{"id":"n-7b2cbe30a868f205","revision":0,"first_index":0,"fingerprint":"f6-ef50924260c73c5c","name":"mobius_x_over_x_plus_one of Triangle of central factorial numbers |t(2n,2n-2k)| read by rows.","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":true,"rows":[["1"],["1","2"],["1","7","10"],["1","17","80","100"],["1","34","369","1460","1700"],["1","60","1253","11054","39660","44200"],["1","97","3473","57415","448658","1511620","1635400"],["1","147","8323","231065","3319408","23944520","77216400","81770000"],["1","212","17878","772060","18338633","239706040","1633610200","5100836000","5315050000"],["1","294","35262","2238056","81647553","1743473946","21289505480","139056872400","423583602000","435834100000"],["1","395","64956","5799518","307691209","9989876799","197380374026","2289296925880","14468327714400","43217777902000","44019244100000"],["1","517","113146","13724150","1015232405","47528204297","1416145343504","26369702557052","293762552671760","1808353759058800","5316588148144000","5370347780200000"],["1","662","188111","30130320","3005234155","194736903022","8307734966569","231710777365132","4117369423444300","44403923896464000","267527883211670000","776275629261080000","778700428129000000"],["1","832","300651","62109190","8127388555","705626709372","41413008480309","1644025721681862","43508201575516740","744356725881995000","7816194945610550000","46256015775244980000","132745557402512600000","132379072781930000000"],["1","1029","464555","121337437","20362898985","2306722254707","180421470226593","9802388392302735","367381268746843554","9315472436258792780","154454469944363565000","1586046420060523330000","9245180665125773660000","26283253881076912200000","26078677338040210000000"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","18","198","3564","96228","3656664","186489864","12308331024","1021591474992","104202330449184","12816886645249632","1871265450206446272","319986391985302312512","63357305613089857877376"],"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","1"]]},"terms":[{"coeff":{"coeffs":[["1","2","1"],["0","0","0"],["0","1","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":189911,"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-28bca084b8ef187b"]},{"id":189912,"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-28bca084b8ef187b"]},{"id":189913,"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-28bca084b8ef187b"]}],"region_exact":{},"incoming_edges":[{"id":"e-28bca084b8ef187b","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-7b2cbe30a868f205","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":"292e97de2b5f3aeb011635972e68326f587ead6ffa75703d1b6d5229d78a1330","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-e734a631f8279507","role":"sequence","name":"Triangle of central factorial numbers |t(2n,2n-2k)| read by rows.","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":[],"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,   7,    10;\n  3: 1,  17,    80,     100;\n  4: 1,  34,   369,    1460,     1700;\n  5: 1,  60,  1253,   11054,    39660,      44200;\n  6: 1,  97,  3473,   57415,   448658,    1511620,     1635400;\n  7: 1, 147,  8323,  231065,  3319408,   23944520,    77216400,     81770000;\n  8: 1, 212, 17878,  772060, 18338633,  239706040,  1633610200,   5100836000,   5315050000;\n  9: 1, 294, 35262, 2238056, 81647553, 1743473946, 21289505480, 139056872400, 423583602000, 435834100000;\n","recurrence_sentence":"The row polynomials satisfy (1+x)*P(n,x) = (1+(n^2+2)*x+(n^2+1)*x^2)*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 + (n^2 + 2)*x + (n^2 + 1)*x^2)*P[n - 1, x])/(1 + x)]];\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 + (n^2 + 2)*x + (n^2 + 1)*x^2)*v[i - 1])/(1 + x)); 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-e734a631f8279507","steps":["x \u21a6 x/(x+1)"],"edges":["e-28bca084b8ef187b"],"text":"A008955 \u2192 x \u21a6 x/(x+1)"},"headline":"A008955 \u2192 x \u21a6 x/(x+1)","chain":null}