{"id":"n-c1f0668fd5bd5aaf","revision":0,"first_index":0,"fingerprint":"f6-1c406f8a3bd5550a","name":"reversal of basis_map_A013611 of Riordan array (1, 2+2x).","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"],["8","2"],["64","40","6"],["512","512","160","16"],["4096","5632","2752","576","44"],["32768","57344","38400","12416","1952","120"],["262144","557056","475136","209408","50560","6368","328"],["2097152","5242880","5439488","3047424","999424","192512","20224","896"],["16777216","48234496","58982400","40173568","16715776","4362240","698880","62976","2448"],["134217728","436207616","614465536","493355008","249331712","82419712","17854464","2448384","193152","6688"],["1073741824","3892314112","6207569920","5744099328","3420717056","1372094464","376004608","69611520","8343552","585344","18272"],["8589934592","34359738368","61203283968","64156073984","44029706240","20799553536","6910246912","1616576512","261267456","27811840","1756672","49920"],["68719476736","300647710720","591631745024","693100347392","538917732352","293309775872","114699272192","32505331712","6632194048","951001088","91046912","5229568","136384"],["549755813888","2611340115968","5626407157760","7286412017664","6333197713408","3904863469568","1758669570048","586322280448","144821321728","26195689472","3375407104","293607424","15462912","372608"],["4398046511104","22539988369408","52776558133248","74878459838464","71966472011776","49602845736960","25303397171200","9712131112960","2820656332800","617276375040","100267032576","11730051072","934848512","45456896","1017984"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","10","110","1200","13100","143000","1561000","17040000","186010000","2030500000","22165100000","241956000000","2641211000000","28831670000000","314728810000000"],"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","-8","-3"],["16","12","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-16","-16","-3"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":186309,"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-8db0a0a17f0ae346"]},{"id":186310,"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-8db0a0a17f0ae346"]},{"id":186311,"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-8db0a0a17f0ae346"]}],"region_exact":{},"incoming_edges":[{"id":"e-8db0a0a17f0ae346","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-c1f0668fd5bd5aaf","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":"05b6410c926b9d70b955822dfdf4db1ff2cb0888569699d6ccedda3707a58ea1","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-8588fa6ee4014167","role":"sequence","name":"basis_map_A013611 of Riordan array (1, 2+2x).","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-dd3e6ebe3cc3a319","kind":"fingerprint_related","node_a_id":"n-8588fa6ee4014167","node_b_id":"n-c1f0668fd5bd5aaf","fingerprint":"f6-1c406f8a3bd5550a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1c406f8a3bd5550a"},"other_id":"n-8588fa6ee4014167","other":{"id":"n-8588fa6ee4014167","name":"basis_map_A013611 of Riordan array (1, 2+2x).","first_index":0,"fingerprint":"f6-1c406f8a3bd5550a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","8"],["6","40","64"]],"card_rows":[["1"],["2","8"],["6","40","64"],["16","160","512","512"]],"row_sums_preview":["1","10","110","1200","13100","143000","1561000","17040000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f5a001953ee28e8f","steps":["basis A013611"],"edges":["e-5d40644b540aa393"],"text":"A099040 \u2192 basis A013611"},"headline":"A099040 \u2192 basis A013611","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:         8,         2;\n  2:        64,        40,         6;\n  3:       512,       512,       160,        16;\n  4:      4096,      5632,      2752,       576,        44;\n  5:     32768,     57344,     38400,     12416,      1952,      120;\n  6:    262144,    557056,    475136,    209408,     50560,     6368,      328;\n  7:   2097152,   5242880,   5439488,   3047424,    999424,   192512,    20224,     896;\n  8:  16777216,  48234496,  58982400,  40173568,  16715776,  4362240,   698880,   62976,   2448;\n  9: 134217728, 436207616, 614465536, 493355008, 249331712, 82419712, 17854464, 2448384, 193152, 6688;\n","recurrence_sentence":"The row polynomials satisfy 2*n*P(n,x) = (16*n+(12*n-8)*x+(3*n-3)*x^2)*P(n-1,x) - (16*x+16*x^2+3*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[((16*n + (12*n - 8)*x + (3*n - 3)*x^2)*P[n - 1, x] - (16*x + 16*x^2 + 3*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] = ((16*n + (12*n - 8)*x + (3*n - 3)*x^2)*v[i - 1] - (16*x + 16*x^2 + 3*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-f5a001953ee28e8f","steps":["basis A013611","reverse"],"edges":["e-5d40644b540aa393","e-8db0a0a17f0ae346"],"text":"A099040 \u2192 basis A013611 \u2192 reverse"},"headline":"A099040 \u2192 basis A013611 \u2192 reverse","chain":null}
