{"id":"n-c2e960ae82fc763f","revision":0,"first_index":0,"fingerprint":"f6-40f50691bbfd11c0","name":"reversal of basis_map_A013609 of Triangle of coefficients of polynomials v(n,x) jointly generated with A207606; see Formula section.","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"],["4","3"],["8","18","8"],["16","60","66","21"],["32","168","304","222","55"],["64","432","1096","1296","710","144"],["128","1056","3456","5712","5004","2196","377"],["256","2496","10016","21360","26048","18108","6632","987"],["512","5760","27392","71712","112720","108648","62584","19674","2584"],["1024","13056","71808","223104","430560","534096","425064","208992","57558","6765"],["2048","29184","182272","655872","1503040","2288160","2339696","1584528","679508","166530","17711"],["4096","64512","451072","1845504","4901888","8862336","11113312","9654048","5687536","2162592","477454","46368"],["8192","141312","1093632","5015040","15158016","31774464","47310592","50359872","37997520","19803264","6763176","1358568","121393"],["16384","307200","2607104","13246464","44910080","107177472","185042688","233789184","215961344","143945040","67248608","20844072","3841036","317811"],["32768","663552","6127616","34172928","128465920","344070144","676459776","990931968","1085979648","885414240","528334816","223630704","63451112","10800162","832040"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","7","34","163","781","3742","17929","85903","411586","1972027","9448549","45270718","216905041","1039254487","4979367394"],"nonn":true,"definitions":[{"position":0,"source":"derived:reversal","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["4","3"]],"recurrence":{"denominator":{"coeffs":[["32"],["-64"],["32"]]},"terms":[{"coeff":{"coeffs":[["64","168","88","30"],["-128","-312","-164","-45"],["64","144","76","15"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","32","120","160","60"],["0","-32","-120","-112","-30"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","0","12","36","15"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":178184,"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-a0027df1f09e1324"]},{"id":178185,"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-a0027df1f09e1324"]},{"id":178186,"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-a0027df1f09e1324"]}],"region_exact":{},"incoming_edges":[{"id":"e-a0027df1f09e1324","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-c2e960ae82fc763f","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":"6fb9491f99dea63b221f6a2870d541a63d78b95ce46457a2c49f268806d9f35b","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-b4785bc61afd940f","role":"sequence","name":"basis_map_A013609 of Triangle of coefficients of polynomials v(n,x) jointly generated with A207606; see Formula section.","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-5947ff95844856a0","kind":"fingerprint_related","node_a_id":"n-b4785bc61afd940f","node_b_id":"n-c2e960ae82fc763f","fingerprint":"f6-40f50691bbfd11c0","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-40f50691bbfd11c0"},"other_id":"n-b4785bc61afd940f","other":{"id":"n-b4785bc61afd940f","name":"basis_map_A013609 of Triangle of coefficients of polynomials v(n,x) jointly generated with A207606; see Formula section.","first_index":0,"fingerprint":"f6-40f50691bbfd11c0","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","4"],["8","18","8"]],"card_rows":[["1"],["3","4"],["8","18","8"],["21","66","60","16"]],"row_sums_preview":["1","7","34","163","781","3742","17929","85903"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-353fb5a029e4e669","steps":["basis A013609"],"edges":["e-3748e9c6bfc37784"],"text":"A207607 \u2192 basis A013609"},"headline":"A207607 \u2192 basis A013609","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,     3;\n  2:    8,    18,     8;\n  3:   16,    60,    66,     21;\n  4:   32,   168,   304,    222,     55;\n  5:   64,   432,  1096,   1296,    710,    144;\n  6:  128,  1056,  3456,   5712,   5004,   2196,    377;\n  7:  256,  2496, 10016,  21360,  26048,  18108,   6632,    987;\n  8:  512,  5760, 27392,  71712, 112720, 108648,  62584,  19674,  2584;\n  9: 1024, 13056, 71808, 223104, 430560, 534096, 425064, 208992, 57558, 6765;\n","recurrence_sentence":"The row polynomials satisfy (32*n^2-64*n+32)*P(n,x) = (64*n^2-128*n+64+(144*n^2-312*n+168)*x+(76*n^2-164*n+88)*x^2+(15*n^2-45*n+30)*x^3)*P(n-1,x) - ((32*n-32)*x+(120*n-120)*x^2+(112*n-160)*x^3+(30*n-60)*x^4)*d/dx P(n-1,x) + (12*x^3+36*x^4+15*x^5)*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 3*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4 + 3*x;\nP[n_, x_] := P[n, x] = Expand[Together[((64*n^2 - 128*n + 64 + (144*n^2 - 312*n + 168)*x + (76*n^2 - 164*n + 88)*x^2 + (15*n^2 - 45*n + 30)*x^3)*P[n - 1, x] - ((32*n - 32)*x + (120*n - 120)*x^2 + (112*n - 160)*x^3 + (30*n - 60)*x^4)*D[P[n - 1, x], x] + (12*x^3 + 36*x^4 + 15*x^5)*D[P[n - 1, x], {x, 2}])/(32*n^2 - 64*n + 32)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = ((64*n^2 - 128*n + 64 + (144*n^2 - 312*n + 168)*x + (76*n^2 - 164*n + 88)*x^2 + (15*n^2 - 45*n + 30)*x^3)*v[i - 1] - ((32*n - 32)*x + (120*n - 120)*x^2 + (112*n - 160)*x^3 + (30*n - 60)*x^4)*deriv(v[i - 1], x) + (12*x^3 + 36*x^4 + 15*x^5)*deriv(deriv(v[i - 1], x), x))/(32*n^2 - 64*n + 32)); 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-353fb5a029e4e669","steps":["basis A013609","reverse"],"edges":["e-3748e9c6bfc37784","e-a0027df1f09e1324"],"text":"A207607 \u2192 basis A013609 \u2192 reverse"},"headline":"A207607 \u2192 basis A013609 \u2192 reverse","chain":null}
