{"id":"n-503a4b66c4a2e621","revision":0,"first_index":0,"fingerprint":"f6-08e302df1914bf50","name":"mobius_x_over_x_plus_one of basis_map_A013610 of Triangle read by rows: T(n,k) is the number of Schroeder paths of semilength n containing exactly k peaks but no peaks at level one (n >= 1; 0 <= k <= n-1).","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"],["3","6"],["11","43","41"],["45","261","495","306"],["197","1514","4296","5324","2426"],["903","8640","32640","60780","55710","20076"],["4279","48993","231135","574595","793095","575673","171481"],["20793","277179","1568133","4877565","9001965","9850491","5912823","1500666"],["103049","1567444","10340624","38625244","89299574","130777780","118403936","60556360","13386206"],["518859","8867610","66832560","291419100","809859330","1486838808","1802518704","1390770216","619414074","121267476"],["2646723","50209095","425584125","2121870150","6888848400","15212043000","23129966160","23902815780","16060786110","6333508470","1112674026"],["13648869","284576193","2679492915","15035387400","55849483800","144154450440","263744601120","341940477780","307763389890","183085502490","64768540518","10318939956"],["71039373","1614666654","16720196328","104282086920","436186763100","1288714719600","2757034708560","4302252745560","4858738244910","3871857701580","2066012814096","662613483528","96572168916"],["372693519","9171411444","103587872544","710881749864","3306451483764","11004663311928","26957648841552","49201444883448","66890955510522","66889412420688","47809598934240","23124782960136","6782636799900","910896992856"],["1968801519","52148919165","637994151795","4777536490545","24460071110475","90557554844757","249970764794751","522527690389419","831031464814551","1000493352547689","897377746624359","581368788888069","257118983177511","69472483213749","8650566601401"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","9","95","1107","13757","178749","2399251","33009615","463060217","6598306737","95240952039","1389638491371","20462671339125","303702505866405","4538499115369755"],"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":[["6","12","6"],["3","6","3"]]},"terms":[{"coeff":{"coeffs":[["2","-16","-78","-100","-40"],["25","124","213","154","40"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-4","-38","-130","-202","-146","-40"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":183392,"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-d23d3d86265a549b"]},{"id":183393,"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-d23d3d86265a549b"]},{"id":183394,"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-d23d3d86265a549b"]}],"region_exact":{},"incoming_edges":[{"id":"e-d23d3d86265a549b","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-503a4b66c4a2e621","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":"75e751ea6192e17b39bb954bb0f7493c625fde74b25b0065ab60d758c8ddd000","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-cbba416b5dae7a51","role":"sequence","name":"basis_map_A013610 of Triangle read by rows: T(n,k) is the number of Schroeder paths of semilength n containing exactly k peaks but no peaks at level one (n >= 1; 0 <= k <= n-1).","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:      3,       6;\n  2:     11,      43,       41;\n  3:     45,     261,      495,       306;\n  4:    197,    1514,     4296,      5324,      2426;\n  5:    903,    8640,    32640,     60780,     55710,      20076;\n  6:   4279,   48993,   231135,    574595,    793095,     575673,     171481;\n  7:  20793,  277179,  1568133,   4877565,   9001965,    9850491,    5912823,    1500666;\n  8: 103049, 1567444, 10340624,  38625244,  89299574,  130777780,  118403936,   60556360,  13386206;\n  9: 518859, 8867610, 66832560, 291419100, 809859330, 1486838808, 1802518704, 1390770216, 619414074, 121267476;\n","recurrence_sentence":"The row polynomials satisfy (3*n+6+(6*n+12)*x+(3*n+6)*x^2)*P(n,x) = (25*n+2+(124*n-16)*x+(213*n-78)*x^2+(154*n-100)*x^3+(40*n-40)*x^4)*P(n-1,x) - (4+38*x+130*x^2+202*x^3+146*x^4+40*x^5)*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[((25*n + 2 + (124*n - 16)*x + (213*n - 78)*x^2 + (154*n - 100)*x^3 + (40*n - 40)*x^4)*P[n - 1, x] - (4 + 38*x + 130*x^2 + 202*x^3 + 146*x^4 + 40*x^5)*D[P[n - 1, x], x])/(3*n + 6 + (6*n + 12)*x + (3*n + 6)*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] = ((25*n + 2 + (124*n - 16)*x + (213*n - 78)*x^2 + (154*n - 100)*x^3 + (40*n - 40)*x^4)*v[i - 1] - (4 + 38*x + 130*x^2 + 202*x^3 + 146*x^4 + 40*x^5)*deriv(v[i - 1], x))/(3*n + 6 + (6*n + 12)*x + (3*n + 6)*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-57b4b14604f4c0d7","steps":["basis A013610","x \u21a6 x/(x+1)"],"edges":["e-125d456539fb62ba","e-d23d3d86265a549b"],"text":"A126216 \u2192 basis A013610 \u2192 x \u21a6 x/(x+1)"},"headline":"A126216 \u2192 basis A013610 \u2192 x \u21a6 x/(x+1)","chain":null}