{"id":"n-fe40873ed3cfd4a4","revision":1,"first_index":0,"fingerprint":"f6-fb7ccb346707daea","name":"reversal of Coefficient table for Chebyshev's U(2*n,x) polynomial expanded in decreasing powers of (-4*(1-x^2)).","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":[["1"],["3","1"],["5","5","1"],["7","14","7","1"],["9","30","27","9","1"],["11","55","77","44","11","1"],["13","91","182","156","65","13","1"],["15","140","378","450","275","90","15","1"],["17","204","714","1122","935","442","119","17","1"],["19","285","1254","2508","2717","1729","665","152","19","1"],["21","385","2079","5148","7007","5733","2940","952","189","21","1"],["23","506","3289","9867","16445","16744","10948","4692","1311","230","23","1"],["25","650","5005","17875","35750","44200","35700","19380","7125","1750","275","25","1"],["27","819","7371","30888","72930","107406","104652","69768","32319","10395","2277","324","27","1"],["29","1015","10556","51272","140998","243542","281010","224808","127281","51359","14674","2900","377","29","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","11","29","76","199","521","1364","3571","9349","24476","64079","167761","439204","1149851"],"nonn":true,"definitions":[{"position":0,"source":"derived:reversal","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["-1"],["2"]]},"terms":[{"coeff":{"coeffs":[["1","0"],["2","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","4","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":163817,"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-41a825a32145fbe0"]},{"id":163818,"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-41a825a32145fbe0"]},{"id":163819,"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-41a825a32145fbe0"]}],"region_exact":{},"incoming_edges":[{"id":"e-41a825a32145fbe0","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-fe40873ed3cfd4a4","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":"e2cc8dcc4f8ff724796498f42fd9a462ef058652ad33b140fac00c14ab9c5f5c","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-9496336e2b29e28e","role":"sequence","name":"Coefficient table for Chebyshev's U(2*n,x) polynomial expanded in decreasing powers of (-4*(1-x^2)).","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-2d3ab1c08c457d56","kind":"fingerprint_related","node_a_id":"n-7895ffdfea7c95d5","node_b_id":"n-fe40873ed3cfd4a4","fingerprint":"f6-fb7ccb346707daea","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-fb7ccb346707daea"},"other_id":"n-7895ffdfea7c95d5","other":{"id":"n-7895ffdfea7c95d5","name":"reversal of Triangle read by rows: T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.","first_index":1,"fingerprint":"f6-fb7ccb346707daea","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","3"],["1","5","5"]],"card_rows":[["1"],["1","3"],["1","5","5"],["1","7","14","7"]],"row_sums_preview":["1","4","11","29","76","199","521","1364"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-318700a428d7b97b","steps":["reverse"],"edges":["e-2020a36645e8b930"],"text":"A111125 \u2192 reverse"},"headline":"A111125 \u2192 reverse","chain":null}},{"id":"l-31a2ba1829eea63c","kind":"fingerprint_related","node_a_id":"n-9496336e2b29e28e","node_b_id":"n-fe40873ed3cfd4a4","fingerprint":"f6-fb7ccb346707daea","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-fb7ccb346707daea"},"other_id":"n-9496336e2b29e28e","other":{"id":"n-9496336e2b29e28e","name":"Coefficient table for Chebyshev's U(2*n,x) polynomial expanded in decreasing powers of (-4*(1-x^2)).","first_index":0,"fingerprint":"f6-fb7ccb346707daea","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","3"],["1","5","5"]],"card_rows":[["1"],["1","3"],["1","5","5"],["1","7","14","7"]],"row_sums_preview":["1","4","11","29","76","199","521","1364"],"regions":["(-inf, +inf)","(-inf, 0)"],"region_exact":{},"oeis_ids":["A082985"],"derivation":{"root":"n-9496336e2b29e28e","steps":[],"edges":[],"text":"A082985"},"headline":"A082985: Coefficient table for Chebyshev's U(2*n,x) polynomial expanded in decreasing powers of (-4*(1-x^2)).","chain":null}},{"id":"l-78795f1689965877","kind":"fingerprint_related","node_a_id":"n-318700a428d7b97b","node_b_id":"n-fe40873ed3cfd4a4","fingerprint":"f6-fb7ccb346707daea","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-fb7ccb346707daea"},"other_id":"n-318700a428d7b97b","other":{"id":"n-318700a428d7b97b","name":"Triangle read by rows: T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.","first_index":1,"fingerprint":"f6-fb7ccb346707daea","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","1"],["5","5","1"]],"card_rows":[["1"],["3","1"],["5","5","1"],["7","14","7","1"]],"row_sums_preview":["1","4","11","29","76","199","521","1364"],"regions":["(-inf, +inf)","(-inf, 0)","(-4, 0)"],"region_exact":{},"oeis_ids":["A111125"],"derivation":{"root":"n-318700a428d7b97b","steps":[],"edges":[],"text":"A111125"},"headline":"A111125: Triangle read by rows: T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.","chain":null}},{"id":"l-d9796a3502957db8","kind":"fingerprint_related","node_a_id":"n-fe40873ed3cfd4a4","node_b_id":"n-14e292285dab27c9","fingerprint":"f6-fb7ccb346707daea","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-fb7ccb346707daea"},"other_id":"n-14e292285dab27c9","other":{"id":"n-14e292285dab27c9","name":"basis_map_A013611 of Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.","first_index":1,"fingerprint":"f6-fb7ccb346707daea","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["4","12"],["16","80","80"]],"card_rows":[["1"],["4","12"],["16","80","80"],["64","448","896","448"]],"row_sums_preview":["1","16","176","1856","19456","203776","2134016","22347776"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ba8de4e02acfb6ca","steps":["basis A013611"],"edges":["e-8dd7014bf6954b08"],"text":"A091042 \u2192 basis A013611"},"headline":"A091042 \u2192 basis A013611","chain":null}},{"id":"l-04bf6aaf6166627c","kind":"oeis_match","node_a_id":"n-fe40873ed3cfd4a4","node_b_id":null,"fingerprint":null,"oeis_id":"A093960","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":2,"target":"row_sums","terms_compared":14},"in_catalog":false,"oeis_id":"A093960","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-114f571615fab930","kind":"oeis_match","node_a_id":"n-fe40873ed3cfd4a4","node_b_id":null,"fingerprint":null,"oeis_id":"A111125","in_catalog":1,"target":"triangle","row_alignment":"tabl","rows_matched":11,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":11,"target":"triangle","terms_compared":66},"in_catalog":true,"oeis_id":"A111125","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-742df7a095fb65f1","kind":"oeis_match","node_a_id":"n-fe40873ed3cfd4a4","node_b_id":null,"fingerprint":null,"oeis_id":"A135064","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":2,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":1,"target":"row_sums","terms_compared":13},"in_catalog":false,"oeis_id":"A135064","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-941ee6fc8bdc52d7","kind":"oeis_match","node_a_id":"n-fe40873ed3cfd4a4","node_b_id":null,"fingerprint":null,"oeis_id":"A082985","in_catalog":1,"target":"reversed_triangle","row_alignment":"tabl","rows_matched":10,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":10,"target":"reversed_triangle","terms_compared":55},"in_catalog":true,"oeis_id":"A082985","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-fd3ce43350abd707","kind":"oeis_match","node_a_id":"n-fe40873ed3cfd4a4","node_b_id":null,"fingerprint":null,"oeis_id":"A002878","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A002878","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A082985","relation":"match_reversed_triangle"},{"oeis_id":"A002878","relation":"match_row_sums"},{"oeis_id":"A093960","relation":"match_row_sums"},{"oeis_id":"A135064","relation":"match_row_sums"},{"oeis_id":"A111125","relation":"match_triangle"}],"bfiles":[{"node_id":"n-fe40873ed3cfd4a4","oeis_id":"A002878","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":30,"oeis_terms_available":30,"term_count":200,"raw_size":9350},{"node_id":"n-fe40873ed3cfd4a4","oeis_id":"A082985","target":"reversed_triangle","oeis_offset":0,"first_row":0,"rows_used":44,"terms_matched":55,"oeis_terms_agreeing":55,"oeis_terms_available":55,"term_count":990,"raw_size":13291},{"node_id":"n-fe40873ed3cfd4a4","oeis_id":"A111125","target":"triangle","oeis_offset":0,"first_row":0,"rows_used":44,"terms_matched":66,"oeis_terms_agreeing":66,"oeis_terms_available":66,"term_count":990,"raw_size":13291}],"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:  3,   1;\n  2:  5,   5,    1;\n  3:  7,  14,    7,    1;\n  4:  9,  30,   27,    9,    1;\n  5: 11,  55,   77,   44,   11,    1;\n  6: 13,  91,  182,  156,   65,   13,   1;\n  7: 15, 140,  378,  450,  275,   90,  15,   1;\n  8: 17, 204,  714, 1122,  935,  442, 119,  17,  1;\n  9: 19, 285, 1254, 2508, 2717, 1729, 665, 152, 19, 1;\n","recurrence_sentence":"The row polynomials satisfy (2*n-1)*P(n,x) = (2*n+1+n*x)*P(n-1,x) + (4*x+x^2)*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 + 1 + n*x)*P[n - 1, x] + (4*x + x^2)*D[P[n - 1, x], x])/(2*n - 1)]];\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 + 1 + n*x)*v[i - 1] + (4*x + x^2)*deriv(v[i - 1], x))/(2*n - 1)); 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}},{"oeis_id":"A082985","target":"reversed_triangle","first_row":0,"oeis_offset":0,"reversed":true,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0: 1;\n  1: 1,  3;\n  2: 1,  5,   5;\n  3: 1,  7,  14,   7;\n  4: 1,  9,  27,  30,    9;\n  5: 1, 11,  44,  77,   55,   11;\n  6: 1, 13,  65, 156,  182,   91,   13;\n  7: 1, 15,  90, 275,  450,  378,  140,   15;\n  8: 1, 17, 119, 442,  935, 1122,  714,  204,  17;\n  9: 1, 19, 152, 665, 1729, 2717, 2508, 1254, 285, 19;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where (2*n-1)*P(n,x) = (2*n+1+n*x)*P(n-1,x) + (4*x+x^2)*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 + 1 + n*x)*P[n - 1, x] + (4*x + x^2)*D[P[n - 1, x], x])/(2*n - 1)]];\nTable[Reverse[PadRight[CoefficientList[P[n, x], x], n + 1]], {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 + 1 + n*x)*v[i - 1] + (4*x + x^2)*deriv(v[i - 1], x))/(2*n - 1)); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c = Vecrev(c); 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":true,"trim_zeros":"none","regular_triangle":true}},{"oeis_id":"A111125","target":"triangle","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:  1;\n  1:  3,   1;\n  2:  5,   5,    1;\n  3:  7,  14,    7,    1;\n  4:  9,  30,   27,    9,    1;\n  5: 11,  55,   77,   44,   11,    1;\n  6: 13,  91,  182,  156,   65,   13,   1;\n  7: 15, 140,  378,  450,  275,   90,  15,   1;\n  8: 17, 204,  714, 1122,  935,  442, 119,  17,  1;\n  9: 19, 285, 1254, 2508, 2717, 1729, 665, 152, 19, 1;\n","recurrence_sentence":"The row polynomials satisfy (2*n-1)*P(n,x) = (2*n+1+n*x)*P(n-1,x) + (4*x+x^2)*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 + 1 + n*x)*P[n - 1, x] + (4*x + x^2)*D[P[n - 1, x], x])/(2*n - 1)]];\nTable[PadRight[CoefficientList[P[n, x], x], n + 1], {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 + 1 + n*x)*v[i - 1] + (4*x + x^2)*deriv(v[i - 1], x))/(2*n - 1)); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); 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":true}}],"derivation":{"root":"n-9496336e2b29e28e","steps":["reverse"],"edges":["e-41a825a32145fbe0"],"text":"A082985 \u2192 reverse"},"headline":"A082985 \u2192 reverse","chain":null}