{"id":"n-84b60802c7175ea1","revision":0,"first_index":0,"fingerprint":"f6-f83cf33d2c000fbc","name":"reversal of Coefficients of a generalized Jaco-Lucas polynomial (even indices) read by rows.","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":false,"rows":[["2"],["2","3"],["2","8","7"],["2","15","30","18"],["2","24","80","104","47"],["2","35","170","355","340","123"],["2","48","315","932","1410","1068","322"],["2","63","532","2079","4396","5208","3262","843"],["2","80","840","4144","11440","18784","18280","9760","2207"],["2","99","1260","7602","26226","55809","74838","61785","28746","5778"],["2","120","1815","13080","54690","144488","249815","282980","202840","83620","15127"],["2","143","2530","21384","105930","337183","720412","1048135","1027466","650705","240812","39603"],["2","168","3432","33528","193347","725208","1858892","3330408","4180722","3611528","2048532","687768","103682"],["2","195","4550","50765","336050","1460030","4392700","9411350","14508520","16011450","12361960","6349200","1950650","271443"],["2","224","5915","74620","560560","2782780","9662835","24227400","44550660","60245080","59305960","41387920","19421150","5499704","710647"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["2","5","17","65","257","1025","4097","16385","65537","262145","1048577","4194305","16777217","67108865","268435457"],"nonn":true,"definitions":[{"position":0,"source":"derived:reversal","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["2"],["2","3"]],"recurrence":{"denominator":{"coeffs":[["-2"],["2"]]},"terms":[{"coeff":{"coeffs":[["-2","1","5"],["2","-1","-5"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","4","9","5"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":176403,"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-0a98dad8563ea284"]},{"id":176404,"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-0a98dad8563ea284"]},{"id":176405,"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-0a98dad8563ea284"]}],"region_exact":{},"incoming_edges":[{"id":"e-0a98dad8563ea284","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"reversal\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":1}}"},"operation_detail":"reversal, shift 0","conclusion_id":"n-84b60802c7175ea1","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":"reversed","lean_declarations":["RealRooted.X_pow_mul_reverse_eq_zero_or_splits","RealRooted.reflect_eq_zero_or_splits"],"certificate_hash":"9205b76db12f72dcd58c4effd86ac50b35676a7dc14a419734660a9ff76d47f8","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-9b40f045d129e237","role":"sequence","name":"Coefficients of a generalized Jaco-Lucas polynomial (even indices) read by rows.","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-89e1c48bd14de1f7","kind":"fingerprint_related","node_a_id":"n-9b40f045d129e237","node_b_id":"n-84b60802c7175ea1","fingerprint":"f6-f83cf33d2c000fbc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f83cf33d2c000fbc"},"other_id":"n-9b40f045d129e237","other":{"id":"n-9b40f045d129e237","name":"Coefficients of a generalized Jaco-Lucas polynomial (even indices) read by rows.","first_index":0,"fingerprint":"f6-f83cf33d2c000fbc","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["2"],["3","2"],["7","8","2"]],"card_rows":[["2"],["3","2"],["7","8","2"],["18","30","15","2"]],"row_sums_preview":["2","5","17","65","257","1025","4097","16385"],"regions":["(-inf, +inf)","(-inf, 0)"],"region_exact":{},"oeis_ids":["A122076"],"derivation":{"root":"n-9b40f045d129e237","steps":[],"edges":[],"text":"A122076"},"headline":"A122076: Coefficients of a generalized Jaco-Lucas polynomial (even indices) read by rows.","chain":null}},{"id":"l-a1da5ca7820adf85","kind":"fingerprint_related","node_a_id":"n-d9956ec4f2e74f86","node_b_id":"n-84b60802c7175ea1","fingerprint":"f6-f83cf33d2c000fbc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f83cf33d2c000fbc"},"other_id":"n-d9956ec4f2e74f86","other":{"id":"n-d9956ec4f2e74f86","name":"shift_x_plus_one of Triangle read by rows: T(n,0)=1, T(n,n)=2, T(n,m) = T(n-1,m-1) + Sum_{k=0..m} T(n-1-k,m-k).","first_index":0,"fingerprint":"f6-f83cf33d2c000fbc","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","2"],["7","8","2"]],"card_rows":[["1"],["3","2"],["7","8","2"],["18","30","15","2"]],"row_sums_preview":["1","5","17","65","257","1025","4097","16385"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2496e2f555756362","steps":["basis A007318"],"edges":["e-7438a94267d81529"],"text":"A084534 \u2192 basis A007318"},"headline":"A084534 \u2192 basis A007318","chain":null}},{"id":"l-c4ee1e3adae90463","kind":"fingerprint_related","node_a_id":"n-84b60802c7175ea1","node_b_id":"n-0f17d4387a7e32e2","fingerprint":"f6-f83cf33d2c000fbc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f83cf33d2c000fbc"},"other_id":"n-0f17d4387a7e32e2","other":{"id":"n-0f17d4387a7e32e2","name":"shift_x_plus_one of basis_map_A013611 of Triangle read by rows: T(n, k) = binomial(2n, 2k).","first_index":1,"fingerprint":"f6-f83cf33d2c000fbc","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["6","4"],["56","64","16"]],"card_rows":[["1"],["6","4"],["56","64","16"],["576","960","480","64"]],"row_sums_preview":["1","10","136","2080","32896","524800","8390656","134225920"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f7935c36a3c0ec8c","steps":["basis A013611","x \u21a6 x+1"],"edges":["e-11143fa0a471b62a","e-2193a834a995fc61"],"text":"A086645 \u2192 basis A013611 \u2192 x \u21a6 x+1"},"headline":"A086645 \u2192 basis A013611 \u2192 x \u21a6 x+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: 2;\n  1: 2,  3;\n  2: 2,  8,    7;\n  3: 2, 15,   30,   18;\n  4: 2, 24,   80,  104,    47;\n  5: 2, 35,  170,  355,   340,   123;\n  6: 2, 48,  315,  932,  1410,  1068,   322;\n  7: 2, 63,  532, 2079,  4396,  5208,  3262,   843;\n  8: 2, 80,  840, 4144, 11440, 18784, 18280,  9760,  2207;\n  9: 2, 99, 1260, 7602, 26226, 55809, 74838, 61785, 28746, 5778;\n","recurrence_sentence":"The row polynomials satisfy (2*n-2)*P(n,x) = (2*n-2-(n-1)*x-(5*n-5)*x^2)*P(n-1,x) + (4*x+9*x^2+5*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 2, P(1,x) = 2 + 3*x.","mathematica":"P[0, x_] := 2;\nP[1, x_] := 2 + 3*x;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n - 2 - (n - 1)*x - (5*n - 5)*x^2)*P[n - 1, x] + (4*x + 9*x^2 + 5*x^3)*D[P[n - 1, x], x])/(2*n - 2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 2; v[2] = 2 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n - 2 - (n - 1)*x - (5*n - 5)*x^2)*v[i - 1] + (4*x + 9*x^2 + 5*x^3)*deriv(v[i - 1], x))/(2*n - 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-9b40f045d129e237","steps":["reverse"],"edges":["e-0a98dad8563ea284"],"text":"A122076 \u2192 reverse"},"headline":"A122076 \u2192 reverse","chain":null}