{"id":"n-8742b407df6b9fe8","revision":1,"first_index":0,"fingerprint":"f6-9c43feb754fbba47","name":"basis_map_A008277 of Triangle T(n,k)=Sum_{i=0..n} |stirling1(n,n-i)|*binomial(i,k), k=0..n-1.","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"],["3","1"],["15","13","2"],["105","175","65","6"],["945","2695","1805","386","24"],["10395","47845","51380","19061","2674","120"],["135135","971425","1571570","899171","213304","21204","720"],["2027025","22317295","52375505","43371146","15466269","2553328","189468","5040"],["34459425","574148575","1906930025","2197221026","1104816349","269269848","32742044","1883664","40320"],["654729075","16386294925","75723748100","118234537421","80539888429","27176617413","4813597844","449138844","20622096","362880"],["13749310575","514522433425","3268536621350","6787198889471","6095278568379","2734938720613","661667599494","89007613244","6574220136","246500640","3628800"],["316234143225","17645005152775","152743894728425","416037040645226","483042851865574","280373284459983","88972742281709","16179104918834","1708822180780","102389499576","3194137440","39916800"],["7905853580625","656660000371375","7696190903526125","27211186216099886","40254572673929614","29628243401485443","11994835414845569","2831195343533414","400916742900388","34123669612968","1691823923712","44596586880","479001600"],["213458046676875","26369176111403125","416434697891712500","1895988772008432101","3533856662049738024","3249107837697745738","1644624882778095504","490013910929935249","89325632785885210","10125726081972436","709224060991368","29575130777856","667412922240","6227020800"],["6190283353629375","1136862985443810625","24105946245387256250","140445288583082654171","326921659082345781454","371135669200773160398","231330121481693232284","85256515046995113679","19455108975828908096","2820046401708120520","261619031455155296","15341385786226800","545532176571264","10657682300160","87178291200"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","30","351","5855","131475","3812529","138305076","6121511276","323999537027","20171724106127","1457124603930347","120717567119397599","11356800229213309026","1202890924980890211572"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A008277","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","0"],["2","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0"],["0","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":166046,"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-451612e307ad81d9"]},{"id":166047,"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-451612e307ad81d9"]},{"id":166048,"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-451612e307ad81d9"]}],"region_exact":{},"incoming_edges":[{"id":"e-451612e307ad81d9","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-8742b407df6b9fe8","premise_count":1,"theorem":"ray basis maps: for L f = (a + b t) f + kappa (t - h) f', b > 0, kappa >= 0, and g real-rooted with every root <= h, of multiplicity m at h, T(u^k) = L^k g sends every real polynomial with all roots in S to one with all roots in (-inf, h] if and only if S lies in (-inf, a + b h + kappa m] (RealRooted rayBasisMap_maps_realRootedIn_iff; the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"ac66ae973d9962d46b36bd936597cea97b916aadcdc4488bcb917397d12a8e9a","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-7fd16a32c438d522","role":"sequence","name":"Triangle T(n,k)=Sum_{i=0..n} |stirling1(n,n-i)|*binomial(i,k), k=0..n-1.","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:         3,           1;\n  2:        15,          13,           2;\n  3:       105,         175,          65,            6;\n  4:       945,        2695,        1805,          386,          24;\n  5:     10395,       47845,       51380,        19061,        2674,         120;\n  6:    135135,      971425,     1571570,       899171,      213304,       21204,        720;\n  7:   2027025,    22317295,    52375505,     43371146,    15466269,     2553328,     189468,      5040;\n  8:  34459425,   574148575,  1906930025,   2197221026,  1104816349,   269269848,   32742044,   1883664,    40320;\n  9: 654729075, 16386294925, 75723748100, 118234537421, 80539888429, 27176617413, 4813597844, 449138844, 20622096, 362880;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2*n+1+n*x)*P(n-1,x) + n*x*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[(2*n + 1 + n*x)*P[n - 1, x] + n*x*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] = (2*n + 1 + n*x)*v[i - 1] + n*x*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}}],"derivation":{"root":"n-7fd16a32c438d522","steps":["basis A008277"],"edges":["e-451612e307ad81d9"],"text":"A059364 \u2192 basis A008277"},"headline":"A059364 \u2192 basis A008277","chain":null}