{"id":"n-054881b8a1092a7d","revision":1,"first_index":0,"fingerprint":"f6-c294b27e66113b73","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)|.","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"],["1","4"],["3","20","32"],["15","136","400","384"],["105","1192","4976","9088","6144"],["945","12828","68624","181312","237056","122880"],["10395","163788","1062736","3641408","6959104","7041024","2949120"],["135135","2420304","18401632","77094912","192427776","286388224","235487232","82575360"],["2027025","40628880","353474208","1745275904","5353453824","10453512192","12696731648","8774221824","2642411520"],["34459425","763663860","7471701216","42394761856","153838647552","370434044928","592170876928","606244110336","360792981504","95126814720"],["654729075","15887990340","172508877504","1104368523904","4618724777728","13191792755712","26068608458752","35205473173504","31104831062016","16239128739840","3805072588800"],["13749310575","362455876440","4321758002544","30782129612160","145585435384064","480251538089984","1127879658885120","1886333708828672","2202242271936512","1709634270265344","794428188917760","167423193907200"],["316234143225","8996452065720","116798316127632","915433365201792","4826007235217152","18033886274504704","48993305982676992","97523898929545216","141195590278316032","145029217269055488","100334293317844992","41983286527918080","8036313307545600"],["7905853580625","241355477090700","3387773410608240","28959346568681664","168252715870921984","701799533093909504","2162594735841169408","4985749384337833984","8601132501294252032","10967901126198820864","10049876630937010176","6266965415725891584","2384038732140380160","417888291992371200"],["213458046676875","6959325681963900","104985788803501680","971617668348466368","6164546736361066752","28370739482307187712","97690831720970506240","255720538584227004416","511432543057863507968","777796750479846277120","885549132102433243136","732001157557071642624","415319109048440193024","144789152883655311360","23401744351572787200"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","55","935","21505","623645","21827575","894930575","42061737025","2229272062325","131527051677175","8549258359016375","606997343490162625","46738795448742522125","3879320022245629336375"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013611","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-1","0"],["2","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":168083,"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-e6ead7ccbecb2d32"]},{"id":168084,"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-e6ead7ccbecb2d32"]},{"id":168085,"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-e6ead7ccbecb2d32"]}],"region_exact":{},"incoming_edges":[{"id":"e-e6ead7ccbecb2d32","operation":{"map":"A013611","operation":"basis_map"},"operation_detail":"A013611","conclusion_id":"n-054881b8a1092a7d","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":"a65b2f6e8572d585cb5171b1cef845ea3fccc433e625302ac79a42dd265bf0d1","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-849de7332df28657","role":"sequence","name":"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":[{"id":"l-42589ec516ef09f6","kind":"oeis_match","node_a_id":"n-054881b8a1092a7d","node_b_id":null,"fingerprint":null,"oeis_id":"A008543","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":"A008543","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A008543","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-054881b8a1092a7d","oeis_id":"A008543","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":18,"oeis_terms_available":18,"term_count":200,"raw_size":49468}],"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,         4;\n  2:        3,        20,         32;\n  3:       15,       136,        400,         384;\n  4:      105,      1192,       4976,        9088,         6144;\n  5:      945,     12828,      68624,      181312,       237056,       122880;\n  6:    10395,    163788,    1062736,     3641408,      6959104,      7041024,      2949120;\n  7:   135135,   2420304,   18401632,    77094912,    192427776,    286388224,    235487232,     82575360;\n  8:  2027025,  40628880,  353474208,  1745275904,   5353453824,  10453512192,  12696731648,   8774221824,   2642411520;\n  9: 34459425, 763663860, 7471701216, 42394761856, 153838647552, 370434044928, 592170876928, 606244110336, 360792981504, 95126814720;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2*n-1+4*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[(2*n - 1 + 4*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] = (2*n - 1 + 4*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"],"edges":["e-e6ead7ccbecb2d32"],"text":"A088996 \u2192 basis A013611"},"headline":"A088996 \u2192 basis A013611","chain":null}
