{"id":"n-bac12d13055658fe","revision":2,"first_index":0,"fingerprint":"f6-73b18cc810b5ad72","name":"5-Stirling numbers of the second kind.","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"],["5","1"],["25","11","1"],["125","91","18","1"],["625","671","217","26","1"],["3125","4651","2190","425","35","1"],["15625","31031","19981","5590","740","45","1"],["78125","201811","170898","64701","12250","1190","56","1"],["390625","1288991","1398097","688506","174951","24150","1806","68","1"],["1953125","8124571","11075670","6906145","2263065","416451","44016","2622","81","1"],["9765625","50700551","85654261","66324830","27273730","6427575","900627","75480","3675","95","1"],["48828125","313968931","650280378","616252901","311788400","91549480","16334472","1806387","123255","5005","110","1"],["244140625","1932641711","4865931577","5580303586","3422348501","1227283200","271228672","38011116","3408702","193325","6655","126","1"],["1220703125","11839990891","35994162750","49508360265","36381440095","15695180501","4210798592","727362064","82324242","6115252","293150","8671","143","1"],["6103515625","72260648471","263799130141","432061044870","376941321120","193333245105","62013965013","12939143360","1797577210","167937770","10512502","431886","11102","161","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","6","37","235","1540","10427","73013","529032","3967195","30785747","247126450","2050937445","17585497797","155666739742","1421428484337"],"nonn":true,"definitions":[{"position":0,"source":"oeis_catalog:A193685","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["5","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":159561,"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":false,"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-9b675b6342ad9d47"]},{"id":159562,"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-2a53ff5ba197ee63","e-9b675b6342ad9d47"]},{"id":159563,"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-2a53ff5ba197ee63","e-9b675b6342ad9d47"]},{"id":159564,"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-9b675b6342ad9d47"]},{"id":159565,"position":4,"claim":"basis_map_preserver","from_index":null,"region_key":null,"provenance":"edges","claim_data":{"claim":"basis_map_preserver","input_interval":"(-inf, 5]","output_interval":"(-inf, 0]","statement":"source (-inf, a + b h + kappa m] = (-inf, 5] with m = 0, target (-inf, 0]"},"label":"Basis map preserves real-rootedness","description":"Basis map u^k -> P_k sends roots in (-inf, 5] to roots in (-inf, 0]: source (-inf, a + b h + kappa m] = (-inf, 5] with m = 0, target (-inf, 0]","region_exact":null,"edges":["e-fbf038f8fefbf6e8"]},{"id":159566,"position":5,"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-2a53ff5ba197ee63"]}],"region_exact":{},"incoming_edges":[{"id":"e-2a53ff5ba197ee63","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-bac12d13055658fe","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":"d40654bfb6e1b81f1854b4abdb93628c8a51c7cf25b302c52d58e72794b43ba4","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-3dcaebc50e901d48","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*4^(i-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]"}]},{"id":"e-fbf038f8fefbf6e8","operation":{"operation":"basis_map_theorem"},"operation_detail":null,"conclusion_id":"n-bac12d13055658fe","premise_count":0,"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":"a80495d482805d3f5d895ae3941fd46bc08b8377df7da28ce7c66141f944c937","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"basis_map_preserver","input_interval":"(-inf, 5]","output_interval":"(-inf, 0]","statement":"source (-inf, a + b h + kappa m] = (-inf, 5] with m = 0, target (-inf, 0]"}],"premises":[]},{"id":"e-9b675b6342ad9d47","operation":{"operation":"theorem_check","theorem":"liu_wang"},"operation_detail":"liu_wang","conclusion_id":"n-bac12d13055658fe","premise_count":0,"theorem":"L. L. Liu and Y. Wang, A unified approach to polynomial sequences with only real zeros, Adv. in Appl. Math. 38 (2007) 542-560, Theorem 2.1 with g = f' (as in their Corollary 2.4), with nonnegativity of the coefficients replaced by an exact induction keeping every root in (-inf, 0)","lean_route":null,"lean_declarations":[],"certificate_hash":"4ee70252a91ce40c400b1ed2242a92f383b2a5e6a2ba858f4a7e107669100e20","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":false,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"previous_interlaces_next"}],"premises":[]}],"used_by":[{"id":"e-76715f1adaa360d7","operation":"obreschkoff_combination","operation_detail":null,"conclusion_id":"n-b1b9b86e187f6cb4","conclusion_name":"obreschkoff_combination of 5-Stirling numbers of the second kind."},{"id":"e-76715f1adaa360d7","operation":"obreschkoff_combination","operation_detail":null,"conclusion_id":"n-b1b9b86e187f6cb4","conclusion_name":"obreschkoff_combination of 5-Stirling numbers of the second kind."}],"used_by_total":2,"links":[{"id":"l-38e0ab38788ce550","kind":"fingerprint_related","node_a_id":"n-bac12d13055658fe","node_b_id":"n-f0b949b616a94e3a","fingerprint":"f6-73b18cc810b5ad72","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-73b18cc810b5ad72"},"other_id":"n-f0b949b616a94e3a","other":{"id":"n-f0b949b616a94e3a","name":"basis_map_A008277 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.","first_index":0,"fingerprint":"f6-73b18cc810b5ad72","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["225","99","9"],["3375","2457","486","27"],["50625","54351","17577","2106","81"]],"card_rows":[["225","99","9"],["3375","2457","486","27"],["50625","54351","17577","2106","81"],["759375","1130193","532170","103275","8505","243"]],"row_sums_preview":["333","6345","124740","2533761","53226477","1156992984","26028766395","605955858201"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-615df6ea54d3bbef","steps":["basis A008277"],"edges":["e-f14e261f691e9869"],"text":"A038329 \u2192 basis A008277"},"headline":"A038329 \u2192 basis A008277","chain":null}},{"id":"l-b78e401bd9434c93","kind":"oeis_match","node_a_id":"n-bac12d13055658fe","node_b_id":null,"fingerprint":null,"oeis_id":"A196834","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":"A196834","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-df255b6ef103e8b9","kind":"oeis_match","node_a_id":"n-bac12d13055658fe","node_b_id":null,"fingerprint":null,"oeis_id":"A193685","in_catalog":1,"target":"triangle","row_alignment":"tabl","rows_matched":11,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":11,"target":"triangle","terms_compared":66},"in_catalog":true,"oeis_id":"A193685","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A193685","relation":"identification"},{"oeis_id":"A196834","relation":"match_row_sums"},{"oeis_id":"A193685","relation":"match_triangle"}],"bfiles":[{"node_id":"n-bac12d13055658fe","oeis_id":"A193685","target":"triangle","oeis_offset":0,"first_row":0,"rows_used":44,"terms_matched":66,"oeis_terms_agreeing":66,"oeis_terms_available":66,"term_count":990,"raw_size":24816},{"node_id":"n-bac12d13055658fe","oeis_id":"A196834","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":22,"oeis_terms_available":22,"term_count":200,"raw_size":26613}],"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,       1;\n  2:      25,      11,        1;\n  3:     125,      91,       18,       1;\n  4:     625,     671,      217,      26,       1;\n  5:    3125,    4651,     2190,     425,      35,      1;\n  6:   15625,   31031,    19981,    5590,     740,     45,     1;\n  7:   78125,  201811,   170898,   64701,   12250,   1190,    56,    1;\n  8:  390625, 1288991,  1398097,  688506,  174951,  24150,  1806,   68,  1;\n  9: 1953125, 8124571, 11075670, 6906145, 2263065, 416451, 44016, 2622, 81, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (5+x)*P(n-1,x) + 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[(5 + x)*P[n - 1, x] + 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] = (5 + x)*v[i - 1] + 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}},{"oeis_id":"A193685","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:       5,       1;\n  2:      25,      11,        1;\n  3:     125,      91,       18,       1;\n  4:     625,     671,      217,      26,       1;\n  5:    3125,    4651,     2190,     425,      35,      1;\n  6:   15625,   31031,    19981,    5590,     740,     45,     1;\n  7:   78125,  201811,   170898,   64701,   12250,   1190,    56,    1;\n  8:  390625, 1288991,  1398097,  688506,  174951,  24150,  1806,   68,  1;\n  9: 1953125, 8124571, 11075670, 6906145, 2263065, 416451, 44016, 2622, 81, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (5+x)*P(n-1,x) + 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[(5 + x)*P[n - 1, x] + x*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] = (5 + x)*v[i - 1] + x*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-bac12d13055658fe","steps":[],"edges":[],"text":"A193685"},"headline":"A193685: 5-Stirling numbers of the second kind.","chain":null}
