{"id":"n-8ac1223942f7b8c5","revision":1,"first_index":0,"fingerprint":"f6-07a6a8e112979ca6","name":"reversal of Triangle of coefficients of Chebyshev's S(n,x+4) polynomials (exponents of x in increasing order).","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"],["1","4"],["1","8","15"],["1","12","46","56"],["1","16","93","232","209"],["1","20","156","592","1091","780"],["1","24","235","1200","3366","4912","2911"],["1","28","330","2120","8010","17784","21468","10864"],["1","32","441","3416","16255","48624","89238","91824","40545"],["1","36","568","5152","29589","111524","275724","430992","386373","151316"],["1","40","711","7392","49756","226464","705565","1485264","2021103","1604984","564719"],["1","44","870","10200","78756","420336","1581832","4196000","7686435","9258404","6598282","2107560"],["1","48","1045","13640","118845","727968","3213420","10296864","23764870","38518880","41610795","26895704","7865521"],["1","52","1236","17776","172535","1193148","6046536","22730208","63370494","129382360","187999880","184080480","108850055","29354524"],["1","56","1443","22672","242594","1869648","10700283","46188384","151077906","372567472","681764450","897561120","803561180","437859040","109552575"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","24","115","551","2640","12649","60605","290376","1391275","6665999","31938720","153027601","733199285","3512968824"],"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":[["0"],["2"]]},"terms":[{"coeff":{"coeffs":[["0","-4","-12"],["2","12","12"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-1","-8","-12"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":165645,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-1/2, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-1/2"},"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-1/2, 0]","region_exact":null,"edges":["e-73dd335e950d1f81"]},{"id":165646,"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-73dd335e950d1f81"]},{"id":165647,"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-73dd335e950d1f81"]}],"region_exact":{},"incoming_edges":[{"id":"e-73dd335e950d1f81","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-8ac1223942f7b8c5","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":"f1cf545ca8d704eebec03aaf59102be1f188fe9b0e4041f67fd6d7c1c20e60ac","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-1/2"},"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-30cb3cad216ed189","role":"sequence","name":"Triangle of coefficients of Chebyshev's S(n,x+4) polynomials (exponents of x in increasing order).","uses":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-6"},"upper":{"closed":false,"value":"-2"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-6, -2)"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-a3c982b8de6ade10","kind":"fingerprint_related","node_a_id":"n-30cb3cad216ed189","node_b_id":"n-8ac1223942f7b8c5","fingerprint":"f6-07a6a8e112979ca6","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-07a6a8e112979ca6"},"other_id":"n-30cb3cad216ed189","other":{"id":"n-30cb3cad216ed189","name":"Triangle of coefficients of Chebyshev's S(n,x+4) polynomials (exponents of x in increasing order).","first_index":0,"fingerprint":"f6-07a6a8e112979ca6","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["4","1"],["15","8","1"]],"card_rows":[["1"],["4","1"],["15","8","1"],["56","46","12","1"]],"row_sums_preview":["1","5","24","115","551","2640","12649","60605"],"regions":["(-inf, +inf)","(-inf, 0)","(-6, -2)","(-inf, 0]"],"region_exact":{},"oeis_ids":["A207823"],"derivation":{"root":"n-30cb3cad216ed189","steps":[],"edges":[],"text":"A207823"},"headline":"A207823: Triangle of coefficients of Chebyshev's S(n,x+4) polynomials (exponents of x in increasing order).","chain":null}},{"id":"l-b3f0d7cb627c1b8e","kind":"oeis_match","node_a_id":"n-8ac1223942f7b8c5","node_b_id":null,"fingerprint":null,"oeis_id":"A207823","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":"A207823","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-ef56821c1a8bc041","kind":"oeis_match","node_a_id":"n-8ac1223942f7b8c5","node_b_id":null,"fingerprint":null,"oeis_id":"A004254","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":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A004254","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A207823","relation":"match_reversed_triangle"},{"oeis_id":"A004254","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-8ac1223942f7b8c5","oeis_id":"A207823","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":18930}],"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: 1,  4;\n  2: 1,  8,  15;\n  3: 1, 12,  46,   56;\n  4: 1, 16,  93,  232,   209;\n  5: 1, 20, 156,  592,  1091,    780;\n  6: 1, 24, 235, 1200,  3366,   4912,   2911;\n  7: 1, 28, 330, 2120,  8010,  17784,  21468,  10864;\n  8: 1, 32, 441, 3416, 16255,  48624,  89238,  91824,  40545;\n  9: 1, 36, 568, 5152, 29589, 111524, 275724, 430992, 386373, 151316;\n","recurrence_sentence":"The row polynomials satisfy 2*n*P(n,x) = (2*n+(12*n-4)*x+(12*n-12)*x^2)*P(n-1,x) - (x+8*x^2+12*x^3)*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 + (12*n - 4)*x + (12*n - 12)*x^2)*P[n - 1, x] - (x + 8*x^2 + 12*x^3)*D[P[n - 1, x], x])/(2*n)]];\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 + (12*n - 4)*x + (12*n - 12)*x^2)*v[i - 1] - (x + 8*x^2 + 12*x^3)*deriv(v[i - 1], x))/(2*n)); 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":"A207823","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:      4,      1;\n  2:     15,      8,      1;\n  3:     56,     46,     12,      1;\n  4:    209,    232,     93,     16,      1;\n  5:    780,   1091,    592,    156,     20,     1;\n  6:   2911,   4912,   3366,   1200,    235,    24,    1;\n  7:  10864,  21468,  17784,   8010,   2120,   330,   28,   1;\n  8:  40545,  91824,  89238,  48624,  16255,  3416,  441,  32,  1;\n  9: 151316, 386373, 430992, 275724, 111524, 29589, 5152, 568, 36, 1;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where 2*n*P(n,x) = (2*n+(12*n-4)*x+(12*n-12)*x^2)*P(n-1,x) - (x+8*x^2+12*x^3)*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 + (12*n - 4)*x + (12*n - 12)*x^2)*P[n - 1, x] - (x + 8*x^2 + 12*x^3)*D[P[n - 1, x], x])/(2*n)]];\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 + (12*n - 4)*x + (12*n - 12)*x^2)*v[i - 1] - (x + 8*x^2 + 12*x^3)*deriv(v[i - 1], x))/(2*n)); 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}}],"derivation":{"root":"n-30cb3cad216ed189","steps":["reverse"],"edges":["e-73dd335e950d1f81"],"text":"A207823 \u2192 reverse"},"headline":"A207823 \u2192 reverse","chain":null}
