{"id":"n-a36b2e24b7a3dbb7","revision":1,"first_index":0,"fingerprint":"f6-9d05ffe77ba48b7a","name":"Coefficient triangle for polynomials used for e.g.f.s for unsigned Stirling1 diagonals.","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":false,"rows":[["1"],["1","1"],["2","5","3"],["6","26","35","15"],["24","154","340","315","105"],["120","1044","3304","4900","3465","945"],["720","8028","33740","70532","78750","45045","10395"],["5040","69264","367884","1008980","1571570","1406790","675675","135135"],["40320","663696","4302216","14777620","29957620","37147110","27747720","11486475","2027025"],["362880","6999840","53961336","224655816","566780500","909508600","936275340","599999400","218243025","34459425"],["3628800","80627040","724534272","3568061640","10855452608","21577556000","28623034440","25180395240","14128364250","4583103525","654729075"],["39916800","1007441280","10386470016","59371808496","212784652080","508072685120","831832481480","939838699800","721686035070","360100991250","105411381075","13749310575"],["479001600","13575738240","158507316864","1035987707664","4295131560720","12041790080320","23610220313680","32829919962840","32290655997600","21995841517650","9881171199900","2635284526875","316234143225"],["6227020800","196287356160","2567670088320","18953413075584","89593039854864","289769457978000","665346407405680","1105192585216720","1334702175206400","1162012289337900","711175450986000","290513766242700","71152682225625","7905853580625"],["87178291200","3031488633600","44027031755520","363290743698048","1934882739672480","7118294418969552","18812547849695600","36515972437664720","52572492144091920","56089730509312500","43794493478435700","24331054979731500","9112813893933750","2063427784543125","213458046676875"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","2","10","82","938","13778","247210","5240338","128149802","3551246162","109979486890","3764281873042","141104799067178","5749087305575378","252969604725106090"],"nonn":true,"definitions":[{"position":0,"source":"oeis_catalog:A112486","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","0"],["1","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":159072,"position":0,"claim":"real_rooted","from_index":0,"region_key":"[-1, 0)","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":false,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in [-1, 0)","region_exact":null,"edges":["e-c24cc7aceb1eaafd"]},{"id":159073,"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-c24cc7aceb1eaafd"]},{"id":159074,"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-c24cc7aceb1eaafd"]},{"id":159075,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"previous_interlaces_next"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 1","region_exact":null,"edges":["e-c24cc7aceb1eaafd"]}],"region_exact":{},"incoming_edges":[{"id":"e-c24cc7aceb1eaafd","operation":{"operation":"theorem_check","theorem":"ma_wang_interval"},"operation_detail":"ma_wang_interval","conclusion_id":"n-a36b2e24b7a3dbb7","premise_count":0,"theorem":"S.-M. Ma and Y. Wang, q-Eulerian polynomials and polynomials with only real zeros, Electron. J. Combin. 15 (2008) R17, Theorem 2 and Corollary 3 (arXiv:math/0611826), in the closed two-endpoint form of the RealRooted library (https://github.com/PerAlexandersson/RealRooted, RealRooted.affineEndpointStep_closed), after the change of variable x = U + (U - L) y, with the endpoint multiplicities tracked exactly through the induction","lean_route":null,"lean_declarations":["RealRooted.affineEndpointStep_closed","RealRooted.affineEndpointStep_roots_mem_Icc_iff","RealRooted.affineEndpointStep_rootMultiplicity_zero","RealRooted.affineEndpointStep_rootMultiplicity_neg_one","RealRooted.MaWang.endpointStep","RealRooted.MaWang.affineEndpointSequence","RealRooted.normalizedAffineEndpointSequence_roots_mem_Icc_iff"],"certificate_hash":"2d33f8e72b3d0771e2527ee7aa80e28b4e4e1a35fbc73149c3c0be9345a1d7ac","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":false,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"previous_interlaces_next"}],"premises":[]}],"used_by":[{"id":"e-fa5fc5a793cceca0","operation":"obreschkoff_combination","operation_detail":null,"conclusion_id":"n-6234241a3daf2f61","conclusion_name":"obreschkoff_combination of Coefficient triangle for polynomials used for e.g.f.s for unsigned Stirling1 diagonals."},{"id":"e-fa5fc5a793cceca0","operation":"obreschkoff_combination","operation_detail":null,"conclusion_id":"n-6234241a3daf2f61","conclusion_name":"obreschkoff_combination of Coefficient triangle for polynomials used for e.g.f.s for unsigned Stirling1 diagonals."}],"used_by_total":2,"links":[{"id":"l-9599627eb1fcd6ea","kind":"oeis_match","node_a_id":"n-a36b2e24b7a3dbb7","node_b_id":null,"fingerprint":null,"oeis_id":"A032034","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":1,"target":"row_sums","terms_compared":14},"in_catalog":false,"oeis_id":"A032034","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-a675c2346f629067","kind":"oeis_match","node_a_id":"n-a36b2e24b7a3dbb7","node_b_id":null,"fingerprint":null,"oeis_id":"A112487","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A112487","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-bb2842a7a614eca6","kind":"oeis_match","node_a_id":"n-a36b2e24b7a3dbb7","node_b_id":null,"fingerprint":null,"oeis_id":"A112486","in_catalog":1,"target":"triangle","row_alignment":"tabl","rows_matched":8,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":8,"target":"triangle","terms_compared":41},"in_catalog":true,"oeis_id":"A112486","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A112486","relation":"identification"},{"oeis_id":"A032034","relation":"match_row_sums"},{"oeis_id":"A112487","relation":"match_row_sums"},{"oeis_id":"A112486","relation":"match_triangle"}],"bfiles":[{"node_id":"n-a36b2e24b7a3dbb7","oeis_id":"A112486","target":"triangle","oeis_offset":0,"first_row":0,"rows_used":44,"terms_matched":41,"oeis_terms_agreeing":41,"oeis_terms_available":41,"term_count":990,"raw_size":46180},{"node_id":"n-a36b2e24b7a3dbb7","oeis_id":"A112487","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":19,"oeis_terms_available":19,"term_count":200,"raw_size":44060}],"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:      1,       1;\n  2:      2,       5,        3;\n  3:      6,      26,       35,        15;\n  4:     24,     154,      340,       315,       105;\n  5:    120,    1044,     3304,      4900,      3465,       945;\n  6:    720,    8028,    33740,     70532,     78750,     45045,     10395;\n  7:   5040,   69264,   367884,   1008980,   1571570,   1406790,    675675,    135135;\n  8:  40320,  663696,  4302216,  14777620,  29957620,  37147110,  27747720,  11486475,   2027025;\n  9: 362880, 6999840, 53961336, 224655816, 566780500, 909508600, 936275340, 599999400, 218243025, 34459425;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (n+n*x)*P(n-1,x) + (x+x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(n + n*x)*P[n - 1, x] + (x + x^2)*D[P[n - 1, x], 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] = (n + n*x)*v[i - 1] + (x + 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 - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}},{"oeis_id":"A112486","target":"triangle","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:      1;\n  1:      1,       1;\n  2:      2,       5,        3;\n  3:      6,      26,       35,        15;\n  4:     24,     154,      340,       315,       105;\n  5:    120,    1044,     3304,      4900,      3465,       945;\n  6:    720,    8028,    33740,     70532,     78750,     45045,     10395;\n  7:   5040,   69264,   367884,   1008980,   1571570,   1406790,    675675,    135135;\n  8:  40320,  663696,  4302216,  14777620,  29957620,  37147110,  27747720,  11486475,   2027025;\n  9: 362880, 6999840, 53961336, 224655816, 566780500, 909508600, 936275340, 599999400, 218243025, 34459425;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (n+n*x)*P(n-1,x) + (x+x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(n + n*x)*P[n - 1, x] + (x + x^2)*D[P[n - 1, x], x]];\nTable[PadRight[CoefficientList[P[n, x], x], n + 1], {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] = (n + n*x)*v[i - 1] + (x + x^2)*deriv(v[i - 1], x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); 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":true}}],"derivation":{"root":"n-a36b2e24b7a3dbb7","steps":[],"edges":[],"text":"A112486"},"headline":"A112486: Coefficient triangle for polynomials used for e.g.f.s for unsigned Stirling1 diagonals.","chain":null}
