{"id":"n-829809e4a0f73a58","revision":0,"first_index":0,"fingerprint":"f6-e0b26d9c45ca69c1","name":"reversal of basis_map_A013611 of Triangle read by rows: coefficients of 1; 1(X+2); 1(X+2)(X+3); 1(X+2)(X+3)(X+4); ....","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":true,"rows":[["1"],["8","3"],["96","68","12"],["1536","1568","532","60"],["30720","40576","20048","4392","360"],["737280","1188864","765184","245744","39384","2520"],["20643840","39186432","30936064","13002304","3068704","385632","20160"],["660602880","1439760384","1342631936","694498304","215219264","39958560","4115808","181440"],["23781703680","58437402624","62732353536","38428258304","14692876544","3590700800","547754688","47689920","1814400"],["951268147200","2599094845440","3152105570304","2227186221056","1010425903104","305249673984","61407896320","7932898368","597165120","19958400"],["41855798476800","125775390965760","169881783238656","135821460570112","71184974389248","25556096492544","6364943525888","1085942284032","121470045696","8044151040","239500800"],["2009078326886400","6581344146554880","9789405678010368","8727893289467904","5182557758095360","2152097298702336","637746543645696","134869495470080","19947811885824","1965229843968","116070001920","3113510400"],["104472072998092800","370356992197263360","601187913308307456","590902130544476160","391683509473509376","184464868145856512","63292182451408896","15941665375483904","2925459154643968","381461318287872","33548857915392","1786882567680","43589145600"],["5850436087893196800","22307072658018140160","39221878028224167936","42108338010115276800","30797808488683667456","16205285258270605312","6311335239466745856","1842115997798232064","402950693292320768","65243721143780352","7600655817580032","603298292520960","29244230668800","653837184000"],["351026165273591808000","1432031336887379558400","2710225844221740318720","3154050329058503294976","2521601917482864476160","1465082051315174998016","637964678500334436352","211508323699361857536","53650897562310959104","10361834361303953408","1499938887355287552","157808390632538112","11407426520463360","507137921740800","10461394944000"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","11","176","3696","96096","2978976","107243136","4396968576","202260554496","10315288279296","577656143640576","35237024762075136","2325643634296958976","165120698035084087296","12549173050666390634496"],"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","2"],["4","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":191665,"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-d196d8cff312ff57"]},{"id":191666,"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-d196d8cff312ff57"]},{"id":191667,"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-d196d8cff312ff57"]}],"region_exact":{},"incoming_edges":[{"id":"e-d196d8cff312ff57","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-829809e4a0f73a58","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":"c1467dbc85a329ad36bab61c7ffa328c140c2ca1e5767aab6029a99a40d682fd","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-2f0c73de4c291a40","role":"sequence","name":"basis_map_A013611 of Triangle read by rows: coefficients of 1; 1(X+2); 1(X+2)(X+3); 1(X+2)(X+3)(X+4); ....","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-ca9058d21ad6e595","kind":"fingerprint_related","node_a_id":"n-2f0c73de4c291a40","node_b_id":"n-829809e4a0f73a58","fingerprint":"f6-e0b26d9c45ca69c1","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e0b26d9c45ca69c1"},"other_id":"n-2f0c73de4c291a40","other":{"id":"n-2f0c73de4c291a40","name":"basis_map_A013611 of Triangle read by rows: coefficients of 1; 1(X+2); 1(X+2)(X+3); 1(X+2)(X+3)(X+4); ....","first_index":0,"fingerprint":"f6-e0b26d9c45ca69c1","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3","8"],["12","68","96"]],"card_rows":[["1"],["3","8"],["12","68","96"],["60","532","1568","1536"]],"row_sums_preview":["1","11","176","3696","96096","2978976","107243136","4396968576"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2feb6dc01f424b0a","steps":["basis A013611"],"edges":["e-39cad9e518a532bb"],"text":"A145324 \u2192 basis A013611"},"headline":"A145324 \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,             3;\n  2:           96,            68,            12;\n  3:         1536,          1568,           532,            60;\n  4:        30720,         40576,         20048,          4392,           360;\n  5:       737280,       1188864,        765184,        245744,         39384,         2520;\n  6:     20643840,      39186432,      30936064,      13002304,       3068704,       385632,       20160;\n  7:    660602880,    1439760384,    1342631936,     694498304,     215219264,     39958560,     4115808,     181440;\n  8:  23781703680,   58437402624,   62732353536,   38428258304,   14692876544,   3590700800,   547754688,   47689920,   1814400;\n  9: 951268147200, 2599094845440, 3152105570304, 2227186221056, 1010425903104, 305249673984, 61407896320, 7932898368, 597165120, 19958400;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (4*n+4+(n+2)*x)*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*n + 4 + (n + 2)*x)*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*n + 4 + (n + 2)*x)*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-2feb6dc01f424b0a","steps":["basis A013611","reverse"],"edges":["e-39cad9e518a532bb","e-d196d8cff312ff57"],"text":"A145324 \u2192 basis A013611 \u2192 reverse"},"headline":"A145324 \u2192 basis A013611 \u2192 reverse","chain":null}