{"id":"n-a9419a0f5f0cae19","revision":0,"first_index":0,"fingerprint":"f6-000cdfc9e19dff85","name":"shift_x_plus_one of Triangle T(n,k) read by rows: coefficient of [x^k] of the polynomial p_n(x)=(5-x)*p_{n-1}(x)-p_{n-2}(x), p_0=1, p_1=5-x.","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"],["4","-1"],["15","-8","1"],["56","-46","12","-1"],["209","-232","93","-16","1"],["780","-1091","592","-156","20","-1"],["2911","-4912","3366","-1200","235","-24","1"],["10864","-21468","17784","-8010","2120","-330","28","-1"],["40545","-91824","89238","-48624","16255","-3416","441","-32","1"],["151316","-386373","430992","-275724","111524","-29589","5152","-568","36","-1"],["564719","-1604984","2021103","-1485264","705565","-226464","49756","-7392","711","-40","1"],["2107560","-6598282","9258404","-7686435","4196000","-1581832","420336","-78756","10200","-870","44","-1"],["7865521","-26895704","41610795","-38518880","23764870","-10296864","3213420","-727968","118845","-13640","1045","-48","1"],["29354524","-108850055","184080480","-187999880","129382360","-63370494","22730208","-6046536","1193148","-172535","17776","-1236","52","-1"],["109552575","-437859040","803561180","-897561120","681764450","-372567472","151077906","-46188384","10700283","-1869648","242594","-22672","1443","-56","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","8","21","55","144","377","987","2584","6765","17711","46368","121393","317811","832040"],"nonn":false,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0"],["2"]]},"terms":[{"coeff":{"coeffs":[["4","-1"],["4","-1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-12","8","-1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":180003,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(2, 6)","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"2"},"upper":{"closed":false,"value":"6"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (2, 6)","region_exact":null,"edges":["e-f2a95836fb5b09ee"]},{"id":180004,"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-f2a95836fb5b09ee"]},{"id":180005,"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-f2a95836fb5b09ee"]}],"region_exact":{},"incoming_edges":[{"id":"e-f2a95836fb5b09ee","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-a9419a0f5f0cae19","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":"d536006ae1c3e4413820927b75e56f2112cdb1d9bf3a8ff2b9c27d2306a6acd7","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"2"},"upper":{"closed":false,"value":"6"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-490435c99149594f","role":"sequence","name":"Triangle T(n,k) read by rows: coefficient of [x^k] of the polynomial p_n(x)=(5-x)*p_{n-1}(x)-p_{n-2}(x), p_0=1, p_1=5-x.","uses":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"3"},"upper":{"closed":false,"value":"7"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (3, 7)"}]}],"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:      4,      -1;\n  2:     15,      -8,      1;\n  3:     56,     -46,     12,      -1;\n  4:    209,    -232,     93,     -16,      1;\n  5:    780,   -1091,    592,    -156,     20,     -1;\n  6:   2911,   -4912,   3366,   -1200,    235,    -24,    1;\n  7:  10864,  -21468,  17784,   -8010,   2120,   -330,   28,   -1;\n  8:  40545,  -91824,  89238,  -48624,  16255,  -3416,  441,  -32,  1;\n  9: 151316, -386373, 430992, -275724, 111524, -29589, 5152, -568, 36, -1;\n","recurrence_sentence":"The row polynomials satisfy 2*n*P(n,x) = (4*n+4-(n+1)*x)*P(n-1,x) - (12-8*x+x^2)*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[((4*n + 4 - (n + 1)*x)*P[n - 1, x] - (12 - 8*x + x^2)*D[P[n - 1, x], x])/(2*n)]];\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] = ((4*n + 4 - (n + 1)*x)*v[i - 1] - (12 - 8*x + x^2)*deriv(v[i - 1], x))/(2*n)); 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-490435c99149594f","steps":["x \u21a6 x+1"],"edges":["e-f2a95836fb5b09ee"],"text":"A179900 \u2192 x \u21a6 x+1"},"headline":"A179900 \u2192 x \u21a6 x+1","chain":null}
