{"id":"n-74e8b5eac5b04b50","revision":0,"first_index":0,"fingerprint":"f6-b0bdfa529c7b31c5","name":"shift_x_plus_one of 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"],["5","4"],["24","38","15"],["115","272","214","56"],["551","1734","2043","1068","209"],["2640","10372","16278","12756","4991","780"],["12649","59584","116816","122004","71591","22378","2911"],["60605","332864","782754","1021600","799190","374736","97516","10864"],["290376","1821858","4996593","7823796","7649935","4782876","1867266","416184","40545"],["1391275","9816720","30761310","56185380","65919715","51519404","26821656","8969352","1748217","151316"],["6665999","52246090","184142745","384333300","526045975","493369338","321102943","143199792","41878314","7252174","564719"],["31938720","275295176","1077912060","2530712760","3958511290","4331486376","3383211888","1886278664","735682161","191157024","29781442","2107560"],["153027601","1438652672","6195440080","16160347160","28436552170","35561682452","32408117350","21685465856","10574144620","3664305170","856587925","121281956","7865521"],["733199285","7466201148","35071369956","100621664064","196726300620","276775399224","288282620628","225075609864","131721135147","57069304320","17792382654","3779934012","490458867","29354524"],["3512968824","38519095006","195995881347","613403920152","1319165472786","2062187064648","2416546665612","2156445148824","1472554921509","765733001922","298479787665","84570437700","16465013025","1971595090","109552575"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","9","77","657","5605","47817","407933","3480129","29689429","253284345","2160801389","18434075121","157263470533","1341634934313","11445660526685"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0"],["2"]]},"terms":[{"coeff":{"coeffs":[["-16","-28","-12"],["26","36","12"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-21","-53","-44","-12"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":177570,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-d94660deea8779b6"]},{"id":177571,"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-d94660deea8779b6"]},{"id":177572,"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-d94660deea8779b6"]}],"region_exact":{},"incoming_edges":[{"id":"e-d94660deea8779b6","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-74e8b5eac5b04b50","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":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"dbc2e59ae91224b19b32ff59b1d8e3374ecc0af8909cdd79ca599cb3d5bc89da","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}}],"premises":[{"position":0,"node_id":"n-8ac1223942f7b8c5","role":"sequence","name":"reversal of 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":{"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-995e9b5d6ba44d76","kind":"fingerprint_related","node_a_id":"n-5038739ef66fadea","node_b_id":"n-74e8b5eac5b04b50","fingerprint":"f6-b0bdfa529c7b31c5","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b0bdfa529c7b31c5"},"other_id":"n-5038739ef66fadea","other":{"id":"n-5038739ef66fadea","name":"mobius_x_over_x_plus_one of Triangle of coefficients of Chebyshev's S(n,x+4) polynomials (exponents of x in increasing order).","first_index":0,"fingerprint":"f6-b0bdfa529c7b31c5","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["4","5"],["15","38","24"]],"card_rows":[["1"],["4","5"],["15","38","24"],["56","214","272","115"]],"row_sums_preview":["1","9","77","657","5605","47817","407933","3480129"],"regions":["[-1, -2/3)"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-30cb3cad216ed189","steps":["x \u21a6 x/(x+1)"],"edges":["e-23455da8188ecb13"],"text":"A207823 \u2192 x \u21a6 x/(x+1)"},"headline":"A207823 \u2192 x \u21a6 x/(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:       1;\n  1:       5,       4;\n  2:      24,      38,       15;\n  3:     115,     272,      214,       56;\n  4:     551,    1734,     2043,     1068,      209;\n  5:    2640,   10372,    16278,    12756,     4991,      780;\n  6:   12649,   59584,   116816,   122004,    71591,    22378,     2911;\n  7:   60605,  332864,   782754,  1021600,   799190,   374736,    97516,   10864;\n  8:  290376, 1821858,  4996593,  7823796,  7649935,  4782876,  1867266,  416184,   40545;\n  9: 1391275, 9816720, 30761310, 56185380, 65919715, 51519404, 26821656, 8969352, 1748217, 151316;\n","recurrence_sentence":"The row polynomials satisfy 2*n*P(n,x) = (26*n-16+(36*n-28)*x+(12*n-12)*x^2)*P(n-1,x) - (21+53*x+44*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[((26*n - 16 + (36*n - 28)*x + (12*n - 12)*x^2)*P[n - 1, x] - (21 + 53*x + 44*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] = ((26*n - 16 + (36*n - 28)*x + (12*n - 12)*x^2)*v[i - 1] - (21 + 53*x + 44*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}}],"derivation":{"root":"n-30cb3cad216ed189","steps":["reverse","x \u21a6 x+1"],"edges":["e-73dd335e950d1f81","e-d94660deea8779b6"],"text":"A207823 \u2192 reverse \u2192 x \u21a6 x+1"},"headline":"A207823 \u2192 reverse \u2192 x \u21a6 x+1","chain":null}