{"id":"n-3ffff24fcb4f40a0","revision":1,"first_index":1,"fingerprint":"f6-605cb6decd8257f5","name":"Triangle read by rows: T(n,k) is the number of permutations of 1..n arranged in a circle with exactly k descents.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"irregular","degree_slope":null,"degree_intercept":null,"palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1"],["0","2"],["0","3","3"],["0","4","16","4"],["0","5","55","55","5"],["0","6","156","396","156","6"],["0","7","399","2114","2114","399","7"],["0","8","960","9528","19328","9528","960","8"],["0","9","2223","38637","140571","140571","38637","2223","9"],["0","10","5020","146080","882340","1561900","882340","146080","5020","10"],["0","11","11143","526240","5007112","14413894","14413894","5007112","526240","11143","11"],["0","12","24432","1831644","26441856","116857368","188690976","116857368","26441856","1831644","24432","12"],["0","13","53079","6217523","132439905","862140162","2112659718","2112659718","862140162","132439905","6217523","53079","13"],["0","14","114492","20716164","637468300","5925941490","21078701112","31852408056","21078701112","5925941490","637468300","20716164","114492","14"]],"degrees":[0,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["1","1","2","6","24","120","720","5040","40320","362880","3628800","39916800","479001600","6227020800","87178291200"],"nonn":true,"definitions":[{"position":0,"source":"oeis_catalog:A334218","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["1"]],"recurrence":{"denominator":{"coeffs":[["1"],["-1/2"]]},"terms":[{"coeff":{"coeffs":[["0","-1"],["0","3/2"],["0","-1/2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1/2","-1/2"],["0","-1/2","1/2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":159919,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-9baf8deff42ffdae"]},{"id":159920,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-9baf8deff42ffdae"]},{"id":159921,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-9baf8deff42ffdae"]},{"id":159922,"position":3,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"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 >= 2","region_exact":null,"edges":["e-9baf8deff42ffdae"]}],"region_exact":{},"incoming_edges":[{"id":"e-9baf8deff42ffdae","operation":{"operation":"theorem_check","theorem":"liu_wang_derivative_lag"},"operation_detail":"liu_wang_derivative_lag","conclusion_id":"n-3ffff24fcb4f40a0","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 the interlacers f' and the lag row (strict form, as in the derivative-lag theorems of the RealRooted library, https://github.com/PerAlexandersson/RealRooted), with nonnegativity of the coefficients replaced by an exact induction keeping every root in (-inf, 0)","lean_route":"halfGrowth","lean_declarations":["RealRooted.derivLag_strictInterl_of_nonnegCoeffs","RealRooted.derivLag_strictInterl_and_noCommonRoot","RealRooted.derivLag_eval_zero_pos"],"certificate_hash":"447d3e61ec8dd9e298d858fbb4b916ce19bdd0e612e5095f4b4d11a30fadbca9","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"previous_interlaces_next"}],"premises":[]}],"used_by":[{"id":"e-33c9a461744a57dc","operation":"obreschkoff_combination","operation_detail":null,"conclusion_id":"n-ffeb6f77baa8bcee","conclusion_name":"obreschkoff_combination of Triangle read by rows: T(n,k) is the number of permutations of 1..n arranged in a circle with exactly k descents."},{"id":"e-33c9a461744a57dc","operation":"obreschkoff_combination","operation_detail":null,"conclusion_id":"n-ffeb6f77baa8bcee","conclusion_name":"obreschkoff_combination of Triangle read by rows: T(n,k) is the number of permutations of 1..n arranged in a circle with exactly k descents."},{"id":"e-404a8e815c57c4e9","operation":"preserver_image","operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-73486cfabd571954","conclusion_name":"theta_plus_one_squared of Triangle read by rows: T(n,k) is the number of permutations of 1..n arranged in a circle with exactly k descents."},{"id":"e-5885e4696854d233","operation":"preserver_image","operation_detail":"affine, shift 0","conclusion_id":"n-5a739e9e90502f2d","conclusion_name":"shift_x_plus_one of Triangle read by rows: T(n,k) is the number of permutations of 1..n arranged in a circle with exactly k descents."},{"id":"e-5bbae1ad83a2b739","operation":"preserver_image","operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-1cc5242f219c3841","conclusion_name":"derivative of Triangle read by rows: T(n,k) is the number of permutations of 1..n arranged in a circle with exactly k descents."},{"id":"e-63c86030c8ae0f8e","operation":"preserver_image","operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-0aa2174fd7ab3229","conclusion_name":"one_plus_derivative of Triangle read by rows: T(n,k) is the number of permutations of 1..n arranged in a circle with exactly k descents."},{"id":"e-ad337510ccdadb92","operation":"preserver_image","operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-371ab4c129aed7f5","conclusion_name":"theta_plus_one of Triangle read by rows: T(n,k) is the number of permutations of 1..n arranged in a circle with exactly k descents."}],"used_by_total":7,"links":[{"id":"l-42abbe9232560540","kind":"fingerprint_related","node_a_id":"n-41809911e05bb4ba","node_b_id":"n-3ffff24fcb4f40a0","fingerprint":"f6-605cb6decd8257f5","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-605cb6decd8257f5"},"other_id":"n-41809911e05bb4ba","other":{"id":"n-41809911e05bb4ba","name":"Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 2*x + 2.","first_index":0,"fingerprint":"f6-605cb6decd8257f5","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","2"],["4","16","4"]],"card_rows":[["1"],["2","2"],["4","16","4"],["8","88","88","8"]],"row_sums_preview":["1","4","24","192","1920","23040","322560","5160960"],"regions":["(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":["A257609"],"derivation":{"root":"n-41809911e05bb4ba","steps":[],"edges":[],"text":"A257609"},"headline":"A257609: Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 2*x + 2.","chain":null}},{"id":"l-5297f57ea8caf959","kind":"fingerprint_related","node_a_id":"n-efd1fdf07e5468d8","node_b_id":"n-3ffff24fcb4f40a0","fingerprint":"f6-605cb6decd8257f5","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-605cb6decd8257f5"},"other_id":"n-efd1fdf07e5468d8","other":{"id":"n-efd1fdf07e5468d8","name":"Euler's triangle: triangle of Eulerian numbers T(n,k) (n>=0, 0 <= k <= n) read by rows.","first_index":1,"fingerprint":"f6-605cb6decd8257f5","polytool_family":null,"degree_pattern":"irregular","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1"],["1","1"]],"card_rows":[["1"],["1"],["1","1"],["1","4","1"]],"row_sums_preview":["1","1","2","6","24","120","720","5040"],"regions":["(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":["A173018"],"derivation":{"root":"n-efd1fdf07e5468d8","steps":[],"edges":[],"text":"A173018"},"headline":"A173018: Euler's triangle: triangle of Eulerian numbers T(n,k) (n>=0, 0 <= k <= n) read by rows.","chain":null}},{"id":"l-6997b2e9abbc32f8","kind":"fingerprint_related","node_a_id":"n-3ffff24fcb4f40a0","node_b_id":"n-e24b36ff0794dfbd","fingerprint":"f6-605cb6decd8257f5","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-605cb6decd8257f5"},"other_id":"n-e24b36ff0794dfbd","other":{"id":"n-e24b36ff0794dfbd","name":"Triangle of Eulerian numbers T(n,k), 0 <= k <= n, read by rows.","first_index":0,"fingerprint":"f6-605cb6decd8257f5","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["0","1"],["0","1","1"]],"card_rows":[["1"],["0","1"],["0","1","1"],["0","1","4","1"]],"row_sums_preview":["1","1","2","6","24","120","720","5040"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":["A123125"],"derivation":{"root":"n-efd1fdf07e5468d8","steps":["reverse (shift +1)"],"edges":["e-61bc78adfe1eb25a"],"text":"A173018 \u2192 reverse (shift +1)"},"headline":"A123125: Triangle of Eulerian numbers T(n,k), 0 <= k <= n, read by rows.","chain":"A173018 \u2192 reverse (shift +1)"}},{"id":"l-c33c38feaffacbaf","kind":"fingerprint_related","node_a_id":"n-3ffff24fcb4f40a0","node_b_id":"n-7d47aefd76b8e894","fingerprint":"f6-605cb6decd8257f5","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-605cb6decd8257f5"},"other_id":"n-7d47aefd76b8e894","other":{"id":"n-7d47aefd76b8e894","name":"reversal of Triangle of Eulerian numbers T(n,k), 0 <= k <= n, read by rows.","first_index":0,"fingerprint":"f6-605cb6decd8257f5","polytool_family":null,"degree_pattern":"irregular","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1"],["1","1"]],"card_rows":[["1"],["1"],["1","1"],["1","4","1"]],"row_sums_preview":["1","1","2","6","24","120","720","5040"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-efd1fdf07e5468d8","steps":["reverse (shift +1)","reverse"],"edges":["e-61bc78adfe1eb25a","e-52ca2f2e23126004"],"text":"A173018 \u2192 reverse (shift +1) \u2192 reverse"},"headline":"A173018 \u2192 reverse (shift +1) \u2192 reverse","chain":null}},{"id":"l-e1ebfe6ad53e3ded","kind":"fingerprint_related","node_a_id":"n-dfedc6a36c97eba4","node_b_id":"n-3ffff24fcb4f40a0","fingerprint":"f6-605cb6decd8257f5","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-605cb6decd8257f5"},"other_id":"n-dfedc6a36c97eba4","other":{"id":"n-dfedc6a36c97eba4","name":"Triangle read by rows: T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(n) = 3*n + 3.","first_index":0,"fingerprint":"f6-605cb6decd8257f5","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","3"],["9","36","9"]],"card_rows":[["1"],["3","3"],["9","36","9"],["27","297","297","27"]],"row_sums_preview":["1","6","54","648","9720","174960","3674160","88179840"],"regions":["(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":["A257620"],"derivation":{"root":"n-dfedc6a36c97eba4","steps":[],"edges":[],"text":"A257620"},"headline":"A257620: Triangle read by rows: T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(n) = 3*n + 3.","chain":null}},{"id":"l-e61fd4f86072b227","kind":"fingerprint_related","node_a_id":"n-ff270bcb802a85af","node_b_id":"n-3ffff24fcb4f40a0","fingerprint":"f6-605cb6decd8257f5","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-605cb6decd8257f5"},"other_id":"n-ff270bcb802a85af","other":{"id":"n-ff270bcb802a85af","name":"eulerian","first_index":1,"fingerprint":"f6-605cb6decd8257f5","polytool_family":"eulerian","degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","1"],["1","4","1"]],"card_rows":[["1"],["1","1"],["1","4","1"],["1","11","11","1"]],"row_sums_preview":["1","2","6","24","120","720","5040","40320"],"regions":["(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":["A008292"],"derivation":{"root":"n-ff270bcb802a85af","steps":[],"edges":[],"text":"A008292"},"headline":"A008292: Eulerian","chain":null}},{"id":"l-0c3eba23e80cd3e6","kind":"oeis_match","node_a_id":"n-3ffff24fcb4f40a0","node_b_id":null,"fingerprint":null,"oeis_id":"A000142","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":0,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A000142","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-0c7ae04775b6066d","kind":"oeis_match","node_a_id":"n-3ffff24fcb4f40a0","node_b_id":null,"fingerprint":null,"oeis_id":"A370360","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":2,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":0,"target":"row_sums","terms_compared":14},"in_catalog":false,"oeis_id":"A370360","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-6412ccdfedd50caa","kind":"oeis_match","node_a_id":"n-3ffff24fcb4f40a0","node_b_id":null,"fingerprint":null,"oeis_id":"A338112","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":4,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":2,"target":"row_sums","terms_compared":12},"in_catalog":false,"oeis_id":"A338112","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-aef2c195b112b42f","kind":"oeis_match","node_a_id":"n-3ffff24fcb4f40a0","node_b_id":null,"fingerprint":null,"oeis_id":"A159333","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":0,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":-2,"oeis_term_position":2,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A159333","oeis_nonn":false},"other_id":null,"other":null},{"id":"l-b9407229b840e214","kind":"oeis_match","node_a_id":"n-3ffff24fcb4f40a0","node_b_id":null,"fingerprint":null,"oeis_id":"A124355","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":3,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":2,"target":"row_sums","terms_compared":13},"in_catalog":false,"oeis_id":"A124355","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-d4de96fc8142e1ef","kind":"oeis_match","node_a_id":"n-3ffff24fcb4f40a0","node_b_id":null,"fingerprint":null,"oeis_id":"A104150","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":0,"oeis_term_position":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A104150","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-da451dd3a72bdd5c","kind":"oeis_match","node_a_id":"n-3ffff24fcb4f40a0","node_b_id":null,"fingerprint":null,"oeis_id":"A354074","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":4,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":1,"target":"row_sums","terms_compared":12},"in_catalog":false,"oeis_id":"A354074","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-f18ccdc299486bbe","kind":"oeis_match","node_a_id":"n-3ffff24fcb4f40a0","node_b_id":null,"fingerprint":null,"oeis_id":"A358185","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":0,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":2,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A358185","oeis_nonn":false},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A334218","relation":"identification"},{"oeis_id":"A000142","relation":"match_row_sums"},{"oeis_id":"A104150","relation":"match_row_sums"},{"oeis_id":"A124355","relation":"match_row_sums"},{"oeis_id":"A159333","relation":"match_row_sums"},{"oeis_id":"A338112","relation":"match_row_sums"},{"oeis_id":"A354074","relation":"match_row_sums"},{"oeis_id":"A358185","relation":"match_row_sums"},{"oeis_id":"A370360","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-3ffff24fcb4f40a0","oeis_id":"A000142","target":"row_sums","oeis_offset":0,"first_row":1,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":23,"oeis_terms_available":23,"term_count":200,"raw_size":34064}],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 1;\n   2: 1;\n   3: 0, 2;\n   4: 0, 3,    3;\n   5: 0, 4,   16,     4;\n   6: 0, 5,   55,    55,      5;\n   7: 0, 6,  156,   396,    156,      6;\n   8: 0, 7,  399,  2114,   2114,    399,     7;\n   9: 0, 8,  960,  9528,  19328,   9528,   960,    8;\n  10: 0, 9, 2223, 38637, 140571, 140571, 38637, 2223, 9;\n","recurrence_sentence":"The row polynomials satisfy -((1/2)*n-1)*P(n,x) = -((1/2)*n^2-(3/2)*n+1)*x*P(n-1,x) - (((1/2)*n-1/2)*x-((1/2)*n-1/2)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-((1/2)*n^2 - (3/2)*n + 1)*x*P[n - 1, x] - (((1/2)*n - 1/2)*x - ((1/2)*n - 1/2)*x^2)*D[P[n - 1, x], x])/(-(1/2)*n + 1)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1; for(n = 3, N, my(i = n); v[i] = (-((1/2)*n^2 - (3/2)*n + 1)*x*v[i - 1] - (((1/2)*n - 1/2)*x - ((1/2)*n - 1/2)*x^2)*deriv(v[i - 1], x))/(-(1/2)*n + 1)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-3ffff24fcb4f40a0","steps":[],"edges":[],"text":"A334218"},"headline":"A334218: Triangle read by rows: T(n,k) is the number of permutations of 1..n arranged in a circle with exactly k descents.","chain":null}
