{"id":"n-45a2b5cb11c37f75","revision":0,"first_index":0,"fingerprint":"f6-6367169308476412","name":"mobius_x_over_x_plus_one of one_plus_derivative of 3n + 1 preceded by n zeros.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["1"],["4","8"],["0","14","21"],["0","0","30","40"],["0","0","0","52","65"],["0","0","0","0","80","96"],["0","0","0","0","0","114","133"],["0","0","0","0","0","0","154","176"],["0","0","0","0","0","0","0","200","225"],["0","0","0","0","0","0","0","0","252","280"],["0","0","0","0","0","0","0","0","0","310","341"],["0","0","0","0","0","0","0","0","0","0","374","408"],["0","0","0","0","0","0","0","0","0","0","0","444","481"],["0","0","0","0","0","0","0","0","0","0","0","0","520","560"],["0","0","0","0","0","0","0","0","0","0","0","0","0","602","645"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","12","35","70","117","176","247","330","425","532","651","782","925","1080","1247"],"nonn":true,"definitions":[{"position":0,"source":"derived:mobius_x_over_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["-2","-4","-2"],["3","6","3"]]},"terms":[{"coeff":{"coeffs":[["8","26","29","12","1"],["-4","-10","-9","-4","-1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","4","13","15","7","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":172243,"position":0,"claim":"real_rooted","from_index":0,"region_key":"[-1, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in [-1, 0]","region_exact":null,"edges":["e-2a5f1b2327c23b03"]},{"id":172244,"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-2a5f1b2327c23b03"]},{"id":172245,"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-2a5f1b2327c23b03"]}],"region_exact":{},"incoming_edges":[{"id":"e-2a5f1b2327c23b03","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"mobius\",\"a\":\"1\",\"b\":\"0\",\"c\":\"1\",\"e\":\"1\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":1}}"},"operation_detail":"mobius, shift 0","conclusion_id":"n-45a2b5cb11c37f75","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":"f1573d30acdaf85ebb870b3c9fec2635ebd15327d21dab45162139771cf2300c","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-bd4ec312ebef253e","role":"sequence","name":"one_plus_derivative of 3n + 1 preceded by n zeros.","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":[{"id":"l-4f41a47b861729d9","kind":"fingerprint_related","node_a_id":"n-56c785d2b08d9301","node_b_id":"n-45a2b5cb11c37f75","fingerprint":"f6-6367169308476412","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6367169308476412"},"other_id":"n-56c785d2b08d9301","other":{"id":"n-56c785d2b08d9301","name":"theta_plus_one of Triangle A127647 * A097806, read by rows.","first_index":0,"fingerprint":"f6-6367169308476412","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","2"],["0","4","6"]],"card_rows":[["1"],["1","2"],["0","4","6"],["0","0","9","12"]],"row_sums_preview":["1","3","10","21","45","88","169","315"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a009f9b98df3ca1d","steps":["\u03b8+1"],"edges":["e-6c1bd56651cfb1a7"],"text":"A128540 \u2192 \u03b8+1"},"headline":"A128540 \u2192 \u03b8+1","chain":null}},{"id":"l-87bc084e0b9d6f6a","kind":"fingerprint_related","node_a_id":"n-45a2b5cb11c37f75","node_b_id":"n-8a3afca20b1dac35","fingerprint":"f6-6367169308476412","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6367169308476412"},"other_id":"n-8a3afca20b1dac35","other":{"id":"n-8a3afca20b1dac35","name":"mobius_x_over_x_plus_one of one_plus_derivative of Triangle read by rows: n-th row is n^2 preceded by (n-1) zeros.","first_index":0,"fingerprint":"f6-6367169308476412","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","8"],["0","18","27"]],"card_rows":[["1"],["4","8"],["0","18","27"],["0","0","48","64"]],"row_sums_preview":["1","12","45","112","225","396","637","960"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7b3743413970d91c","steps":["\u03b8+1","D+1","x \u21a6 x/(x+1)"],"edges":["e-06738ea5bb413735","e-2dc8b6d950a061ae","e-29ed467f74ea37ef"],"text":"image under \u03b8+1 \u2192 \u03b8+1 \u2192 D+1 \u2192 x \u21a6 x/(x+1)"},"headline":"image under \u03b8+1 \u2192 \u03b8+1 \u2192 D+1 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-8d478685c4645744","kind":"fingerprint_related","node_a_id":"n-0ae5dc943046865a","node_b_id":"n-45a2b5cb11c37f75","fingerprint":"f6-6367169308476412","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-6367169308476412"},"other_id":"n-0ae5dc943046865a","other":{"id":"n-0ae5dc943046865a","name":"theta_plus_one of Riordan array (1+x, 1) read by rows.","first_index":0,"fingerprint":"f6-6367169308476412","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","2"],["0","2","3"]],"card_rows":[["1"],["1","2"],["0","2","3"],["0","0","3","4"]],"row_sums_preview":["1","3","5","7","9","11","13","15"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-38c9f89c1db150a9","steps":["\u03b8+1"],"edges":["e-aeaa859dbac70c25"],"text":"A097806 \u2192 \u03b8+1"},"headline":"A097806 \u2192 \u03b8+1","chain":null}}],"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,  8;\n  2: 0, 14, 21;\n  3: 0,  0, 30, 40;\n  4: 0,  0,  0, 52, 65;\n  5: 0,  0,  0,  0, 80,  96;\n  6: 0,  0,  0,  0,  0, 114, 133;\n  7: 0,  0,  0,  0,  0,   0, 154, 176;\n  8: 0,  0,  0,  0,  0,   0,   0, 200, 225;\n  9: 0,  0,  0,  0,  0,   0,   0,   0, 252, 280;\n","recurrence_sentence":"The row polynomials satisfy (3*n-2+(6*n-4)*x+(3*n-2)*x^2)*P(n,x) = -(4*n-8+(10*n-26)*x+(9*n-29)*x^2+(4*n-12)*x^3+(n-1)*x^4)*P(n-1,x) + (4*x+13*x^2+15*x^3+7*x^4+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[(-(4*n - 8 + (10*n - 26)*x + (9*n - 29)*x^2 + (4*n - 12)*x^3 + (n - 1)*x^4)*P[n - 1, x] + (4*x + 13*x^2 + 15*x^3 + 7*x^4 + x^5)*D[P[n - 1, x], x])/(3*n - 2 + (6*n - 4)*x + (3*n - 2)*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] = (-(4*n - 8 + (10*n - 26)*x + (9*n - 29)*x^2 + (4*n - 12)*x^3 + (n - 1)*x^4)*v[i - 1] + (4*x + 13*x^2 + 15*x^3 + 7*x^4 + x^5)*deriv(v[i - 1], x))/(3*n - 2 + (6*n - 4)*x + (3*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 - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-ccc86c3dcf2ac685","steps":["D+1","x \u21a6 x/(x+1)"],"edges":["e-133ff869421f85a2","e-2a5f1b2327c23b03"],"text":"A131431 \u2192 D+1 \u2192 x \u21a6 x/(x+1)"},"headline":"A131431 \u2192 D+1 \u2192 x \u21a6 x/(x+1)","chain":null}