{"id":"n-ff270bcb802a85af","revision":2,"first_index":1,"fingerprint":"f6-605cb6decd8257f5","name":"eulerian","polytool_family":"eulerian","interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1","1"],["1","4","1"],["1","11","11","1"],["1","26","66","26","1"],["1","57","302","302","57","1"],["1","120","1191","2416","1191","120","1"],["1","247","4293","15619","15619","4293","247","1"],["1","502","14608","88234","156190","88234","14608","502","1"],["1","1013","47840","455192","1310354","1310354","455192","47840","1013","1"],["1","2036","152637","2203488","9738114","15724248","9738114","2203488","152637","2036","1"],["1","4083","478271","10187685","66318474","162512286","162512286","66318474","10187685","478271","4083","1"],["1","8178","1479726","45533450","423281535","1505621508","2275172004","1505621508","423281535","45533450","1479726","8178","1"],["1","16369","4537314","198410786","2571742175","12843262863","27971176092","27971176092","12843262863","2571742175","198410786","4537314","16369","1"],["1","32752","13824739","848090912","15041229521","102776998928","311387598411","447538817472","311387598411","102776998928","15041229521","848090912","13824739","32752","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","2","6","24","120","720","5040","40320","362880","3628800","39916800","479001600","6227020800","87178291200","1307674368000"],"nonn":true,"definitions":[{"position":0,"source":"named_family:eulerian","source_kind":"named_family","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","-1"],["0","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1","-1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":158161,"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":false,"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-ab3cdcfae289b48d"]},{"id":158162,"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-6c46c948823f0a8c","e-ab3cdcfae289b48d","e-c61ecf4ef1c45372"]},{"id":158163,"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-6c46c948823f0a8c","e-ab3cdcfae289b48d","e-c61ecf4ef1c45372"]},{"id":158164,"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-ab3cdcfae289b48d"]},{"id":158165,"position":4,"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-6c46c948823f0a8c","e-c61ecf4ef1c45372"]}],"region_exact":{},"incoming_edges":[{"id":"e-6c46c948823f0a8c","operation":{"operation":"parent_provers","route":"root_domain"},"operation_detail":"root_domain","conclusion_id":"n-ff270bcb802a85af","premise_count":0,"theorem":"polytool parent provers, route root_domain: every parent row P_n, n >= 1, is zero or real-rooted with coefficients of one sign","lean_route":null,"lean_declarations":[],"certificate_hash":"9d12cd77b9ee0d013bd3da840c9003151ce5085bf526c6e28203ff8bed34becf","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[]},{"id":"e-c61ecf4ef1c45372","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"reversal\",\"schedule\":{\"first_index\":1,\"first_degree\":0,\"step\":1}}"},"operation_detail":"reversal, shift 0","conclusion_id":"n-ff270bcb802a85af","premise_count":1,"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":"reversed","lean_declarations":["RealRooted.X_pow_mul_reverse_eq_zero_or_splits","RealRooted.reflect_eq_zero_or_splits"],"certificate_hash":"4c40a3ab00f1ce67f6f473f13ba06210b6aa3b371384c2cd91b3b145372c59da","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-ff270bcb802a85af","role":"sequence","name":"eulerian","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]},{"id":"e-ab3cdcfae289b48d","operation":{"operation":"theorem_check","theorem":"liu_wang"},"operation_detail":"liu_wang","conclusion_id":"n-ff270bcb802a85af","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":"3b401216ecbf6036382b55b4f9f71f3915b47662bd6a7e242606f2c4929da12f","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":false,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"previous_interlaces_next"}],"premises":[]}],"used_by":[{"id":"e-00ceaf3777abe095","operation":"gamma_polynomial","operation_detail":"scale 1/4, shift 1","conclusion_id":"n-3dbfcbca0c0c8d13","conclusion_name":"Triangle read by rows: T(n,k) (n >= 1, 0 <= k <= ceiling(n/2)-1) = number of permutations of [n] with k peaks."},{"id":"e-8b7786e9ad6af8fd","operation":"gamma_polynomial","operation_detail":"scale 1/4, shift 0","conclusion_id":"n-92480a234eee36bb","conclusion_name":"peak"},{"id":"e-c5d35f047ff7eb16","operation":"gamma_polynomial","operation_detail":"scale 1, shift 0","conclusion_id":"n-0942305fd4723770","conclusion_name":"gamma_polynomial of eulerian"},{"id":"e-e49ea542bfed6bbf","operation":"gamma_polynomial","operation_detail":"scale 1, shift 1","conclusion_id":"n-991b9bffcf8ddedb","conclusion_name":"Triangle T(n,k) (n >= 1, 0 <= k <= floor((n-1)/2)) read by rows, where T(n,k) = (k+1)T(n-1,k) + (2n-4k)T(n-1,k-1)."},{"id":"e-ea229ed6e1820773","operation":"gamma_polynomial","operation_detail":"scale 1/2, shift 1","conclusion_id":"n-b4814297dd192d81","conclusion_name":"Triangle read by rows: coefficients d(n,k) of Andr\u00e9 polynomials D(x,n) = Sum_{k>0} d(n,k)*x^k."},{"id":"e-8f0a401a806157c3","operation":"obreschkoff_combination","operation_detail":null,"conclusion_id":"n-1567559463beb931","conclusion_name":"obreschkoff_combination of eulerian"},{"id":"e-8f0a401a806157c3","operation":"obreschkoff_combination","operation_detail":null,"conclusion_id":"n-1567559463beb931","conclusion_name":"obreschkoff_combination of eulerian"},{"id":"e-0215d14e0b41ed17","operation":"preserver_image","operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-84d0c5dbe02497ed","conclusion_name":"derivative of eulerian"},{"id":"e-16a5ff84df4996fb","operation":"preserver_image","operation_detail":"affine, shift 0","conclusion_id":"n-76d25dafcdd8c9cc","conclusion_name":"shift_x_plus_one of eulerian"},{"id":"e-3a7fca57ac79b222","operation":"preserver_image","operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-ecf4607c39fc2a4b","conclusion_name":"theta_plus_one_squared of eulerian"},{"id":"e-667ea58aa192e8e7","operation":"preserver_image","operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-1df15b470497821b","conclusion_name":"one_plus_derivative of eulerian"},{"id":"e-6afbce5b42114ab6","operation":"preserver_image","operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-4f741935109d06a2","conclusion_name":"derivative of Triangle of Eulerian numbers T(n,k), 0 <= k <= n, read by rows."},{"id":"e-b7f464e4dcf953bf","operation":"preserver_image","operation_detail":"mobius, shift 0","conclusion_id":"n-d938c1937a93c4df","conclusion_name":"Triangle of numbers T(n,k) = k!*Stirling2(n,k) read by rows (n >= 1, 1 <= k <= n)."},{"id":"e-c61ecf4ef1c45372","operation":"preserver_image","operation_detail":"reversal, shift 0","conclusion_id":"n-ff270bcb802a85af","conclusion_name":"eulerian"},{"id":"e-fb10cc40ca2d4795","operation":"preserver_image","operation_detail":"euler_polynomial, shift 2","conclusion_id":"n-8293244ef059e34c","conclusion_name":"Triangle with T(n,k) = k*E(n,k) where E(n,k) are Eulerian numbers A008292."}],"used_by_total":15,"links":[{"id":"l-1c7f1b04041ddcd2","kind":"fingerprint_related","node_a_id":"n-ff270bcb802a85af","node_b_id":"n-41809911e05bb4ba","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-46d871f0eef05a09","kind":"fingerprint_related","node_a_id":"n-ff270bcb802a85af","node_b_id":"n-efd1fdf07e5468d8","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-6076ee430a1bd46b","kind":"fingerprint_related","node_a_id":"n-ff270bcb802a85af","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-651b6013eac52c1c","kind":"fingerprint_related","node_a_id":"n-ff270bcb802a85af","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-e0375a6775c509d9","kind":"fingerprint_related","node_a_id":"n-ff270bcb802a85af","node_b_id":"n-dfedc6a36c97eba4","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-3ffff24fcb4f40a0","other":{"id":"n-3ffff24fcb4f40a0","name":"Triangle read by rows: T(n,k) is the number of permutations of 1..n arranged in a circle with exactly k descents.","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"],["0","2"]],"card_rows":[["1"],["1"],["0","2"],["0","3","3"]],"row_sums_preview":["1","1","2","6","24","120","720","5040"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":["A334218"],"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}},{"id":"l-2a844624b2930c78","kind":"oeis_match","node_a_id":"n-ff270bcb802a85af","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":2,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":2,"target":"row_sums","terms_compared":14},"in_catalog":false,"oeis_id":"A124355","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-67d380cab3d7589a","kind":"oeis_match","node_a_id":"n-ff270bcb802a85af","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":3,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":2,"target":"row_sums","terms_compared":13},"in_catalog":false,"oeis_id":"A338112","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-686878f38fac3b35","kind":"oeis_match","node_a_id":"n-ff270bcb802a85af","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":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A000142","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-69a51bea201655dd","kind":"oeis_match","node_a_id":"n-ff270bcb802a85af","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":3,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A358185","oeis_nonn":false},"other_id":null,"other":null},{"id":"l-776c3c2e38eae769","kind":"oeis_match","node_a_id":"n-ff270bcb802a85af","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":1,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A370360","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-a8bd890c2dd532d4","kind":"oeis_match","node_a_id":"n-ff270bcb802a85af","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":3,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":1,"target":"row_sums","terms_compared":13},"in_catalog":false,"oeis_id":"A354074","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-b7d340f2cf53ab62","kind":"oeis_match","node_a_id":"n-ff270bcb802a85af","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":2,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A104150","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-becd4f7342f52f66","kind":"oeis_match","node_a_id":"n-ff270bcb802a85af","node_b_id":null,"fingerprint":null,"oeis_id":"A008292","in_catalog":1,"target":"reversed_triangle","row_alignment":"tabl","rows_matched":10,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":10,"target":"reversed_triangle","terms_compared":55},"in_catalog":true,"oeis_id":"A008292","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-d4f49bbd2396ce87","kind":"oeis_match","node_a_id":"n-ff270bcb802a85af","node_b_id":null,"fingerprint":null,"oeis_id":"A008292","in_catalog":1,"target":"triangle","row_alignment":"tabl","rows_matched":10,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":10,"target":"triangle","terms_compared":55},"in_catalog":true,"oeis_id":"A008292","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-f6999dac4ac1ee46","kind":"oeis_match","node_a_id":"n-ff270bcb802a85af","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":3,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A159333","oeis_nonn":false},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A008292","relation":"identification"},{"oeis_id":"A008292","relation":"match_reversed_triangle"},{"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"},{"oeis_id":"A008292","relation":"match_triangle"}],"bfiles":[{"node_id":"n-ff270bcb802a85af","oeis_id":"A008292","target":"triangle","oeis_offset":1,"first_row":1,"rows_used":44,"terms_matched":55,"oeis_terms_agreeing":55,"oeis_terms_available":55,"term_count":990,"raw_size":29042},{"node_id":"n-ff270bcb802a85af","oeis_id":"A008292","target":"reversed_triangle","oeis_offset":1,"first_row":1,"rows_used":44,"terms_matched":55,"oeis_terms_agreeing":55,"oeis_terms_available":55,"term_count":990,"raw_size":29042}],"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,    1;\n   3: 1,    4,     1;\n   4: 1,   11,    11,      1;\n   5: 1,   26,    66,     26,       1;\n   6: 1,   57,   302,    302,      57,       1;\n   7: 1,  120,  1191,   2416,    1191,     120,      1;\n   8: 1,  247,  4293,  15619,   15619,    4293,    247,     1;\n   9: 1,  502, 14608,  88234,  156190,   88234,  14608,   502,    1;\n  10: 1, 1013, 47840, 455192, 1310354, 1310354, 455192, 47840, 1013, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+(n-1)*x)*P(n-1,x) + (x-x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(1 + (n - 1)*x)*P[n - 1, x] + (x - x^2)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (1 + (n - 1)*x)*v[i - 1] + (x - x^2)*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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}},{"oeis_id":"A008292","target":"triangle","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 1;\n   2: 1,    1;\n   3: 1,    4,     1;\n   4: 1,   11,    11,      1;\n   5: 1,   26,    66,     26,       1;\n   6: 1,   57,   302,    302,      57,       1;\n   7: 1,  120,  1191,   2416,    1191,     120,      1;\n   8: 1,  247,  4293,  15619,   15619,    4293,    247,     1;\n   9: 1,  502, 14608,  88234,  156190,   88234,  14608,   502,    1;\n  10: 1, 1013, 47840, 455192, 1310354, 1310354, 455192, 47840, 1013, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+(n-1)*x)*P(n-1,x) + (x-x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(1 + (n - 1)*x)*P[n - 1, x] + (x - x^2)*D[P[n - 1, x], x]];\nTable[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (1 + (n - 1)*x)*v[i - 1] + (x - x^2)*deriv(v[i - 1], x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); 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":true}},{"oeis_id":"A008292","target":"reversed_triangle","first_row":1,"oeis_offset":1,"reversed":true,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 1;\n   2: 1,    1;\n   3: 1,    4,     1;\n   4: 1,   11,    11,      1;\n   5: 1,   26,    66,     26,       1;\n   6: 1,   57,   302,    302,      57,       1;\n   7: 1,  120,  1191,   2416,    1191,     120,      1;\n   8: 1,  247,  4293,  15619,   15619,    4293,    247,     1;\n   9: 1,  502, 14608,  88234,  156190,   88234,  14608,   502,    1;\n  10: 1, 1013, 47840, 455192, 1310354, 1310354, 455192, 47840, 1013, 1;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+(n-1)*x)*P(n-1,x) + (x-x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(1 + (n - 1)*x)*P[n - 1, x] + (x - x^2)*D[P[n - 1, x], x]];\nTable[Reverse[PadRight[CoefficientList[P[n, x], x], n]], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (1 + (n - 1)*x)*v[i - 1] + (x - x^2)*deriv(v[i - 1], x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c = Vecrev(c); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":true,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-ff270bcb802a85af","steps":[],"edges":[],"text":"A008292"},"headline":"A008292: Eulerian","chain":null}