{"id":"n-4d1e2be69e19d889","revision":0,"first_index":1,"fingerprint":"f6-702b8764f2c773f5","name":"mobius_x_over_x_plus_one of Triangle T(n,k) with diagonals T(n,n-k) = binomial(n, 2*k).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["0","1"],["0","1","2"],["0","0","3","4"],["0","0","1","8","8"],["0","0","0","5","20","16"],["0","0","0","1","18","48","32"],["0","0","0","0","7","56","112","64"],["0","0","0","0","1","32","160","256","128"],["0","0","0","0","0","9","120","432","576","256"],["0","0","0","0","0","1","50","400","1120","1280","512"],["0","0","0","0","0","0","11","220","1232","2816","2816","1024"],["0","0","0","0","0","0","1","72","840","3584","6912","6144","2048"],["0","0","0","0","0","0","0","13","364","2912","9984","16640","13312","4096"],["0","0","0","0","0","0","0","1","98","1568","9408","26880","39424","28672","8192"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","1","3","7","17","41","99","239","577","1393","3363","8119","19601","47321","114243"],"nonn":true,"definitions":[{"position":0,"source":"derived:mobius_x_over_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["0","1"]],"recurrence":{"denominator":{"coeffs":[["-2","-4","-2"],["1","2","1"]]},"terms":[{"coeff":{"coeffs":[["2","4","2"],["-1","-2","-1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2","6","6","2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":171851,"position":0,"claim":"real_rooted","from_index":1,"region_key":"[-1, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in [-1, 0]","region_exact":null,"edges":["e-c830e34032c85a28"]},{"id":171852,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-c830e34032c85a28"]},{"id":171853,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-c830e34032c85a28"]}],"region_exact":{},"incoming_edges":[{"id":"e-c830e34032c85a28","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"mobius\",\"a\":\"1\",\"b\":\"0\",\"c\":\"1\",\"e\":\"1\",\"schedule\":{\"first_index\":1,\"first_degree\":0,\"step\":1}}"},"operation_detail":"mobius, shift 0","conclusion_id":"n-4d1e2be69e19d889","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":"fTransform","lean_declarations":["RealRooted.splits_fPolynomial"],"certificate_hash":"04a21c47a6c602f9a4a989106a8b6c586de45984e4b204e1017f3d591d3e27d7","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-cf23198864b5678a","role":"sequence","name":"Triangle T(n,k) with diagonals T(n,n-k) = binomial(n, 2*k).","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-036cc21daa87eba5","kind":"fingerprint_related","node_a_id":"n-4d1e2be69e19d889","node_b_id":"n-97deebd28987ca5a","fingerprint":"f6-702b8764f2c773f5","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-702b8764f2c773f5"},"other_id":"n-97deebd28987ca5a","other":{"id":"n-97deebd28987ca5a","name":"shift_x_plus_one of veronese_section_2_0 of Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-702b8764f2c773f5","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["4"],["20","10"]],"card_rows":[["1"],["4"],["20","10"],["80","60"]],"row_sums_preview":["1","4","30","140","595","2296","8316","28680"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a820c1859f68e71c","steps":["Veronese 2","basis A007318"],"edges":["e-9985dc3bbd6a1733","e-05c29514f8b31a01"],"text":"A178820 \u2192 Veronese 2 \u2192 basis A007318"},"headline":"A178820 \u2192 Veronese 2 \u2192 basis A007318","chain":null}},{"id":"l-21a0b233942ac987","kind":"fingerprint_related","node_a_id":"n-4d1e2be69e19d889","node_b_id":"n-66ad0c588cd0c2e0","fingerprint":"f6-702b8764f2c773f5","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-702b8764f2c773f5"},"other_id":"n-66ad0c588cd0c2e0","other":{"id":"n-66ad0c588cd0c2e0","name":"shift_x_plus_one of veronese_section_2_0 of Triangle whose (i,j)-th entry is binomial(i,j)*2^i.","first_index":0,"fingerprint":"f6-702b8764f2c773f5","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2"],["8","4"]],"card_rows":[["1"],["2"],["8","4"],["32","24"]],"row_sums_preview":["1","2","12","56","272","1312","6336","30592"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d28ad9841fc73317","steps":["Veronese 2","basis A007318"],"edges":["e-01a436d0e1f57933","e-4c7f7e6e93d0f4c2"],"text":"A038208 \u2192 Veronese 2 \u2192 basis A007318"},"headline":"A038208 \u2192 Veronese 2 \u2192 basis A007318","chain":null}},{"id":"l-2e8537f347f45f35","kind":"fingerprint_related","node_a_id":"n-4d1e2be69e19d889","node_b_id":"n-f47ef477093d1825","fingerprint":"f6-702b8764f2c773f5","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-702b8764f2c773f5"},"other_id":"n-f47ef477093d1825","other":{"id":"n-f47ef477093d1825","name":"shift_x_plus_one of veronese_section_2_0 of Triangle read by rows: T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-702b8764f2c773f5","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["6"],["42","21"]],"card_rows":[["1"],["6"],["42","21"],["224","168"]],"row_sums_preview":["1","6","63","392","2142","10332","45738","189288"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-97228a88c81561dc","steps":["Veronese 2","x \u21a6 x+1"],"edges":["e-496b1966a8762df1","e-3dd245e74534b1c0"],"text":"A178822 \u2192 Veronese 2 \u2192 x \u21a6 x+1"},"headline":"A178822 \u2192 Veronese 2 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-500a4ed7ac195a24","kind":"fingerprint_related","node_a_id":"n-4d1e2be69e19d889","node_b_id":"n-5b714ee1f71b153f","fingerprint":"f6-702b8764f2c773f5","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-702b8764f2c773f5"},"other_id":"n-5b714ee1f71b153f","other":{"id":"n-5b714ee1f71b153f","name":"shift_x_plus_one of veronese_section_2_0 of Exponential Riordan array [x*exp(x),x].","first_index":0,"fingerprint":"f6-702b8764f2c773f5","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["1"],["2"]],"card_rows":[["1"],["2"],["6","3"],["16","12"]],"row_sums_preview":["0","1","2","9","28","85","246","693"],"regions":["(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7b5887f058255aa1","steps":["Veronese 2","x \u21a6 x+1"],"edges":["e-30ed1b2c5535cdc0","e-0b9e68fa7443398c"],"text":"A003506 \u2192 Veronese 2 \u2192 x \u21a6 x+1"},"headline":"A003506 \u2192 Veronese 2 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-9eef6ade7664ab1d","kind":"fingerprint_related","node_a_id":"n-4d1e2be69e19d889","node_b_id":"n-777834828b7583c2","fingerprint":"f6-702b8764f2c773f5","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-702b8764f2c773f5"},"other_id":"n-777834828b7583c2","other":{"id":"n-777834828b7583c2","name":"shift_x_plus_one of veronese_section_2_0 of basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-702b8764f2c773f5","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":0,"preview_rows":[["1"],["-1"],["2","1"]],"card_rows":[["1"],["-1"],["2","1"],["-4","-3"]],"row_sums_preview":["1","-1","3","-7","17","-41","99","-239"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-74c1e94669ff21ff","steps":["Veronese 2","x \u21a6 x+1"],"edges":["e-969b818b29eb86a6","e-9f1fd6b6bc67e5d0"],"text":"basis-map coefficients of A113214, shifted \u2192 Veronese 2 \u2192 x \u21a6 x+1"},"headline":"basis-map coefficients of A113214, shifted \u2192 Veronese 2 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-a5da937d6e0d6a1b","kind":"fingerprint_related","node_a_id":"n-495335f6a4ca3289","node_b_id":"n-4d1e2be69e19d889","fingerprint":"f6-702b8764f2c773f5","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-702b8764f2c773f5"},"other_id":"n-495335f6a4ca3289","other":{"id":"n-495335f6a4ca3289","name":"Riordan triangle ((1-x)/(1-2*x), x^2/(1-2*x)).","first_index":0,"fingerprint":"f6-702b8764f2c773f5","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1"],["2","1"]],"card_rows":[["1"],["1"],["2","1"],["4","3"]],"row_sums_preview":["1","1","3","7","17","41","99","239"],"regions":["(-inf, 0)","(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":["A201701"],"derivation":{"root":"n-495335f6a4ca3289","steps":[],"edges":[],"text":"A201701"},"headline":"A201701: Riordan triangle ((1-x)/(1-2*x), x^2/(1-2*x)).","chain":null}},{"id":"l-bd91bb7d2eb193f2","kind":"fingerprint_related","node_a_id":"n-4d1e2be69e19d889","node_b_id":"n-fa2ea18f13501719","fingerprint":"f6-702b8764f2c773f5","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-702b8764f2c773f5"},"other_id":"n-fa2ea18f13501719","other":{"id":"n-fa2ea18f13501719","name":"shift_x_plus_one of veronese_section_2_0 of Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-702b8764f2c773f5","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["5"],["30","15"]],"card_rows":[["1"],["5"],["30","15"],["140","105"]],"row_sums_preview":["1","5","45","245","1190","5166","20790","78870"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-79986ab5023c2764","steps":["Veronese 2","x \u21a6 x+1"],"edges":["e-8ba085c2fb60bb1b","e-68de572e63ababc1"],"text":"A178821 \u2192 Veronese 2 \u2192 x \u21a6 x+1"},"headline":"A178821 \u2192 Veronese 2 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-c1c4d68a22f51d32","kind":"fingerprint_related","node_a_id":"n-4d1e2be69e19d889","node_b_id":"n-d7283fa0e9ac22b4","fingerprint":"f6-702b8764f2c773f5","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-702b8764f2c773f5"},"other_id":"n-d7283fa0e9ac22b4","other":{"id":"n-d7283fa0e9ac22b4","name":"shift_x_plus_one of Table: T(n,k) = n!*binomial(n+1,2*k).","first_index":0,"fingerprint":"f6-702b8764f2c773f5","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","1"],["8","6"]],"card_rows":[["1"],["2","1"],["8","6"],["48","48","6"]],"row_sums_preview":["1","3","14","102","984","11880","172080","2908080"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-90f6949024558b8d","steps":["basis A007318"],"edges":["e-26a3e60cc630357d"],"text":"A228955 \u2192 basis A007318"},"headline":"A228955 \u2192 basis A007318","chain":null}},{"id":"l-c30be0ba53cbbc83","kind":"fingerprint_related","node_a_id":"n-4d1e2be69e19d889","node_b_id":"n-0d4f187a7cfebed3","fingerprint":"f6-702b8764f2c773f5","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-702b8764f2c773f5"},"other_id":"n-0d4f187a7cfebed3","other":{"id":"n-0d4f187a7cfebed3","name":"shift_x_plus_one of veronese_section_2_0 of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).","first_index":0,"fingerprint":"f6-702b8764f2c773f5","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["2"],["2"],["4","2"]],"card_rows":[["2"],["2"],["4","2"],["8","6"]],"row_sums_preview":["2","2","6","14","34","82","198","478"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-089a9b4f824a1c20","steps":["Veronese 2","x \u21a6 x+1"],"edges":["e-e811ab86eff6d185","e-1f4bc87c26595050"],"text":"A028326 \u2192 Veronese 2 \u2192 x \u21a6 x+1"},"headline":"A028326 \u2192 Veronese 2 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-d730d44810bff1a3","kind":"fingerprint_related","node_a_id":"n-4d1e2be69e19d889","node_b_id":"n-cd85e79f8f3e8a20","fingerprint":"f6-702b8764f2c773f5","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-702b8764f2c773f5"},"other_id":"n-cd85e79f8f3e8a20","other":{"id":"n-cd85e79f8f3e8a20","name":"shift_x_plus_one of Triangle read by rows: T(n,k) is the number of circular binary words of length n having k occurrences of 01 (0 <= k <= floor(n/2)).","first_index":0,"fingerprint":"f6-702b8764f2c773f5","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2"],["4","2"]],"card_rows":[["1"],["2"],["4","2"],["8","6"]],"row_sums_preview":["1","2","6","14","34","82","198","478"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ced86a7ce48c966e","steps":["basis A007318"],"edges":["e-6dc4734e89bdcd0a"],"text":"A119462 \u2192 basis A007318"},"headline":"A119462 \u2192 basis A007318","chain":null}}],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 1;\n   2: 0, 1;\n   3: 0, 1, 2;\n   4: 0, 0, 3, 4;\n   5: 0, 0, 1, 8,  8;\n   6: 0, 0, 0, 5, 20, 16;\n   7: 0, 0, 0, 1, 18, 48,  32;\n   8: 0, 0, 0, 0,  7, 56, 112,  64;\n   9: 0, 0, 0, 0,  1, 32, 160, 256, 128;\n  10: 0, 0, 0, 0,  0,  9, 120, 432, 576, 256;\n","recurrence_sentence":"The row polynomials satisfy (n-2+(2*n-4)*x+(n-2)*x^2)*P(n,x) = -(n-2+(2*n-4)*x+(n-2)*x^2)*P(n-1,x) + (2*x+6*x^2+6*x^3+2*x^4)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n - 2 + (2*n - 4)*x + (n - 2)*x^2)*P[n - 1, x] + (2*x + 6*x^2 + 6*x^3 + 2*x^4)*D[P[n - 1, x], x])/(n - 2 + (2*n - 4)*x + (n - 2)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = x; for(n = 3, N, my(i = n); v[i] = (-(n - 2 + (2*n - 4)*x + (n - 2)*x^2)*v[i - 1] + (2*x + 6*x^2 + 6*x^3 + 2*x^4)*deriv(v[i - 1], x))/(n - 2 + (2*n - 4)*x + (n - 2)*x^2)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-cf23198864b5678a","steps":["x \u21a6 x/(x+1)"],"edges":["e-c830e34032c85a28"],"text":"A098158 \u2192 x \u21a6 x/(x+1)"},"headline":"A098158 \u2192 x \u21a6 x/(x+1)","chain":null}