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A Fibonacci binary word is a binary word having no 00 subword.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["2"],["3","1"],["5","3"],["7","7","1"],["9","15","4"],["11","27","12","1"],["13","43","31","5"],["15","63","67","18","1"],["17","87","126","54","6"],["19","115","214","135","25","1"],["21","147","337","291","85","7"],["23","183","501","560","240","33","1"],["25","223","712","988","581","125","8"],["27","267","976","1629","1246","392","42","1"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","2","4","8","15","28","51","92","164","290","509","888","1541","2662","4580"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["6","-24","-10","8"],["-13","31","-9","4"],["9","-8","4","0"],["-2","1","0","0"]]},"terms":[{"coeff":{"coeffs":[["2","-11","-73","46","-8"],["-7","0","59","-30","8"],["7","14","-18","8","0"],["-2","-3","2","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1","-6","-31","32","-16"],["0","-3","-4","28","-16","0"],["0","2","7","-4","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":171217,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-1f4d8b5f13666237"]},{"id":171218,"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-1f4d8b5f13666237"]},{"id":171219,"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-1f4d8b5f13666237"]}],"region_exact":{},"incoming_edges":[{"id":"e-1f4d8b5f13666237","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-01db3e9311c87670","premise_count":1,"theorem":"Laguerre's theorem on differential operators: if phi is a real polynomial with only real roots, then phi(D) maps every real-rooted polynomial to a real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"1b76e6a20cdbd0089d3e390e5bc1c36c16decebeba5518fdd710508fff25d20b","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-a5d9cd4cf05da467","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of Fibonacci binary words of length n and having k 01 subwords (0 <= k <= floor(n/2)). A Fibonacci binary word is a binary word having no 00 subword.","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:  2;\n  2:  3,  1;\n  3:  5,  3;\n  4:  7,  7,   1;\n  5:  9, 15,   4;\n  6: 11, 27,  12,  1;\n  7: 13, 43,  31,  5;\n  8: 15, 63,  67, 18, 1;\n  9: 17, 87, 126, 54, 6;\n","recurrence_sentence":"The row polynomials satisfy -(2*n^3-9*n^2+13*n-6-(n^3-8*n^2+31*n-24)*x-(4*n^2-9*n-10)*x^2-(4*n+8)*x^3)*P(n,x) = -(2*n^3-7*n^2+7*n-2+(3*n^3-14*n^2+11)*x-(2*n^3-18*n^2+59*n-73)*x^2-(8*n^2-30*n+46)*x^3-(8*n-8)*x^4)*P(n-1,x) + ((2*n^2-3*n+1)*x+(7*n^2-4*n-6)*x^2-(4*n^2-28*n+31)*x^3-(16*n-32)*x^4-16*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[(-(2*n^3 - 7*n^2 + 7*n - 2 + (3*n^3 - 14*n^2 + 11)*x - (2*n^3 - 18*n^2 + 59*n - 73)*x^2 - (8*n^2 - 30*n + 46)*x^3 - (8*n - 8)*x^4)*P[n - 1, x] + ((2*n^2 - 3*n + 1)*x + (7*n^2 - 4*n - 6)*x^2 - (4*n^2 - 28*n + 31)*x^3 - (16*n - 32)*x^4 - 16*x^5)*D[P[n - 1, x], x])/(-2*n^3 + 9*n^2 - 13*n + 6 + (n^3 - 8*n^2 + 31*n - 24)*x + (4*n^2 - 9*n - 10)*x^2 + (4*n + 8)*x^3)]];\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] = (-(2*n^3 - 7*n^2 + 7*n - 2 + (3*n^3 - 14*n^2 + 11)*x - (2*n^3 - 18*n^2 + 59*n - 73)*x^2 - (8*n^2 - 30*n + 46)*x^3 - (8*n - 8)*x^4)*v[i - 1] + ((2*n^2 - 3*n + 1)*x + (7*n^2 - 4*n - 6)*x^2 - (4*n^2 - 28*n + 31)*x^3 - (16*n - 32)*x^4 - 16*x^5)*deriv(v[i - 1], x))/(-2*n^3 + 9*n^2 - 13*n + 6 + (n^3 - 8*n^2 + 31*n - 24)*x + (4*n^2 - 9*n - 10)*x^2 + (4*n + 8)*x^3)); 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-a5d9cd4cf05da467","steps":["D+1"],"edges":["e-1f4d8b5f13666237"],"text":"A129710 \u2192 D+1"},"headline":"A129710 \u2192 D+1","chain":null}