{"id":"n-6494b967cfaa76e5","revision":0,"first_index":0,"fingerprint":"f6-c64e29f1cee5cd50","name":"reversal of basis_map_A013609 of Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":2,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["4","10","5"],["16","80","140","100","25"],["64","480","1440","2200","1800","750","125"],["256","2560","10880","25600","36400","32000","17000","5000","625"],["1024","12800","70400","224000","456000","620000","570000","350000","137500","31250","3125"],["4096","61440","414720","1664000","4416000","8160000","10760000","10200000","6900000","3250000","1012500","187500","15625"],["16384","286720","2293760","11110400","36377600","85120000","146720000","189200000","183400000","133000000","71050000","27125000","7000000","1093750","78125"],["65536","1310720","12124160","68812800","268083200","759808000","1619968000","2649600000","3359200000","3312000000","2531200000","1484000000","654500000","210000000","46250000","6250000","390625"],["262144","5898240","61931520","403046400","1821081600","6064128000","15418368000","30597120000","48032640000","60088000000","60040800000","47808000000","30114000000","14805000000","5557500000","1537500000","295312500","35156250","1953125"],["1048576","26214400","308019200","2260992000","11624448000","44482560000","131420160000","306892800000","575193600000","873664000000","1081206400000","1092080000000","898740000000","599400000000","320850000000","135750000000","44343750000","10781250000","1835937500","195312500","9765625"],["4194304","115343360","1499463680","12255232000","70647808000","305479680000","1028628480000","2764185600000","6026803200000","10781056000000","15937433600000","19548704000000","19921792000000","16845400000000","11771100000000","6748500000000","3139125000000","1165312500000","336875000000","73046875000","11171875000","1074218750","48828125"],["16777216","503316480","7172259840","64592281600","412640870400","1989672960000","7522549760000","22870425600000","56892211200000","117213184000000","201694310400000","291474432000000","354861376000000","364343040000000","315147360000000","228932000000000","138897000000000","69795000000000","28696250000000","9487500000000","2459531250000","481250000000","66796875000","5859375000","244140625"],["67108864","2181038080","33806090240","332608307200","2332347596800","12408061952000","52050132992000","176655564800000","493885849600000","1152126976000000","2263370137600000","3768906752000000","5342661376000000","6463358080000000","6678326720000000","5888916800000000","4420644800000000","2812810000000000","1507220000000000","673887500000000","248194375000000","73957812500000","17377343750000","3097656250000","393554687500","31738281250","1220703125"],["268435456","9395240960","157370286080","1679399321600","12824503910400","74618765312000","343942889472000","1289163898880000","4002349711360000","10430644224000000","23044179558400000","43469963264000000","70376563712000000","98124579840000000","118060194560000000","122655724800000000","109963380800000000","84902272000000000","56260204000000000","31831800000000000","15267752500000000","6147212500000000","2050059375000000","555953125000000","119437500000000","19550781250000","2290039062500","170898437500","6103515625"]],"degrees":[0,2,4,6,8,10,12,14,16,18,20,22,24,26,28],"row_sums":["1","19","361","6859","130321","2476099","47045881","893871739","16983563041","322687697779","6131066257801","116490258898219","2213314919066161","42052983462257059","799006685782884121"],"nonn":true,"definitions":[{"position":0,"source":"derived:reversal","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["4","10","5"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":190078,"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-bce9343cd42e809d"]},{"id":190079,"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-bce9343cd42e809d"]},{"id":190080,"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-bce9343cd42e809d"]}],"region_exact":{},"incoming_edges":[{"id":"e-bce9343cd42e809d","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"reversal\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":2}}"},"operation_detail":"reversal, shift 0","conclusion_id":"n-6494b967cfaa76e5","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":"d68fcf9657ed88437af9280934fca6c001690ec6c9057e9b9c53aeb0d74b041d","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-feae95e8ddee939b","role":"sequence","name":"basis_map_A013609 of Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.","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-2308a412828007cc","kind":"fingerprint_related","node_a_id":"n-feae95e8ddee939b","node_b_id":"n-6494b967cfaa76e5","fingerprint":"f6-c64e29f1cee5cd50","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-c64e29f1cee5cd50"},"other_id":"n-feae95e8ddee939b","other":{"id":"n-feae95e8ddee939b","name":"basis_map_A013609 of Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.","first_index":0,"fingerprint":"f6-c64e29f1cee5cd50","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["5","10","4"],["25","100","140","80","16"]],"card_rows":[["1"],["5","10","4"],["25","100","140","80","16"],["125","750","1800","2200","1440","480","64"]],"row_sums_preview":["1","19","361","6859","130321","2476099","47045881","893871739"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-b30e7a5b4198051a","steps":["basis A013609"],"edges":["e-89275777750acd05"],"text":"A272866 \u2192 basis A013609"},"headline":"A272866 \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,      10,        5;\n  2:     16,      80,      140,       100,         25;\n  3:     64,     480,     1440,      2200,       1800,        750,         125;\n  4:    256,    2560,    10880,     25600,      36400,      32000,       17000,        5000,         625;\n  5:   1024,   12800,    70400,    224000,     456000,     620000,      570000,      350000,      137500,       31250,        3125;\n  6:   4096,   61440,   414720,   1664000,    4416000,    8160000,    10760000,    10200000,     6900000,     3250000,     1012500,      187500,       15625;\n  7:  16384,  286720,  2293760,  11110400,   36377600,   85120000,   146720000,   189200000,   183400000,   133000000,    71050000,    27125000,     7000000,     1093750,      78125;\n  8:  65536, 1310720, 12124160,  68812800,  268083200,  759808000,  1619968000,  2649600000,  3359200000,  3312000000,  2531200000,  1484000000,   654500000,   210000000,   46250000,    6250000,    390625;\n  9: 262144, 5898240, 61931520, 403046400, 1821081600, 6064128000, 15418368000, 30597120000, 48032640000, 60088000000, 60040800000, 47808000000, 30114000000, 14805000000, 5557500000, 1537500000, 295312500, 35156250, 1953125;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (4+10*x+5*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(4 + 10*x + 5*x^2)*P[n - 1, x]];\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] = (4 + 10*x + 5*x^2)*v[i - 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}}],"derivation":{"root":"n-b30e7a5b4198051a","steps":["basis A013609","reverse"],"edges":["e-89275777750acd05","e-bce9343cd42e809d"],"text":"A272866 \u2192 basis A013609 \u2192 reverse"},"headline":"A272866 \u2192 basis A013609 \u2192 reverse","chain":null}
