{"id":"n-447b8fa6e8cf1eaa","revision":0,"first_index":0,"fingerprint":"f6-47aff865bb54210c","name":"basis_map_A013610 of basis_map_A013611 of Triangle T(n, k) read by rows: T(n, k) = Sum_{j=0..n} binomial(j, n-k) * |Stirling1(n, n-j)|.","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"],["5","12"],["55","252","288"],["935","6264","13968","10368"],["21505","188952","621936","908928","497664"],["623645","6769908","29373264","63675072","68967936","29859840"],["21827575","281849220","1515497616","4343502528","6998482944","6010785792","2149908480"],["894930575","13389334320","85810736736","305385403392","651792013056","834314784768","593052254208","180592312320"],["42061737025","715212048240","5318480721312","22590944686080","59951223339264","101784828137472","107967675285504","65420855083008","17336861982720"],["2229272062325","42448906155420","359122379439456","1771715986263936","5617236863077632","11869328011926528","16715048228978688","15127814250233856","7984306034049024","1872381094133760"],["131527051677175","2771998110648780","26282089125578304","147625928722306944","544022893273252608","1374358776272980992","2410507206940925952","2898346828241240064","2286411766036955136","1068587208639774720","224685731296051200"],["8549258359016375","197541448013557800","2074239543768228720","13064921131526287488","54848110654105936128","161144342369813108736","338098326919193677824","506579495151882829824","531198546120245772288","371264521678463434752","155658084074693591040","29658516531078758400"],["606997343490162625","15256536012660961800","175716976121496562320","1226299894640991347328","5775564499381306863360","19339376242447985522688","47209766512515838783488","84653303232147570524160","110662544076408577327104","102852371680486295076864","64513815091001979568128","24520518780462468956160","4270826380475341209600"],["46738795448742522125","1269444858559359428100","15910226779330345339440","121836940162309797466176","636021250016355278661888","2390120032571978755931136","6652094715285605327867904","13883027924827833780584448","21726931198098481455955968","25182989495317182783946752","21012533744163014458736640","11952235100291918922252288","4154054561048746430300160","666248915354153228697600"],["3879320022245629336375","113216040895815576249300","1533815558922391047094320","12785384132399211206718528","73258369698625534103254272","305231532706221923557481472","953664026840797673209466880","2269843229928691898870317056","4135683980813250035982532608","5740312569391871055668183040","5974782535978816907778195456","4522141182343615699614695424","2352762025416088332653297664","753179826230584118272327680","111929817779497742421196800"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","17","595","31535","2238985","199269665","21321854155","2665231769375","381128143020625","61361631026320625","10983731953711391875","2163795194881144199375","465215966899446002865625","108395320287570918667690625","27207225392180300585590346875"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013610","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-1","0"],["6","12"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":203691,"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-25878a451c897200"]},{"id":203692,"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-25878a451c897200"]},{"id":203693,"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-25878a451c897200"]},{"id":203694,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-25878a451c897200"]}],"region_exact":{},"incoming_edges":[{"id":"e-25878a451c897200","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-447b8fa6e8cf1eaa","premise_count":1,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"75c60ec5f520f4f3c4ecdf7441f1adab8801ba79a73e7d56cb4db67bcab49914","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-054881b8a1092a7d","role":"sequence","name":"basis_map_A013611 of Triangle T(n, k) read by rows: T(n, k) = Sum_{j=0..n} binomial(j, n-k) * |Stirling1(n, n-j)|.","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":[],"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:             5,             12;\n  2:            55,            252,             288;\n  3:           935,           6264,           13968,            10368;\n  4:         21505,         188952,          621936,           908928,           497664;\n  5:        623645,        6769908,        29373264,         63675072,         68967936,          29859840;\n  6:      21827575,      281849220,      1515497616,       4343502528,       6998482944,        6010785792,        2149908480;\n  7:     894930575,    13389334320,     85810736736,     305385403392,     651792013056,      834314784768,      593052254208,      180592312320;\n  8:   42061737025,   715212048240,   5318480721312,   22590944686080,   59951223339264,   101784828137472,   107967675285504,    65420855083008,   17336861982720;\n  9: 2229272062325, 42448906155420, 359122379439456, 1771715986263936, 5617236863077632, 11869328011926528, 16715048228978688, 15127814250233856, 7984306034049024, 1872381094133760;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (6*n-1+12*n*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[(6*n - 1 + 12*n*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] = (6*n - 1 + 12*n*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-849de7332df28657","steps":["basis A013611","basis A013610"],"edges":["e-3d3e7193ca830199","e-25878a451c897200"],"text":"A088996 \u2192 basis A013611 \u2192 basis A013610"},"headline":"A088996 \u2192 basis A013611 \u2192 basis A013610","chain":null}