{"id":"n-bc4e6b211c5c1593","revision":0,"first_index":0,"fingerprint":"f6-d98c1510b99a0b49","name":"reversal of basis_map_A013611 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"],["16","20","5"],["256","640","560","200","25"],["4096","15360","23040","17600","7200","1500","125"],["65536","327680","696320","819200","582400","256000","68000","10000","625"],["1048576","6553600","18022400","28672000","29184000","19840000","9120000","2800000","550000","62500","3125"],["16777216","125829120","424673280","851968000","1130496000","1044480000","688640000","326400000","110400000","26000000","4050000","375000","15625"],["268435456","2348810240","9395240960","22754099200","37250662400","43581440000","37560320000","24217600000","11737600000","4256000000","1136800000","217000000","28000000","2187500","78125"],["4294967296","42949672960","198642237440","563714457600","1098068787200","1556086784000","1658847232000","1356595200000","859955200000","423936000000","161996800000","47488000000","10472000000","1680000000","185000000","12500000","390625"],["68719476736","773094113280","4058744094720","13207024435200","29836600934400","49677336576000","63153635328000","62662901760000","49185423360000","30765056000000","15370444800000","6119424000000","1927296000000","473760000000","88920000000","12300000000","1181250000","70312500","1953125"],["1099511627776","13743895347200","80745385164800","296352743424000","761819824128000","1457604526080000","2153187901440000","2514065817600000","2355992985600000","1789263872000000","1107155353600000","559144960000000","230077440000000","76723200000000","20534400000000","4344000000000","709500000000","86250000000","7343750000","390625000","9765625"],["17592186044416","241892558110720","1572301627719680","6425271074816000","18519898980352000","40039832616960000","67412196065280000","90576833740800000","98743143628800000","88318410752000000","65279728025600000","40035745792000000","20399915008000000","8624844800000000","3013401600000000","863808000000000","200904000000000","37290000000000","5390000000000","584375000000","44687500000","2148437500","48828125"],["281474976710656","4222124650659840","30082638135951360","135459832542003200","432685313320550400","1043161656852480000","1971991284285440000","2997672424243200000","3728487953203200000","3840841613312000000","3304559581593600000","2387758546944000000","1453512196096000000","746174545920000000","322710896640000000","117213184000000000","35557632000000000","8933760000000000","1836560000000000","303600000000000","39352500000000","3850000000000","267187500000","11718750000","244140625"],["4503599627370496","73183493944770560","567172078071971840","2790120706644377600","9782574854648627200","26021591938760704000","54578520252219392000","92618392757862400000","129469212157542400000","151011586998272000000","148332225337753600000","123499536449536000000","87534163984384000000","52947829391360000000","27354426245120000000","12060501606400000000","4526740275200000000","1440158720000000000","385848320000000000","86257600000000000","15884440000000000","2366650000000000","278037500000000","24781250000000","1574218750000","63476562500","1220703125"],["72057594037927936","1261007895663738880","10560941126183813120","56351290337473331200","215159472197625446400","625947571646365696000","1442601037083967488000","2703572648863989760000","4196767850939023360000","5468661598912512000000","6040893406157209600000","5697695024939008000000","4612198479429632000000","3215346232197120000000","1934298227671040000000","1004795697561600000000","450410007756800000000","173879853056000000000","57610448896000000000","16297881600000000000","3908544640000000000","786843200000000000","131203800000000000","17790500000000000","1911000000000000","156406250000000","9160156250000","341796875000","6103515625"]],"degrees":[0,2,4,6,8,10,12,14,16,18,20,22,24,26,28],"row_sums":["1","41","1681","68921","2825761","115856201","4750104241","194754273881","7984925229121","327381934393961","13422659310152401","550329031716248441","22563490300366186081","925103102315013629321","37929227194915558802161"],"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":[["16","20","5"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":192909,"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-77819ef68070fe35"]},{"id":192910,"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-77819ef68070fe35"]},{"id":192911,"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-77819ef68070fe35"]}],"region_exact":{},"incoming_edges":[{"id":"e-77819ef68070fe35","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-bc4e6b211c5c1593","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":"df187c53d7a2e1db1eba42fe22afc7ca83572d9ffa87058d8d810244de8340f1","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-08dfc533c843ef45","role":"sequence","name":"basis_map_A013611 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-90bb79ff2d858d18","kind":"fingerprint_related","node_a_id":"n-08dfc533c843ef45","node_b_id":"n-bc4e6b211c5c1593","fingerprint":"f6-d98c1510b99a0b49","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-d98c1510b99a0b49"},"other_id":"n-08dfc533c843ef45","other":{"id":"n-08dfc533c843ef45","name":"basis_map_A013611 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-d98c1510b99a0b49","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["5","20","16"],["25","200","560","640","256"]],"card_rows":[["1"],["5","20","16"],["25","200","560","640","256"],["125","1500","7200","17600","23040","15360","4096"]],"row_sums_preview":["1","41","1681","68921","2825761","115856201","4750104241","194754273881"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-b30e7a5b4198051a","steps":["basis A013611"],"edges":["e-a99971f11d0aad5a"],"text":"A272866 \u2192 basis A013611"},"headline":"A272866 \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:          16,           20,             5;\n  2:         256,          640,           560,            200,             25;\n  3:        4096,        15360,         23040,          17600,           7200,           1500,            125;\n  4:       65536,       327680,        696320,         819200,         582400,         256000,          68000,          10000,            625;\n  5:     1048576,      6553600,      18022400,       28672000,       29184000,       19840000,        9120000,        2800000,         550000,          62500,           3125;\n  6:    16777216,    125829120,     424673280,      851968000,     1130496000,     1044480000,      688640000,      326400000,      110400000,       26000000,        4050000,        375000,         15625;\n  7:   268435456,   2348810240,    9395240960,    22754099200,    37250662400,    43581440000,    37560320000,    24217600000,    11737600000,     4256000000,     1136800000,     217000000,      28000000,      2187500,       78125;\n  8:  4294967296,  42949672960,  198642237440,   563714457600,  1098068787200,  1556086784000,  1658847232000,  1356595200000,   859955200000,   423936000000,   161996800000,   47488000000,   10472000000,   1680000000,   185000000,    12500000,     390625;\n  9: 68719476736, 773094113280, 4058744094720, 13207024435200, 29836600934400, 49677336576000, 63153635328000, 62662901760000, 49185423360000, 30765056000000, 15370444800000, 6119424000000, 1927296000000, 473760000000, 88920000000, 12300000000, 1181250000, 70312500, 1953125;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (16+20*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[(16 + 20*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] = (16 + 20*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 A013611","reverse"],"edges":["e-a99971f11d0aad5a","e-77819ef68070fe35"],"text":"A272866 \u2192 basis A013611 \u2192 reverse"},"headline":"A272866 \u2192 basis A013611 \u2192 reverse","chain":null}
