{"id":"n-861b0207a722f305","revision":0,"first_index":0,"fingerprint":"f6-b94fba28f209154b","name":"mobius_x_over_x_plus_one of Triangle read by rows: T(n,k) (1<=k<=n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*k-m+1)*T(n-1,k), where m = 5.","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"],["1","2"],["1","14","14"],["1","86","252","168"],["1","518","3374","5712","2856"],["1","3110","40740","131880","157080","62832"],["1","18662","469910","2598960","5540640","5089392","1696464"],["1","111974","5299644","47222280","160535760","254218272","190003968","54286848"],["1","671846","59079902","819152208","4174029720","10141461792","12754016352","8034453504","2008613376"],["1","4031078","654581844","13815394152","101580211656","355506660432","669193975296","697368849408","379627928064","84361761792"],["1","24186470","7228617830","228901288560","2368046145360","11477937827664","30343693065840","46519485788160","41367392478720","19825014021120","3965002804224"],["1","145118822","79684101420","3749164030920","53620290958080","350523398717184","1250558806387968","2645699666480640","3417284065789440","2649381129077760","1133990802008064","206180145819648"],["1","870712934","877540947374","60942833711760","1189761898645320","10293254767724544","48231454763390976","135263069797718016","237999534357150720","265397462700733440","182354443969065984","70513609870319616","11752268311719936"],["1","5224277606","9659045411652","985615830756648","26021028044651640","293799385731035184","1773092976393682944","6412877065146359808","14762714491158746112","22204263727234068480","21773757389356836864","13431193239129329664","4736164129623134208","728640635326636032"],["1","31345665638","106286069471414","15885761837046192","563197058060547456","8211246645989250864","62898465682942121232","287602549589866807296","842547745547923903488","1641430140081434486784","2154062022037199898624","1884382205637797978112","1054484026537382977536","341732457968192299008","48818922566884614144"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","29","507","12461","395643","15414029","711678747","37991478701","2302113393723","156102726223949","11711067298489947","963056681681840621","86143519347244926843","8326748971432927067789"],"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","2","1"]]},"terms":[{"coeff":{"coeffs":[["1","4","5","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","5","15","15","5"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":188386,"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-1ca198ecd1278a12"]},{"id":188387,"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-1ca198ecd1278a12"]},{"id":188388,"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-1ca198ecd1278a12"]}],"region_exact":{},"incoming_edges":[{"id":"e-1ca198ecd1278a12","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-861b0207a722f305","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":"9821b5e0889cbebade97ddf72bf26c360dd93ca26aafb3f9e8bf933d50378d1a","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-05516f61aeaea779","role":"sequence","name":"Triangle read by rows: T(n,k) (1<=k<=n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*k-m+1)*T(n-1,k), where m = 5.","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-bf2286098c73c37c","kind":"fingerprint_related","node_a_id":"n-59a56c7d8834e636","node_b_id":"n-861b0207a722f305","fingerprint":"f6-b94fba28f209154b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b94fba28f209154b"},"other_id":"n-59a56c7d8834e636","other":{"id":"n-59a56c7d8834e636","name":"shift_x_plus_one of Triangle read by rows: T(n,k) (1<=k<=n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*k-m+1)*T(n-1,k), where m = 5.","first_index":0,"fingerprint":"f6-b94fba28f209154b","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","1"],["14","14","1"]],"card_rows":[["1"],["2","1"],["14","14","1"],["168","252","86","1"]],"row_sums_preview":["1","3","29","507","12461","395643","15414029","711678747"],"regions":["(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-05516f61aeaea779","steps":["x \u21a6 x+1"],"edges":["e-124f2ee3a2f5b8b2"],"text":"A142460 \u2192 x \u21a6 x+1"},"headline":"A142460 \u2192 x \u21a6 x+1","chain":null}}],"oeis":[],"bfiles":[],"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: 1,       2;\n  2: 1,      14,        14;\n  3: 1,      86,       252,         168;\n  4: 1,     518,      3374,        5712,         2856;\n  5: 1,    3110,     40740,      131880,       157080,        62832;\n  6: 1,   18662,    469910,     2598960,      5540640,      5089392,      1696464;\n  7: 1,  111974,   5299644,    47222280,    160535760,    254218272,    190003968,     54286848;\n  8: 1,  671846,  59079902,   819152208,   4174029720,  10141461792,  12754016352,   8034453504,   2008613376;\n  9: 1, 4031078, 654581844, 13815394152, 101580211656, 355506660432, 669193975296, 697368849408, 379627928064, 84361761792;\n","recurrence_sentence":"The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (1+4*x+5*x^2+2*x^3)*P(n-1,x) + (5*x+15*x^2+15*x^3+5*x^4)*d/dx 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[((1 + 4*x + 5*x^2 + 2*x^3)*P[n - 1, x] + (5*x + 15*x^2 + 15*x^3 + 5*x^4)*D[P[n - 1, x], x])/(1 + 2*x + x^2)]];\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] = ((1 + 4*x + 5*x^2 + 2*x^3)*v[i - 1] + (5*x + 15*x^2 + 15*x^3 + 5*x^4)*deriv(v[i - 1], x))/(1 + 2*x + x^2)); 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-05516f61aeaea779","steps":["x \u21a6 x/(x+1)"],"edges":["e-1ca198ecd1278a12"],"text":"A142460 \u2192 x \u21a6 x/(x+1)"},"headline":"A142460 \u2192 x \u21a6 x/(x+1)","chain":null}
