{"id":"n-6e47c52ce477d423","revision":0,"first_index":1,"fingerprint":"f6-570ab0ccd1213152","name":"shift_x_plus_one of Triangle read by rows: T(n, k) = 2^n * [x^k] hypergeom([-n, -n, 1], [1], x).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["28","32","8"],["272","504","288","48"],["3344","8704","8064","3072","384"],["49472","167200","217600","134400","38400","3840"],["852928","3561984","6019200","5222400","2419200","552960","46080"],["16758016","83586944","174537216","196627200","127948800","47416320","9031680","645120"],["369082624","2145026048","5349564416","7446921216","6292070400","3275489280","1011548160","165150720","10321920"],["8996922368","59791385088","173747109888","288876478464","301600309248","203863080960","88438210560","23410114560","3344302080","185794560"],["240294124544","1799384473600","5979138508800","11583140659200","14443823923200","12064012369920","6795436032000","2526806016000","585252864000","74317824000","3715891200"],["6974172532736","58151178139648","217725521305600","482317173043200","700780009881600","699081077882880","486581832253440","234927931392000","76435881984000","15736799232000","1798491340800","81749606400"],["218457925292032","2008561689427968","8373769652109312","20901650045337600","34726836459110400","40364928569180160","33555891738378240","20019366812712960","8457405530112000","2445948223488000","453219817881600","47087773286400","1961990553600"],["7342688736329728","73838778748706816","339446925513326592","943444714137649152","1766189428831027200","2347534144635863040","2273890976063815680","1620270201081692160","845818247837122560","317622563241984000","82673049953894400","13926208949452800","1326305614233600","51011754393600"],["263513011737051136","2878333984641253376","14472400634746535936","44354398267074674688","92457581985489616896","138469251220352532480","153372230782876385280","127337894659573678080","79393239853002915840","36840083683572449280","12450804479085772800","2946166871084236800","454922825682124800","39993215444582400","1428329123020800"],["10054112734940561408","118580855281673011200","647625146544282009600","2170860095211980390400","4989869805045900902400","8321182378694065520640","10385193841526439936000","9859643407470624768000","7162756574601019392000","3969661992650145792000","1657803765760760217600","509351092326236160000","110481257665658880000","15747328581304320000","1285496210718720000","42849873690624000"],["406301504472849907712","5147705720289567440896","30356698952108290867200","110528025010224129638400","277870092187133489971200","510962668036700252405760","710074229648560257761280","759602749551648178176000","631017178078119985152000","407481262910635769856000","203246694023687464550400","77163229824500839219200","21732313272586076160000","4351261840370565120000","575902302401986560000","43878270659198976000","1371195958099968000"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["68","1112","23568","610912","18674752","656551296","26065174784","1152253707776","56095322686464","2980591818594304","171575086226870272","10633375245099491328","705732243803486830592","49930140000179441205248","3750561562330057001730048"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["28","32","8"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["4","2"],["4","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-2","-4","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":191912,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-ed5af84bfccfc67f"]},{"id":191913,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-ed5af84bfccfc67f"]},{"id":191914,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-ed5af84bfccfc67f"]},{"id":191915,"position":3,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2","region_exact":null,"edges":["e-ed5af84bfccfc67f"]}],"region_exact":{},"incoming_edges":[{"id":"e-ed5af84bfccfc67f","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-6e47c52ce477d423","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":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"57b9bb564bbd60293066d5c2581fdbb2e56aeb2386be7a4134f60aebb8e03e92","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"-1"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-2ee8dc15b8ca5b2d","role":"sequence","name":"Triangle read by rows: T(n, k) = 2^n * [x^k] hypergeom([-n, -n, 1], [1], x).","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-424d3bb9afb073ec","kind":"fingerprint_related","node_a_id":"n-e763351afcabc7a7","node_b_id":"n-6e47c52ce477d423","fingerprint":"f6-570ab0ccd1213152","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-570ab0ccd1213152"},"other_id":"n-e763351afcabc7a7","other":{"id":"n-e763351afcabc7a7","name":"mobius_x_over_x_plus_one of basis_map_A008277 of Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.","first_index":0,"fingerprint":"f6-570ab0ccd1213152","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","2"],["2","8","7"]],"card_rows":[["1"],["1","2"],["2","8","7"],["6","36","63","34"]],"row_sums_preview":["1","3","17","139","1473","19091","291793","5129307"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-b9fbfb57d9269e2f","steps":["basis A008277","x \u21a6 x/(x+1)"],"edges":["e-07cfebc08bea7e3c","e-b400a5696574ba31"],"text":"A132393 \u2192 basis A008277 \u2192 x \u21a6 x/(x+1)"},"headline":"A132393 \u2192 basis A008277 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-dda5e1d926f4c18f","kind":"fingerprint_related","node_a_id":"n-12211825af356650","node_b_id":"n-6e47c52ce477d423","fingerprint":"f6-570ab0ccd1213152","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-570ab0ccd1213152"},"other_id":"n-12211825af356650","other":{"id":"n-12211825af356650","name":"shift_x_plus_one of T(n,k) is the number of partial bijections of height k (height(alpha) = |Im(alpha)|) of an n-element set.","first_index":0,"fingerprint":"f6-570ab0ccd1213152","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","1"],["7","8","2"]],"card_rows":[["1"],["2","1"],["7","8","2"],["34","63","36","6"]],"row_sums_preview":["1","3","17","139","1473","19091","291793","5129307"],"regions":["(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-47fe395e9f7d5f86","steps":["x \u21a6 x+1"],"edges":["e-54b278190ec3c985"],"text":"A144084 \u2192 x \u21a6 x+1"},"headline":"A144084 \u2192 x \u21a6 x+1","chain":null}}],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:            28,             32,               8;\n   2:           272,            504,             288,              48;\n   3:          3344,           8704,            8064,            3072,             384;\n   4:         49472,         167200,          217600,          134400,           38400,            3840;\n   5:        852928,        3561984,         6019200,         5222400,         2419200,          552960,           46080;\n   6:      16758016,       83586944,       174537216,       196627200,       127948800,        47416320,         9031680,          645120;\n   7:     369082624,     2145026048,      5349564416,      7446921216,      6292070400,      3275489280,      1011548160,       165150720,       10321920;\n   8:    8996922368,    59791385088,    173747109888,    288876478464,    301600309248,    203863080960,     88438210560,     23410114560,     3344302080,      185794560;\n   9:  240294124544,  1799384473600,   5979138508800,  11583140659200,  14443823923200,  12064012369920,   6795436032000,   2526806016000,   585252864000,    74317824000,    3715891200;\n  10: 6974172532736, 58151178139648, 217725521305600, 482317173043200, 700780009881600, 699081077882880, 486581832253440, 234927931392000, 76435881984000, 15736799232000, 1798491340800, 81749606400;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (4*n+4+(4*n+2)*x)*P(n-1,x) - (2+4*x+2*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 28 + 32*x + 8*x^2.","mathematica":"P[1, x_] := 28 + 32*x + 8*x^2;\nP[n_, x_] := P[n, x] = Expand[(4*n + 4 + (4*n + 2)*x)*P[n - 1, x] - (2 + 4*x + 2*x^2)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 28 + 32*x + 8*x^2; for(n = 2, N, my(i = n); v[i] = (4*n + 4 + (4*n + 2)*x)*v[i - 1] - (2 + 4*x + 2*x^2)*deriv(v[i - 1], x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-2ee8dc15b8ca5b2d","steps":["x \u21a6 x+1"],"edges":["e-ed5af84bfccfc67f"],"text":"A395972 \u2192 x \u21a6 x+1"},"headline":"A395972 \u2192 x \u21a6 x+1","chain":null}
