476","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-b0d6ec97a8d172b8","role":"sequence","name":"Triangle read by rows: coefficients of the partial fraction decomposition of [d^n/dx^n] (x/(1-x))^n/n!.","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-d8ea5c3abf79f432","kind":"fingerprint_related","node_a_id":"n-c7b2a4255e084d17","node_b_id":"n-e924d06af50baffa","fingerprint":"f6-b98f707c8be2e8fb","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b98f707c8be2e8fb"},"other_id":"n-c7b2a4255e084d17","other":{"id":"n-c7b2a4255e084d17","name":"shift_x_plus_one of basis_map_A007318 of Triangle read by rows. T(n, k) = binomial(n, k)*binomial(n - 1, n - k - 1).","first_index":0,"fingerprint":"f6-b98f707c8be2e8fb","polytool_family":null,"degree_pattern":"irregular","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1"],["5","2"]],"card_rows":[["1"],["1"],["5","2"],["25","18","3"]],"row_sums_preview":["1","1","7","46","307","2086","14374","100108"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-b1b19325e0094118","steps":["x \u21a6 x+1","x \u21a6 x+1"],"edges":["e-0de802f77814f187","e-1c6f6cde901894f2"],"text":"A367270 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x+1"},"headline":"A367270 \u2192 x \u21a6 x+1 \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: 0, 1;\n   3: 0, 2,   5;\n   4: 0, 3,  18,   25;\n   5: 0, 4,  42,  132,   129;\n   6: 0, 5,  80,  420,   900,    681;\n   7: 0, 6, 135, 1040,  3630,   5910,   3653;\n   8: 0, 7, 210, 2205, 11060,  28875,  37926,   19825;\n   9: 0, 8, 308, 4200, 28210, 104440, 217644,  239624,  108545;\n  10: 0, 9, 432, 7392, 63504, 311220, 911232, 1579536, 1497096, 598417;\n","recurrence_sentence":"The row polynomials satisfy (2*n^2-7*n+5+(2*n^2-7*n+5)*x)*P(n,x) = -(2*n^2-9*n+9+(6*n^2-27*n+27)*x+(6*n^2-27*n+27)*x^2+(2*n^2-9*n+9)*x^3)*P(n-2,x) + (4*n^2-16*n+16+(16*n^2-64*n+62)*x+(12*n^2-48*n+46)*x^2)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(2*n^2 - 9*n + 9 + (6*n^2 - 27*n + 27)*x + (6*n^2 - 27*n + 27)*x^2 + (2*n^2 - 9*n + 9)*x^3)*P[n - 2, x] + (4*n^2 - 16*n + 16 + (16*n^2 - 64*n + 62)*x + (12*n^2 - 48*n + 46)*x^2)*P[n - 1, x])/(2*n^2 - 7*n + 5 + (2*n^2 - 7*n + 5)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = x; for(n = 3, N, my(i = n); v[i] = (-(2*n^2 - 9*n + 9 + (6*n^2 - 27*n + 27)*x + (6*n^2 - 27*n + 27)*x^2 + (2*n^2 - 9*n + 9)*x^3)*v[i - 2] + (4*n^2 - 16*n + 16 + (16*n^2 - 64*n + 62)*x + (12*n^2 - 48*n + 46)*x^2)*v[i - 1])/(2*n^2 - 7*n + 5 + (2*n^2 - 7*n + 5)*x)); 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-b0d6ec97a8d172b8","steps":["x \u21a6 x/(x+1)"],"edges":["e-1d83a42fbcaa03fe"],"text":"A253283 \u2192 x \u21a6 x/(x+1)"},"headline":"A253283 \u2192 x \u21a6 x/(x+1)","chain":null}