{"id":"n-2b307fcb6ac49d03","revision":0,"first_index":0,"fingerprint":"f6-12a125bb843728b5","name":"theta_plus_one_squared 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","4"],["2","12"],["2","20","9"],["2","28","36"],["2","36","81","16"],["2","44","144","80"],["2","52","225","224","25"],["2","60","324","480","150"],["2","68","441","880","500","36"],["2","76","576","1456","1250","252"],["2","84","729","2240","2625","972","49"],["2","92","900","3264","4900","2772","392"],["2","100","1089","4560","8400","6552","1715","64"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","2","6","14","31","66","135","270","528","1016","1927","3612","6701","12322","22482"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one_squared","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0","0","0","0"],["0","4","-8","0"],["2","-2","-2","2"],["-2","16","11","5"],["0","-6","7","4"],["0","0","-2","1"]]},"terms":[{"coeff":{"coeffs":[["0","0","-20","8","-8"],["0","4","-44","-46","-16"],["2","26","43","11","-2"],["-2","0","48","45","14"],["0","-6","-13","9","10"],["0","0","-2","-3","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-2","-8","-4","-16"],["0","0","-6","-36","-60","-48"],["0","2","10","1","-41","-52"],["0","0","6","27","6","-24"],["0","0","0","2","7","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":171223,"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-98db3ebe0a664f24"]},{"id":171224,"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-98db3ebe0a664f24"]},{"id":171225,"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-98db3ebe0a664f24"]}],"region_exact":{},"incoming_edges":[{"id":"e-98db3ebe0a664f24","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"2\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-2b307fcb6ac49d03","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":"07fd310caa39ddcb53b4dfdef8c3726cf118c5441e0f8e3e078416948bfd05dd","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,  4;\n  3: 2, 12;\n  4: 2, 20,   9;\n  5: 2, 28,  36;\n  6: 2, 36,  81,  16;\n  7: 2, 44, 144,  80;\n  8: 2, 52, 225, 224,  25;\n  9: 2, 60, 324, 480, 150;\n","recurrence_sentence":"The row polynomials satisfy -(2*n^3-2*n^2+(6*n^4-16*n^3+2*n^2-4*n)*x+(2*n^5-7*n^4-11*n^3+2*n^2+8*n)*x^2-(n^5+4*n^4+5*n^3+2*n^2)*x^3)*P(n,x) = -(2*n^3-2*n^2+(6*n^4-26*n^2-4*n)*x+(2*n^5+13*n^4-48*n^3-43*n^2+44*n+20)*x^2+(3*n^5-9*n^4-45*n^3-11*n^2+46*n-8)*x^3-(2*n^5+10*n^4+14*n^3-2*n^2-16*n-8)*x^4)*P(n-1,x) + (2*n^2*x+(6*n^3+10*n^2-6*n-2)*x^2+(2*n^4+27*n^3+n^2-36*n-8)*x^3+(7*n^4+6*n^3-41*n^2-60*n-4)*x^4-(4*n^4+24*n^3+52*n^2+48*n+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 - 2*n^2 + (6*n^4 - 26*n^2 - 4*n)*x + (2*n^5 + 13*n^4 - 48*n^3 - 43*n^2 + 44*n + 20)*x^2 + (3*n^5 - 9*n^4 - 45*n^3 - 11*n^2 + 46*n - 8)*x^3 - (2*n^5 + 10*n^4 + 14*n^3 - 2*n^2 - 16*n - 8)*x^4)*P[n - 1, x] + (2*n^2*x + (6*n^3 + 10*n^2 - 6*n - 2)*x^2 + (2*n^4 + 27*n^3 + n^2 - 36*n - 8)*x^3 + (7*n^4 + 6*n^3 - 41*n^2 - 60*n - 4)*x^4 - (4*n^4 + 24*n^3 + 52*n^2 + 48*n + 16)*x^5)*D[P[n - 1, x], x])/(-2*n^3 + 2*n^2 - (6*n^4 - 16*n^3 + 2*n^2 - 4*n)*x - (2*n^5 - 7*n^4 - 11*n^3 + 2*n^2 + 8*n)*x^2 + (n^5 + 4*n^4 + 5*n^3 + 2*n^2)*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 - 2*n^2 + (6*n^4 - 26*n^2 - 4*n)*x + (2*n^5 + 13*n^4 - 48*n^3 - 43*n^2 + 44*n + 20)*x^2 + (3*n^5 - 9*n^4 - 45*n^3 - 11*n^2 + 46*n - 8)*x^3 - (2*n^5 + 10*n^4 + 14*n^3 - 2*n^2 - 16*n - 8)*x^4)*v[i - 1] + (2*n^2*x + (6*n^3 + 10*n^2 - 6*n - 2)*x^2 + (2*n^4 + 27*n^3 + n^2 - 36*n - 8)*x^3 + (7*n^4 + 6*n^3 - 41*n^2 - 60*n - 4)*x^4 - (4*n^4 + 24*n^3 + 52*n^2 + 48*n + 16)*x^5)*deriv(v[i - 1], x))/(-2*n^3 + 2*n^2 - (6*n^4 - 16*n^3 + 2*n^2 - 4*n)*x - (2*n^5 - 7*n^4 - 11*n^3 + 2*n^2 + 8*n)*x^2 + (n^5 + 4*n^4 + 5*n^3 + 2*n^2)*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":["(\u03b8+1)\u00b2"],"edges":["e-98db3ebe0a664f24"],"text":"A129710 \u2192 (\u03b8+1)\u00b2"},"headline":"A129710 \u2192 (\u03b8+1)\u00b2","chain":null}