{"id":"n-91b6a5a9e4e04cbc","revision":1,"first_index":0,"fingerprint":"f6-0d3e004da859c95a","name":"basis-map coefficients of A Jacobsthal triangle.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":false,"rows":[["1"],["-12","1"],["120","-22","1"],["-1152","336","-32","1"],["10944","-4416","652","-42","1"],["-103680","53568","-10800","1068","-52","1"],["981504","-618624","158688","-21304","1584","-62","1"],["-9289728","6912000","-2156544","367104","-36928","2200","-72","1"],["87920640","-75423744","27744768","-5733120","729616","-58672","2916","-82","1"],["-832094208","808538112","-342779904","83418624","-12868032","1307024","-87536","3732","-92","1"],["7875035136","-8548540416","4105175040","-1150718976","208857984","-25684960","2170128","-124520","4648","-102","1"],["-74530160640","89388417024","-47966846976","15226195968","-3181344768","459972864","-47011584","3399728","-170624","5664","-112","1"],["705361084416","-926276124672","549369298944","-194853593088","46088303616","-7666864128","920658880","-80479584","5086624","-226848","6780","-122","1"],["-6675609157632","9526292250624","-6188726550528","2426250018816","-641108570112","120688634880","-16666546944","1710797248","-130624640","7331616","-294192","7996","-132","1"],["63178643865600","-97349767004160","68753595432960","-29528501944320","8623726387200","-1813190842368","283257656832","-33423286144","2995254784","-202986432","10245504","-373656","9312","-142","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","-11","99","-847","7139","-59895","501787","-4201967","35182323","-294562279","2466173963","-20647543455","172867150819","-1447292702999","12117142414971"],"nonn":false,"definitions":[{"position":0,"source":"derived:basis_map_coefficients_of:oeis_catalog:A113953","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["-12","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-12","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-24","2"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":160277,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(4, 12]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"4"},"upper":{"closed":true,"value":"12"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (4, 12]","region_exact":null,"edges":["e-0ef3f7736abad08b"]},{"id":160278,"position":1,"claim":"one_signed_coefficients","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":true,"claim":"one_signed_coefficients","from_index":0},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 0: the coefficients of P_n(-x) have one sign","region_exact":null,"edges":["e-0ef3f7736abad08b"]},{"id":160279,"position":2,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-0ef3f7736abad08b"]}],"region_exact":{},"incoming_edges":[{"id":"e-0ef3f7736abad08b","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"-12\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-91b6a5a9e4e04cbc","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":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"98be90f0b4526e0c57ad8753252511fe7fa91d7774b3c2f5341d8fa982591eba","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"4"},"upper":{"closed":true,"value":"12"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-ba196dcd01800741","role":"sequence","name":"A Jacobsthal triangle.","uses":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-8"},"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-8, 0]"}]}],"used_by":[{"id":"e-39068ae2f1d878f6","operation":"basis_map","operation_detail":"A038327","conclusion_id":"n-ba196dcd01800741","conclusion_name":"A Jacobsthal triangle."}],"used_by_total":1,"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:        -12,         1;\n  2:        120,       -22,          1;\n  3:      -1152,       336,        -32,        1;\n  4:      10944,     -4416,        652,      -42,         1;\n  5:    -103680,     53568,     -10800,     1068,       -52,       1;\n  6:     981504,   -618624,     158688,   -21304,      1584,     -62,      1;\n  7:   -9289728,   6912000,   -2156544,   367104,    -36928,    2200,    -72,    1;\n  8:   87920640, -75423744,   27744768, -5733120,    729616,  -58672,   2916,  -82,   1;\n  9: -832094208, 808538112, -342779904, 83418624, -12868032, 1307024, -87536, 3732, -92, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(12-x)*P(n-1,x) - (24-2*x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = -12 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := -12 + x;\nP[n_, x_] := P[n, x] = Expand[-(12 - x)*P[n - 1, x] - (24 - 2*x)*P[n - 2, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = -12 + x; for(n = 2, N, my(i = n + 1); v[i] = -(12 - x)*v[i - 1] - (24 - 2*x)*v[i - 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-ba196dcd01800741","steps":["x \u21a6 x\u221212"],"edges":["e-0ef3f7736abad08b"],"text":"A113953 \u2192 x \u21a6 x\u221212"},"headline":"A113953 \u2192 x \u21a6 x\u221212","chain":null}