{"id":"n-530cbe1f55af9e87","revision":0,"first_index":0,"fingerprint":"f6-da2c2ef4620dc58e","name":"theta_plus_one of 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.","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"],["2","2"],["2","6"],["2","10","3"],["2","14","12"],["2","18","27","4"],["2","22","48","20"],["2","26","75","56","5"],["2","30","108","120","30"],["2","34","147","220","100","6"],["2","38","192","364","250","42"],["2","42","243","560","525","162","7"],["2","46","300","816","980","462","56"],["2","50","363","1140","1680","1092","245","8"]],"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:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0","0","0"],["2","4","2"],["-2","4","3"],["0","-2","1"]]},"terms":[{"coeff":{"coeffs":[["0","0","-10","-4"],["2","18","17","-2"],["-2","-4","8","4"],["0","-2","-3","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-2","-10","-8"],["0","2","10","5","-12"],["0","0","2","7","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":171220,"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-98dc23cfe2761ff4"]},{"id":171221,"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-98dc23cfe2761ff4"]},{"id":171222,"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-98dc23cfe2761ff4"]}],"region_exact":{},"incoming_edges":[{"id":"e-98dc23cfe2761ff4","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-530cbe1f55af9e87","premise_count":1,"theorem":"Laguerre's theorem on multiplier sequences: if f is a real polynomial whose roots are all real and <= 0, then (f(k))_(k >= 0) is a multiplier sequence, so x^k -> f(k) x^k maps every real-rooted polynomial to a real-rooted polynomial or zero (Laguerre; see T. Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"certificate_hash":"3607759ee44434fe88861f1eefb2ec45dc98e7f67e83c16f36cd88c4fdbf9e4b","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: 2,  2;\n  3: 2,  6;\n  4: 2, 10,   3;\n  5: 2, 14,  12;\n  6: 2, 18,  27,   4;\n  7: 2, 22,  48,  20;\n  8: 2, 26,  75,  56,  5;\n  9: 2, 30, 108, 120, 30;\n","recurrence_sentence":"The row polynomials satisfy -(2*n^2-2*n+(2*n^3-4*n^2-4*n)*x-(n^3+3*n^2+2*n)*x^2)*P(n,x) = -(2*n^2-2*n+(2*n^3+4*n^2-18*n)*x+(3*n^3-8*n^2-17*n+10)*x^2-(2*n^3+4*n^2-2*n-4)*x^3)*P(n-1,x) + (2*n*x+(2*n^2+10*n-2)*x^2+(7*n^2+5*n-10)*x^3-(4*n^2+12*n+8)*x^4)*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^2 - 2*n + (2*n^3 + 4*n^2 - 18*n)*x + (3*n^3 - 8*n^2 - 17*n + 10)*x^2 - (2*n^3 + 4*n^2 - 2*n - 4)*x^3)*P[n - 1, x] + (2*n*x + (2*n^2 + 10*n - 2)*x^2 + (7*n^2 + 5*n - 10)*x^3 - (4*n^2 + 12*n + 8)*x^4)*D[P[n - 1, x], x])/(-2*n^2 + 2*n - (2*n^3 - 4*n^2 - 4*n)*x + (n^3 + 3*n^2 + 2*n)*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] = (-(2*n^2 - 2*n + (2*n^3 + 4*n^2 - 18*n)*x + (3*n^3 - 8*n^2 - 17*n + 10)*x^2 - (2*n^3 + 4*n^2 - 2*n - 4)*x^3)*v[i - 1] + (2*n*x + (2*n^2 + 10*n - 2)*x^2 + (7*n^2 + 5*n - 10)*x^3 - (4*n^2 + 12*n + 8)*x^4)*deriv(v[i - 1], x))/(-2*n^2 + 2*n - (2*n^3 - 4*n^2 - 4*n)*x + (n^3 + 3*n^2 + 2*n)*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-a5d9cd4cf05da467","steps":["\u03b8+1"],"edges":["e-98dc23cfe2761ff4"],"text":"A129710 \u2192 \u03b8+1"},"headline":"A129710 \u2192 \u03b8+1","chain":null}