{"id":"n-74d5bcdab635c3b8","revision":0,"first_index":0,"fingerprint":"f6-5b052678045ac842","name":"reversal of basis_map_A013610 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"],["6","3"],["54","51","12"],["648","882","399","60"],["9720","17118","11277","3294","360"],["174960","376164","322812","138231","29538","2520"],["3674160","9299124","9788364","5485347","1726146","289224","20160"],["88179840","256246416","318612852","219743604","90795627","22476690","3086856","181440"],["2380855680","7800451632","11165011164","9119205828","4648917969","1514826900","308112012","35767440","1814400"],["71425670400","260202961440","420755302872","396391297644","239778803178","96582904659","25906456260","4462255332","447873840","19958400"],["2357047123200","9443805772320","17007360532056","18129976456716","12669396076602","6064581491883","2013907912488","458131901076","68326900704","6033113280","239500800"],["84853696435200","370618620405120","735034454194176","873774839358504","691787952694980","383027082703614","151340244244047","42673551301080","8415483139332","1105441787232","87052501440","3113510400"],["3309294160972800","15642077945892480","33855004399244544","44367701093700120","39212577906123276","24623087563170666","11264648683368429","3783031920158778","925633560649068","160928993652696","18871232577408","1340161925760","43589145600"],["138990354760857600","706606686142076160","1656541353956658048","2371268511924073200","2312443788462679392","1622358346245017112","842461558149034008","327857070897195111","95622088349642526","20643521143149252","3206526673041576","339355289543040","21933173001600","653837184000"],["6254565964238592000","34021146552567148800","85850067906322830720","133211744699889822768","142000266671605743840","110005226196428640312","63868503656626804152","28232953120758324123","9548707110089035446","2458911865035996756","474590038577258952","66575414798102016","6416677417760640","380353441305600","10461394944000"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","9","117","1989","41769","1044225","30282525","999323325","36974963025","1515973484025","68218806781125","3342721532275125","177164241210581625","10098361749003152625","616000066689192310125"],"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":[["3","2"],["3","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":191118,"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-481020639a95e9e8"]},{"id":191119,"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-481020639a95e9e8"]},{"id":191120,"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-481020639a95e9e8"]}],"region_exact":{},"incoming_edges":[{"id":"e-481020639a95e9e8","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-74d5bcdab635c3b8","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":"eaedf73435435b523a0a76458f855b54d70bd357c31bdb219e8bee7edf598a9e","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-70594f2875298da0","role":"sequence","name":"basis_map_A013610 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-82790876bf7529cd","kind":"fingerprint_related","node_a_id":"n-70594f2875298da0","node_b_id":"n-74d5bcdab635c3b8","fingerprint":"f6-5b052678045ac842","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-5b052678045ac842"},"other_id":"n-70594f2875298da0","other":{"id":"n-70594f2875298da0","name":"basis_map_A013610 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":"f6-5b052678045ac842","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3","6"],["12","51","54"]],"card_rows":[["1"],["3","6"],["12","51","54"],["60","399","882","648"]],"row_sums_preview":["1","9","117","1989","41769","1044225","30282525","999323325"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2feb6dc01f424b0a","steps":["basis A013610"],"edges":["e-45b4054949a52789"],"text":"A145324 \u2192 basis A013610"},"headline":"A145324 \u2192 basis A013610","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:           6,            3;\n  2:          54,           51,           12;\n  3:         648,          882,          399,           60;\n  4:        9720,        17118,        11277,         3294,          360;\n  5:      174960,       376164,       322812,       138231,        29538,        2520;\n  6:     3674160,      9299124,      9788364,      5485347,      1726146,      289224,       20160;\n  7:    88179840,    256246416,    318612852,    219743604,     90795627,    22476690,     3086856,     181440;\n  8:  2380855680,   7800451632,  11165011164,   9119205828,   4648917969,  1514826900,   308112012,   35767440,   1814400;\n  9: 71425670400, 260202961440, 420755302872, 396391297644, 239778803178, 96582904659, 25906456260, 4462255332, 447873840, 19958400;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3*n+3+(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[(3*n + 3 + (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] = (3*n + 3 + (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 A013610","reverse"],"edges":["e-45b4054949a52789","e-481020639a95e9e8"],"text":"A145324 \u2192 basis A013610 \u2192 reverse"},"headline":"A145324 \u2192 basis A013610 \u2192 reverse","chain":null}