{"id":"n-c237629cf5fdbb07","revision":1,"first_index":0,"fingerprint":"f6-2ee8a94f56aa5e4f","name":"Unsigned 2-Stirling numbers of the first kind.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["6","5","1"],["24","26","9","1"],["120","154","71","14","1"],["720","1044","580","155","20","1"],["5040","8028","5104","1665","295","27","1"],["40320","69264","48860","18424","4025","511","35","1"],["362880","663696","509004","214676","54649","8624","826","44","1"],["3628800","6999840","5753736","2655764","761166","140889","16884","1266","54","1"],["39916800","80627040","70290936","34967140","11028590","2310945","326613","30810","1860","65","1"],["479001600","1007441280","924118272","489896616","167310220","38759930","6230301","696333","53130","2640","77","1"],["6227020800","13575738240","13020978816","7292774280","2664929476","671189310","119753843","15282630","1387023","87450","3641","90","1"],["87178291200","196287356160","195869441664","115119818736","44601786944","12061579816","2347743112","333710663","34700952","2611323","138424","4901","104","1"],["1307674368000","3031488633600","3134328981120","1922666722704","784146622896","225525484184","47277726496","7353403057","854224943","73870797","4687683","211939","6461","119","1"],["20922789888000","49811492505600","53180752331520","33896996544384","14469012689040","4392554369840","981969108120","164932175408","21021002145","2036157695","148873725","8078707","315315","8365","135","1"],["355687428096000","867718162483200","953884282141440","629429693586048","279870212258064","89142436976320","21086029207880","3785816090056","522289211873","55635682960","4567011020","286211744","13439062","457520","10660","152","1"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["12","60","360","2520","20160","181440","1814400","19958400","239500800","3113510400","43589145600","653837184000","10461394944000","177843714048000","3201186852864000"],"nonn":true,"definitions":[{"position":0,"source":"oeis_catalog:A143491","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["6","5","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["3","1"],["1","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":159308,"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-84f3431e657be6ed"]},{"id":159309,"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-25f97869157cba20","e-84f3431e657be6ed"]},{"id":159310,"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-25f97869157cba20","e-84f3431e657be6ed"]},{"id":159311,"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-84f3431e657be6ed"]},{"id":159312,"position":4,"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-25f97869157cba20"]},{"id":159313,"position":5,"claim":"basis_map_preserver","from_index":null,"region_key":null,"provenance":"edges","claim_data":{"claim":"basis_map_preserver","input_interval":"(-inf, 0]","output_interval":"(-inf, 0]","statement":"(-inf, 0] -> (-inf, 0] through R_s, s = 1, and U_g"},"label":"Basis map preserves real-rootedness","description":"Basis map u^k -> P_k sends roots in (-inf, 0] to roots in (-inf, 0]: (-inf, 0] -> (-inf, 0] through R_s, s = 1, and U_g","region_exact":null,"edges":["e-aadeee73ad838eee"]}],"region_exact":{},"incoming_edges":[{"id":"e-aadeee73ad838eee","operation":{"operation":"basis_map_theorem"},"operation_detail":null,"conclusion_id":"n-c237629cf5fdbb07","premise_count":0,"theorem":"index-dependent scalar term: P_(k+1) = (L + k s) P_k with L f = (a + b t) f + kappa (t - h) f' gives T = U_g o R_s (RealRooted basisMap_factorization_of_scalarIndex_recurrence), where R_s(u^k) = u (u + s) ... (u + (k-1) s) keeps every root <= 0 for s >= 0 (Su, Yang and Zhang, Australas. J. Combin. 56 (2013), Theorem 4; Brenti, Mem. AMS 413 (1989), Theorem 2.4.3; RealRooted NewtonParametersAffine.preservesNonposRooted) and U_g(u^k) = L^k g sends (-inf, a + b h + kappa m] into (-inf, h] (RealRooted rayBasisMap_maps_realRootedIn_iff)","lean_route":null,"lean_declarations":[],"certificate_hash":"4039205f6e1b251b52637c8396133ff95768059aabb06af45c1b7bf1fe36246e","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"basis_map_preserver","input_interval":"(-inf, 0]","output_interval":"(-inf, 0]","statement":"(-inf, 0] -> (-inf, 0] through R_s, s = 1, and U_g"}],"premises":[]},{"id":"e-25f97869157cba20","operation":{"operation":"parent_provers","route":"root_domain"},"operation_detail":"root_domain","conclusion_id":"n-c237629cf5fdbb07","premise_count":0,"theorem":"polytool parent provers, route root_domain: every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign","lean_route":null,"lean_declarations":[],"certificate_hash":"63d614298564b6696a24288f8596d40d2a381fa42b79bba32f6001e5df56de9f","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[]},{"id":"e-84f3431e657be6ed","operation":{"operation":"theorem_check","theorem":"liu_wang"},"operation_detail":"liu_wang","conclusion_id":"n-c237629cf5fdbb07","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":"ffcdad60c865d1005383e505193f14b4d0b0bac6a642b28122c52a8c6b0c8356","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-120de3004347d127","operation":"basis_map","operation_detail":"A013611","conclusion_id":"n-f2c0c28c5292c0fb","conclusion_name":"basis_map_A013611 of Unsigned 2-Stirling numbers of the first kind."},{"id":"e-13b78624414ee2b1","operation":"basis_map","operation_detail":"A008277","conclusion_id":"n-271026fda29dd6cb","conclusion_name":"basis_map_A008277 of Unsigned 2-Stirling numbers of the first kind."},{"id":"e-36015c56c1237184","operation":"basis_map","operation_detail":"A013610","conclusion_id":"n-0d23d1c7119dd8ef","conclusion_name":"basis_map_A013610 of Unsigned 2-Stirling numbers of the first kind."},{"id":"e-7cc986a9eefcf6a5","operation":"basis_map","operation_detail":"A013609","conclusion_id":"n-578e4bbd7f77169d","conclusion_name":"basis_map_A013609 of Unsigned 2-Stirling numbers of the first kind."},{"id":"e-a33c0ab38a356260","operation":"basis_map","operation_detail":"A007318","conclusion_id":"n-1b45c74901cc1e95","conclusion_name":"basis_map_A007318 of Unsigned 2-Stirling numbers of the first kind."},{"id":"e-a410505258437c48","operation":"obreschkoff_combination","operation_detail":null,"conclusion_id":"n-af7f1e73681bcc05","conclusion_name":"obreschkoff_combination of Unsigned 2-Stirling numbers of the first kind."},{"id":"e-a410505258437c48","operation":"obreschkoff_combination","operation_detail":null,"conclusion_id":"n-af7f1e73681bcc05","conclusion_name":"obreschkoff_combination of Unsigned 2-Stirling numbers of the first kind."}],"used_by_total":7,"links":[{"id":"l-0fba51ddd72a9a02","kind":"fingerprint_related","node_a_id":"n-914700b1060feeb2","node_b_id":"n-c237629cf5fdbb07","fingerprint":"f6-2ee8a94f56aa5e4f","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2ee8a94f56aa5e4f"},"other_id":"n-914700b1060feeb2","other":{"id":"n-914700b1060feeb2","name":"Triangle read by rows: T(n,k) = (-1)^(n+k)*Sum_{j=1..k} s(n,j), where s(n,j) are the signed Stirling numbers of the first kind (n >= 2; 1 <= k <= n-1; s(n,j) = A008275(n,j)).","first_index":0,"fingerprint":"f6-2ee8a94f56aa5e4f","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","1"],["6","5","1"]],"card_rows":[["1"],["2","1"],["6","5","1"],["24","26","9","1"]],"row_sums_preview":["1","3","12","60","360","2520","20160","181440"],"regions":["(-inf, 0)","(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":["A136124"],"derivation":{"root":"n-914700b1060feeb2","steps":[],"edges":[],"text":"A136124"},"headline":"A136124: Triangle read by rows: T(n,k) = (-1)^(n+k)*Sum_{j=1..k} s(n,j), where s(n,j) are the signed Stirling numbers of the first kind (n >= 2; 1 <= k <= n-1; s(n,j) = A008275(n,j)).","chain":null}},{"id":"l-6b0d018869c8aae2","kind":"fingerprint_related","node_a_id":"n-c237629cf5fdbb07","node_b_id":"n-2feb6dc01f424b0a","fingerprint":"f6-2ee8a94f56aa5e4f","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2ee8a94f56aa5e4f"},"other_id":"n-2feb6dc01f424b0a","other":{"id":"n-2feb6dc01f424b0a","name":"Triangle read by rows: coefficients of 1; 1(X+2); 1(X+2)(X+3); 1(X+2)(X+3)(X+4); ....","first_index":0,"fingerprint":"f6-2ee8a94f56aa5e4f","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","2"],["1","5","6"]],"card_rows":[["1"],["1","2"],["1","5","6"],["1","9","26","24"]],"row_sums_preview":["1","3","12","60","360","2520","20160","181440"],"regions":["(-inf, 0)","(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":["A145324"],"derivation":{"root":"n-2feb6dc01f424b0a","steps":[],"edges":[],"text":"A145324"},"headline":"A145324: Triangle read by rows: coefficients of 1; 1(X+2); 1(X+2)(X+3); 1(X+2)(X+3)(X+4); ....","chain":null}},{"id":"l-4365734e20660def","kind":"oeis_match","node_a_id":"n-c237629cf5fdbb07","node_b_id":null,"fingerprint":null,"oeis_id":"A188588","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":3,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A188588","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-b5f0cf524add282d","kind":"oeis_match","node_a_id":"n-c237629cf5fdbb07","node_b_id":null,"fingerprint":null,"oeis_id":"A357711","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":3,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A357711","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A188588","relation":"match_row_sums"},{"oeis_id":"A357711","relation":"match_row_sums"},{"oeis_id":"A143491","relation":"tail"}],"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:         6,          5,         1;\n  1:        24,         26,         9,         1;\n  2:       120,        154,        71,        14,         1;\n  3:       720,       1044,       580,       155,        20,        1;\n  4:      5040,       8028,      5104,      1665,       295,       27,       1;\n  5:     40320,      69264,     48860,     18424,      4025,      511,      35,      1;\n  6:    362880,     663696,    509004,    214676,     54649,     8624,     826,     44,     1;\n  7:   3628800,    6999840,   5753736,   2655764,    761166,   140889,   16884,   1266,    54,    1;\n  8:  39916800,   80627040,  70290936,  34967140,  11028590,  2310945,  326613,  30810,  1860,   65,  1;\n  9: 479001600, 1007441280, 924118272, 489896616, 167310220, 38759930, 6230301, 696333, 53130, 2640, 77, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (n+3+x)*P(n-1,x) for n >= 1, with P(0,x) = 6 + 5*x + x^2.","mathematica":"P[0, x_] := 6 + 5*x + x^2;\nP[n_, x_] := P[n, x] = Expand[(n + 3 + 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] = 6 + 5*x + x^2; for(n = 1, N, my(i = n + 1); v[i] = (n + 3 + 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-c237629cf5fdbb07","steps":[],"edges":[],"text":"A143491"},"headline":"Unsigned 2-Stirling numbers of the first kind.","chain":null}