{"id":"n-8a3afca20b1dac35","revision":0,"first_index":0,"fingerprint":"f6-6367169308476412","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.","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","18","27"],["0","0","48","64"],["0","0","0","100","125"],["0","0","0","0","180","216"],["0","0","0","0","0","294","343"],["0","0","0","0","0","0","448","512"],["0","0","0","0","0","0","0","648","729"],["0","0","0","0","0","0","0","0","900","1000"],["0","0","0","0","0","0","0","0","0","1210","1331"],["0","0","0","0","0","0","0","0","0","0","1584","1728"],["0","0","0","0","0","0","0","0","0","0","0","2028","2197"],["0","0","0","0","0","0","0","0","0","0","0","0","2548","2744"],["0","0","0","0","0","0","0","0","0","0","0","0","0","3150","3375"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","12","45","112","225","396","637","960","1377","1900","2541","3312","4225","5292","6525"],"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":[["0","0","0"],["0","0","0"],["1","2","1"]]},"terms":[{"coeff":{"coeffs":[["8","26","29","12","1"],["-4","-12","-14","-8","-2"],["0","2","5","4","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","4","13","15","7","1"],["0","0","-1","-3","-3","-1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":173234,"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-29ed467f74ea37ef"]},{"id":173235,"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-29ed467f74ea37ef"]},{"id":173236,"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-29ed467f74ea37ef"]}],"region_exact":{},"incoming_edges":[{"id":"e-29ed467f74ea37ef","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-8a3afca20b1dac35","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":"059f2730beef797ce302ac60dee6fff7af5d3d3e380aa3557cb84f6052c323a4","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-fc17ecb54217ecb0","role":"sequence","name":"one_plus_derivative of Triangle read by rows: n-th row is n^2 preceded by (n-1) 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-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-45a2b5cb11c37f75","other":{"id":"n-45a2b5cb11c37f75","name":"mobius_x_over_x_plus_one of one_plus_derivative of 3n + 1 preceded by n 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","14","21"]],"card_rows":[["1"],["4","8"],["0","14","21"],["0","0","30","40"]],"row_sums_preview":["1","12","35","70","117","176","247","330"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"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}},{"id":"l-89812f05c5a8472b","kind":"fingerprint_related","node_a_id":"n-0ae5dc943046865a","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-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}},{"id":"l-a724ff818b194fc2","kind":"fingerprint_related","node_a_id":"n-56c785d2b08d9301","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-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}}],"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, 18, 27;\n  3: 0,  0, 48,  64;\n  4: 0,  0,  0, 100, 125;\n  5: 0,  0,  0,   0, 180, 216;\n  6: 0,  0,  0,   0,   0, 294, 343;\n  7: 0,  0,  0,   0,   0,   0, 448, 512;\n  8: 0,  0,  0,   0,   0,   0,   0, 648, 729;\n  9: 0,  0,  0,   0,   0,   0,   0,   0, 900, 1000;\n","recurrence_sentence":"The row polynomials satisfy (n^2+2*n^2*x+n^2*x^2)*P(n,x) = -(4*n-8-(2*n^2-12*n+26)*x-(5*n^2-14*n+29)*x^2-(4*n^2-8*n+12)*x^3-(n^2-2*n+1)*x^4)*P(n-1,x) + (4*x-(n-13)*x^2-(3*n-15)*x^3-(3*n-7)*x^4-(n-1)*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 - (2*n^2 - 12*n + 26)*x - (5*n^2 - 14*n + 29)*x^2 - (4*n^2 - 8*n + 12)*x^3 - (n^2 - 2*n + 1)*x^4)*P[n - 1, x] + (4*x - (n - 13)*x^2 - (3*n - 15)*x^3 - (3*n - 7)*x^4 - (n - 1)*x^5)*D[P[n - 1, x], x])/(n^2 + 2*n^2*x + 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 - (2*n^2 - 12*n + 26)*x - (5*n^2 - 14*n + 29)*x^2 - (4*n^2 - 8*n + 12)*x^3 - (n^2 - 2*n + 1)*x^4)*v[i - 1] + (4*x - (n - 13)*x^2 - (3*n - 15)*x^3 - (3*n - 7)*x^4 - (n - 1)*x^5)*deriv(v[i - 1], x))/(n^2 + 2*n^2*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 - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"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}