{"id":"n-5f3c91f2e89bfe38","revision":0,"first_index":0,"fingerprint":"f6-0ad613824303cac4","name":"reversal of basis_map_A013611 of Triangle of coefficients in expansion of (1+6x)^n.","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"],["24","7"],["576","336","49"],["13824","12096","3528","343"],["331776","387072","169344","32928","2401"],["7962624","11612160","6773760","1975680","288120","16807"],["191102976","334430208","243855360","94832640","20744640","2420208","117649"],["4586471424","9364045824","8193540096","3982970880","1161699840","203297472","19765032","823543"],["110075314176","256842399744","262193283072","152946081792","55761592320","13011038208","1897443072","158120256","5764801"],["2641807540224","6934744793088","8090535591936","5506058944512","2408900788224","702596063232","136615901184","17076987648","1245197016","40353607"],["63403380965376","184926527815680","242716067758080","188779163811840","96356031528960","33724611035136","8196954071040","1366159011840","149423641920","9684865680","282475249"],["1521681143169024","4882060334333952","7119671320903680","6229712405790720","3633998903377920","1483882885545984","432799174950912","90166494781440","13149280488960","1278402269760","74573465736","1977326743"],["36520347436056576","127821216026198016","205046534042025984","199350796985303040","130823960521605120","61051181576749056","20774360397643776","5193590099410944","946748195205120","122726617896960","10738579065984","569470101984","13841287201"],["876488338465357824","3323351616681148416","5815865329192009728","6219744865941454848","4535230631415644160","2380996081493213184","925942920580694016","270066685169369088","59077087380799488","9572676195962880","1116812222862336","88837335909504","4318481606712","96889010407"],["21035720123168587776","85895857169605066752","162844229217376272384","189984934086938984448","152383749215565643776","88890520375746625536","38889602664389148672","12963200888129716224","3308316893324771328","643283840368705536","93812226720436224","9949781621864448","725504909927616","32554707496752","678223072849"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","31","961","29791","923521","28629151","887503681","27512614111","852891037441","26439622160671","819628286980801","25408476896404831","787662783788549761","24417546297445042591","756943935220796320321"],"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":[["24","7"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":192151,"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-7ad438a6c7558204"]},{"id":192152,"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-7ad438a6c7558204"]},{"id":192153,"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-7ad438a6c7558204"]}],"region_exact":{},"incoming_edges":[{"id":"e-7ad438a6c7558204","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-5f3c91f2e89bfe38","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":"e41943dd7a7833253a5ef2afd0dabfe70433aa232900896c0663e4393d4ff023","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-1cedd000ef39425e","role":"sequence","name":"basis_map_A013611 of Triangle of coefficients in expansion of (1+6x)^n.","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-36a36df62e5bb947","kind":"fingerprint_related","node_a_id":"n-1cedd000ef39425e","node_b_id":"n-5f3c91f2e89bfe38","fingerprint":"f6-0ad613824303cac4","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-0ad613824303cac4"},"other_id":"n-1cedd000ef39425e","other":{"id":"n-1cedd000ef39425e","name":"basis_map_A013611 of Triangle of coefficients in expansion of (1+6x)^n.","first_index":0,"fingerprint":"f6-0ad613824303cac4","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["7","24"],["49","336","576"]],"card_rows":[["1"],["7","24"],["49","336","576"],["343","3528","12096","13824"]],"row_sums_preview":["1","31","961","29791","923521","28629151","887503681","27512614111"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-efce1b6c07b70e1a","steps":["basis A013611"],"edges":["e-83999c64158b130e"],"text":"A013613 \u2192 basis A013611"},"headline":"A013613 \u2192 basis A013611","chain":null}},{"id":"l-ceb0c337fbb51c48","kind":"fingerprint_related","node_a_id":"n-13735c4e8406e5f2","node_b_id":"n-5f3c91f2e89bfe38","fingerprint":"f6-0ad613824303cac4","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-0ad613824303cac4"},"other_id":"n-13735c4e8406e5f2","other":{"id":"n-13735c4e8406e5f2","name":"derivative of basis_map_A013611 of Triangle of coefficients in expansion of (1+6x)^n.","first_index":0,"fingerprint":"f6-0ad613824303cac4","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["24"],["336","1152"]],"card_rows":[["24"],["336","1152"],["3528","24192","41472"],["32928","338688","1161216","1327104"]],"row_sums_preview":["0","24","1488","69192","2859936","110822520","4122597744","149100618408"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-efce1b6c07b70e1a","steps":["basis A013611","D"],"edges":["e-83999c64158b130e","e-f691eabc48bbe128"],"text":"A013613 \u2192 basis A013611 \u2192 D"},"headline":"A013613 \u2192 basis A013611 \u2192 D","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:            24,             7;\n  2:           576,           336,            49;\n  3:         13824,         12096,          3528,           343;\n  4:        331776,        387072,        169344,         32928,          2401;\n  5:       7962624,      11612160,       6773760,       1975680,        288120,        16807;\n  6:     191102976,     334430208,     243855360,      94832640,      20744640,      2420208,       117649;\n  7:    4586471424,    9364045824,    8193540096,    3982970880,    1161699840,    203297472,     19765032,      823543;\n  8:  110075314176,  256842399744,  262193283072,  152946081792,   55761592320,  13011038208,   1897443072,   158120256,    5764801;\n  9: 2641807540224, 6934744793088, 8090535591936, 5506058944512, 2408900788224, 702596063232, 136615901184, 17076987648, 1245197016, 40353607;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (24+7*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[(24 + 7*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] = (24 + 7*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-efce1b6c07b70e1a","steps":["basis A013611","reverse"],"edges":["e-83999c64158b130e","e-7ad438a6c7558204"],"text":"A013613 \u2192 basis A013611 \u2192 reverse"},"headline":"A013613 \u2192 basis A013611 \u2192 reverse","chain":null}