{"id":"n-193bc293afe2c18c","revision":0,"first_index":1,"fingerprint":"f6-d37e58d0d0ef7fc3","name":"mobius_x_over_x_plus_one of Triangle of numbers used for basis change between certain falling factorials.","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","5"],["1","14","43"],["1","27","219","529"],["1","44","666","4084","8501"],["1","65","1570","17530","90005","169021"],["1","90","3165","55460","508635","2304546","4010455"],["1","119","5733","144515","2051875","16351461","67457719","110676833"],["1","152","9604","328664","6645310","81058376","580699252","2226361544","3484717129"],["1","189","15156","675444","18396126","316786806","3440379684","22659789636","81889548489","123320412181"],["1","230","22815","1283160","45230010","1042022772","15856491630","157011698040","965854925685","3324190884310","4847038223171"],["1","275","33055","2289045","101312970","3004238622","60793963230","837936237930","7693411049445","44713244264095","147698428274771","209536628422705"],["1","324","46398","3878380","210476475","7801382952","202199942868","3686407269336","46838340652455","403715781663700","2236271229766926","7132319682005244","9882471447634813"],["1","377","63414","6294574","410891195","18603406107","600239689764","13954340113428","233397870242823","2769489619877695","22622712699507206","120248045391858462","372055492525328989","505033804901100749"],["1","434","84721","9850204","761261501","41324825542","1623087118653","46798973024232","994140043339443","15461875871984542","173063054721848291","1349684919461358844","6921487652028642511","20854972330296659114","27802319803528367791"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","6","58","776","13296","278192","6882352","196688256","6379820032","231646003712","9311040781824","410543550086144","19705513104719872","1022977516608374784","57118032240108365824"],"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","3","3","1"]]},"terms":[{"coeff":{"coeffs":[["1","0","-6","-8","-3","0"],["0","4","10","6","-2","-2"],["0","0","1","3","3","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-1","-3","-3","-1","0"],["0","0","0","-1","-3","-3","-1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":189395,"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-a125ce1800ac1387"]},{"id":189396,"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-a125ce1800ac1387"]},{"id":189397,"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-a125ce1800ac1387"]}],"region_exact":{},"incoming_edges":[{"id":"e-a125ce1800ac1387","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-193bc293afe2c18c","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":"eb2dd0dc701b1bcf0e555c4f077329a9affef63c5ada5453fb6252f975060854","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-ee693a689121a12c","role":"sequence","name":"Triangle of numbers used for basis change between certain falling factorials.","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":[],"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: 1,   5;\n   3: 1,  14,    43;\n   4: 1,  27,   219,    529;\n   5: 1,  44,   666,   4084,     8501;\n   6: 1,  65,  1570,  17530,    90005,    169021;\n   7: 1,  90,  3165,  55460,   508635,   2304546,    4010455;\n   8: 1, 119,  5733, 144515,  2051875,  16351461,   67457719,   110676833;\n   9: 1, 152,  9604, 328664,  6645310,  81058376,  580699252,  2226361544,  3484717129;\n  10: 1, 189, 15156, 675444, 18396126, 316786806, 3440379684, 22659789636, 81889548489, 123320412181;\n","recurrence_sentence":"The row polynomials satisfy (1+3*x+3*x^2+x^3)*P(n,x) = (1+4*n*x+(n^2+10*n-6)*x^2+(3*n^2+6*n-8)*x^3+(3*n^2-2*n-3)*x^4+(n^2-2*n)*x^5)*P(n-1,x) - (x^2+(n+3)*x^3+(3*n+3)*x^4+(3*n+1)*x^5+n*x^6)*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[((1 + 4*n*x + (n^2 + 10*n - 6)*x^2 + (3*n^2 + 6*n - 8)*x^3 + (3*n^2 - 2*n - 3)*x^4 + (n^2 - 2*n)*x^5)*P[n - 1, x] - (x^2 + (n + 3)*x^3 + (3*n + 3)*x^4 + (3*n + 1)*x^5 + n*x^6)*D[P[n - 1, x], x])/(1 + 3*x + 3*x^2 + x^3)]];\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] = ((1 + 4*n*x + (n^2 + 10*n - 6)*x^2 + (3*n^2 + 6*n - 8)*x^3 + (3*n^2 - 2*n - 3)*x^4 + (n^2 - 2*n)*x^5)*v[i - 1] - (x^2 + (n + 3)*x^3 + (3*n + 3)*x^4 + (3*n + 1)*x^5 + n*x^6)*deriv(v[i - 1], x))/(1 + 3*x + 3*x^2 + x^3)); 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-ee693a689121a12c","steps":["x \u21a6 x/(x+1)"],"edges":["e-a125ce1800ac1387"],"text":"A089503 \u2192 x \u21a6 x/(x+1)"},"headline":"A089503 \u2192 x \u21a6 x/(x+1)","chain":null}