{"id":"n-984320118b653f6b","revision":2,"first_index":0,"fingerprint":"f6-2305441cae17e38c","name":"Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k)^2/(n-k+1).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["2","2"],["3","9","3"],["4","24","24","4"],["5","50","100","50","5"],["6","90","300","300","90","6"],["7","147","735","1225","735","147","7"],["8","224","1568","3920","3920","1568","224","8"],["9","324","3024","10584","15876","10584","3024","324","9"],["10","450","5400","25200","52920","52920","25200","5400","450","10"],["11","605","9075","54450","152460","213444","152460","54450","9075","605","11"],["12","792","14520","108900","392040","731808","731808","392040","108900","14520","792","12"],["13","1014","22308","204490","920205","2208492","2944656","2208492","920205","204490","22308","1014","13"],["14","1274","33124","364364","2004002","6012006","10306296","10306296","6012006","2004002","364364","33124","1274","14"],["15","1575","47775","621075","4099095","15030015","32207175","41409225","32207175","15030015","4099095","621075","47775","1575","15"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","15","56","210","792","3003","11440","43758","167960","646646","2496144","9657700","37442160","145422675"],"nonn":true,"definitions":[{"position":0,"source":"oeis_catalog:A132812","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["2","2"],["3","9","3"]],"recurrence":{"denominator":{"coeffs":[["2"],["1"]]},"terms":[{"coeff":{"coeffs":[["3","3"],["3","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","4","0"],["-3","-2","-3"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-1","1","1","-1"],["1","-1","-1","1"]]},"deriv_order":0,"offset":3,"sign":"none"}]},"schema":"polytool.recurrence.v1"}},{"position":1,"source":"derived:root_bound_shift:derived:basis_map_coefficients_of:oeis_catalog:A132812","source_kind":"derived","equality_edge_id":"e-17841f9b7ad2ac5d","recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0"],["2"],["1"]]},"terms":[{"coeff":{"coeffs":[["2","0"],["3","3"],["1","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2","-2"],["0","2","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}},{"position":2,"source":"derived:reversal","source_kind":"derived","equality_edge_id":"e-dee9185888bdea38","recurrence":{"first_index":0,"initial_polynomials":[["1"],["2","2"]],"recurrence":{"denominator":{"coeffs":[["0"],["2"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0","0"],["-1","2","-1"],["-1","2","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1","1"],["3","3"],["2","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":160335,"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-041eea8fc0a7af05","e-a32917d44ac940a7","e-fcd77baa2d5e1258"]},{"id":160336,"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-041eea8fc0a7af05","e-a32917d44ac940a7","e-f8a41d41829f0ed1","e-fcd77baa2d5e1258"]},{"id":160337,"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-041eea8fc0a7af05","e-a32917d44ac940a7","e-f8a41d41829f0ed1","e-fcd77baa2d5e1258"]},{"id":160338,"position":3,"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-f8a41d41829f0ed1"]},{"id":160339,"position":4,"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-f8a41d41829f0ed1"]},{"id":160340,"position":5,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-a32917d44ac940a7"]},{"id":160341,"position":6,"claim":"real_rooted","from_index":0,"region_key":"(-inf, +inf)","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)","region_exact":null,"edges":["e-138e1ca38baf9fe5"]}],"region_exact":{},"incoming_edges":[{"id":"e-a32917d44ac940a7","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-984320118b653f6b","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":"2345cba956c4f310aed18ef1808c39555fb624d80a8d64479bb23397f00f2308","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-3bc20048ee6f0787","role":"sequence","name":"basis-map coefficients of Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k)^2/(n-k+1).","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"1"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 1]"}]},{"id":"e-17841f9b7ad2ac5d","operation":{"operation":"definition_equality"},"operation_detail":null,"conclusion_id":"n-984320118b653f6b","premise_count":0,"theorem":"all-index identity Q_n = P_n certified by exact Ore-algebra division and identifying rows (polytool preserver-image certificate of the identity map)","lean_route":null,"lean_declarations":[],"certificate_hash":"51f556abf2f48d6e039b9636a4ccfcdc4cee4fe51b38542153d4831b8e18ec86","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[],"premises":[]},{"id":"e-dee9185888bdea38","operation":{"operation":"definition_equality"},"operation_detail":null,"conclusion_id":"n-984320118b653f6b","premise_count":0,"theorem":"all-index identity Q_n = P_n certified by exact Ore-algebra division and identifying rows (polytool preserver-image certificate of the identity map)","lean_route":null,"lean_declarations":[],"certificate_hash":"cdc1a386f038720155cd567261ae022cb23909bc22dca87dbc872940f32081a8","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[],"premises":[]},{"id":"e-041eea8fc0a7af05","operation":{"index_shift":1,"operation":"preserver_image","operator":"{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\",\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"euler_polynomial, shift 1","conclusion_id":"n-984320118b653f6b","premise_count":1,"theorem":"Laguerre's theorem on multiplier sequences: if f is a real polynomial whose roots are all real and <= 0, then (f(k))_(k >= 0) is a multiplier sequence, so x^k -> f(k) x^k maps every real-rooted polynomial to a real-rooted polynomial or zero (Laguerre; see T. Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"certificate_hash":"c826c88ecb18a6a1e70a3a9963a253ce2342e70e850a698aabcd639059e7dcca","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-2a05d8340c329b21","role":"sequence","name":"narayana","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-138e1ca38baf9fe5","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-984320118b653f6b","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":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"f8e9f81591158460e930064703c28fab3dd95378ee7298c900950a22a47a296d","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-3bc20048ee6f0787","role":"sequence","name":"basis-map coefficients of Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k)^2/(n-k+1).","uses":{"claim":"real_rooted","from_index":0,"region":{}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)"}]},{"id":"e-fcd77baa2d5e1258","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"reversal\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":1}}"},"operation_detail":"reversal, shift 0","conclusion_id":"n-984320118b653f6b","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":"39dbb52c4bd2c2a01d6b21c6ae80c5ced002ea8e30b3792f418394bd354b60bc","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-984320118b653f6b","role":"sequence","name":"Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k)^2/(n-k+1).","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-f8a41d41829f0ed1","operation":{"operation":"theorem_check","theorem":"liu_wang"},"operation_detail":"liu_wang","conclusion_id":"n-984320118b653f6b","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":"5caeaf14f63d0efeefa2a61f7fa5ab2383cfca1c43e96b2c498be810a4ae236e","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-cc6de87a980d771e","operation":"gamma_polynomial","operation_detail":"scale 1, shift 0","conclusion_id":"n-a8a898e8495cff05","conclusion_name":"gamma_polynomial of Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k)^2/(n-k+1)."},{"id":"e-60496e4e2f616741","operation":"preserver_image","operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-8552357c29e7b09e","conclusion_name":"derivative of Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k)^2/(n-k+1)."},{"id":"e-676bb32d6a1b9c64","operation":"preserver_image","operation_detail":"affine, shift 0","conclusion_id":"n-3bc20048ee6f0787","conclusion_name":"basis-map coefficients of Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k)^2/(n-k+1)."},{"id":"e-6a8f8db3c1596aaf","operation":"preserver_image","operation_detail":"mobius, shift 0","conclusion_id":"n-b2e66fb8e43e7448","conclusion_name":"mobius_x_over_x_plus_one of Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k)^2/(n-k+1)."},{"id":"e-f4ea45816c9a82eb","operation":"preserver_image","operation_detail":"affine, shift 0","conclusion_id":"n-00a555c278779f26","conclusion_name":"shift_x_plus_one of Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k)^2/(n-k+1)."},{"id":"e-fcd77baa2d5e1258","operation":"preserver_image","operation_detail":"reversal, shift 0","conclusion_id":"n-984320118b653f6b","conclusion_name":"Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k)^2/(n-k+1)."}],"used_by_total":6,"links":[{"id":"l-11b607427846863f","kind":"fingerprint_related","node_a_id":"n-c2eaf5212a4dec13","node_b_id":"n-984320118b653f6b","fingerprint":"f6-2305441cae17e38c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2305441cae17e38c"},"other_id":"n-c2eaf5212a4dec13","other":{"id":"n-c2eaf5212a4dec13","name":"Triangle of Narayana numbers T(n,k) = C(n-1,k-1)*C(n,k-1)/k with 1 <= k <= n, read by rows. Also called the Catalan triangle.","first_index":0,"fingerprint":"f6-2305441cae17e38c","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","1"],["1","3","1"]],"card_rows":[["1"],["1","1"],["1","3","1"],["1","6","6","1"]],"row_sums_preview":["1","2","5","14","42","132","429","1430"],"regions":["(-inf, +inf)","(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":["A001263"],"derivation":{"root":"n-c2eaf5212a4dec13","steps":[],"edges":[],"text":"A001263"},"headline":"A001263: Triangle of Narayana numbers T(n,k) = C(n-1,k-1)*C(n,k-1)/k with 1 <= k <= n, read by rows. Also called the Catalan triangle.","chain":null}},{"id":"l-c2f14e081d2a3f18","kind":"fingerprint_related","node_a_id":"n-984320118b653f6b","node_b_id":"n-4ea7eb31fd1a1554","fingerprint":"f6-2305441cae17e38c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2305441cae17e38c"},"other_id":"n-4ea7eb31fd1a1554","other":{"id":"n-4ea7eb31fd1a1554","name":"derivative of Triangle read by rows. T(n, k) = binomial(n, k)*binomial(n - 1, n - k - 1).","first_index":0,"fingerprint":"f6-2305441cae17e38c","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],[],["2"]],"card_rows":[["2"],["6","6"],["12","36","12"],["20","120","120","20"]],"row_sums_preview":["0","0","2","12","60","280","1260","5544"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-b1b19325e0094118","steps":["D"],"edges":["e-2f63263d248c26b4"],"text":"A367270 \u2192 D"},"headline":"A367270 \u2192 D","chain":null}},{"id":"l-f2ffb8a669ba508e","kind":"fingerprint_related","node_a_id":"n-2a05d8340c329b21","node_b_id":"n-984320118b653f6b","fingerprint":"f6-2305441cae17e38c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2305441cae17e38c"},"other_id":"n-2a05d8340c329b21","other":{"id":"n-2a05d8340c329b21","name":"narayana","first_index":1,"fingerprint":"f6-2305441cae17e38c","polytool_family":"narayana","degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","1"],["1","3","1"]],"card_rows":[["1"],["1","1"],["1","3","1"],["1","6","6","1"]],"row_sums_preview":["1","2","5","14","42","132","429","1430"],"regions":["(-inf, +inf)","(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2a05d8340c329b21","steps":[],"edges":[],"text":"narayana"},"headline":"narayana","chain":null}},{"id":"l-3eeffc05ad00a05a","kind":"oeis_match","node_a_id":"n-984320118b653f6b","node_b_id":null,"fingerprint":null,"oeis_id":"A132812","in_catalog":1,"target":"reversed_triangle","row_alignment":"tabl","rows_matched":10,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"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":"A132812","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-6e31a47666360d28","kind":"oeis_match","node_a_id":"n-984320118b653f6b","node_b_id":null,"fingerprint":null,"oeis_id":"A001791","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":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A001791","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-c0b6e837176db6b9","kind":"oeis_match","node_a_id":"n-984320118b653f6b","node_b_id":null,"fingerprint":null,"oeis_id":"A132812","in_catalog":1,"target":"triangle","row_alignment":"tabl","rows_matched":10,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"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":"A132812","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A132812","relation":"identification"},{"oeis_id":"A132812","relation":"match_reversed_triangle"},{"oeis_id":"A001791","relation":"match_row_sums"},{"oeis_id":"A132812","relation":"match_triangle"}],"bfiles":[{"node_id":"n-984320118b653f6b","oeis_id":"A132812","target":"triangle","oeis_offset":1,"first_row":0,"rows_used":44,"terms_matched":55,"oeis_terms_agreeing":55,"oeis_terms_available":55,"term_count":990,"raw_size":16697},{"node_id":"n-984320118b653f6b","oeis_id":"A132812","target":"reversed_triangle","oeis_offset":1,"first_row":0,"rows_used":44,"terms_matched":55,"oeis_terms_agreeing":55,"oeis_terms_available":55,"term_count":990,"raw_size":16697}],"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:  2,   2;\n  2:  3,   9,    3;\n  3:  4,  24,   24,     4;\n  4:  5,  50,  100,    50,     5;\n  5:  6,  90,  300,   300,    90,     6;\n  6:  7, 147,  735,  1225,   735,   147,     7;\n  7:  8, 224, 1568,  3920,  3920,  1568,   224,    8;\n  8:  9, 324, 3024, 10584, 15876, 10584,  3024,  324,   9;\n  9: 10, 450, 5400, 25200, 52920, 52920, 25200, 5400, 450, 10;\n","recurrence_sentence":"The row polynomials satisfy (n+2)*P(n,x) = (3*n+3+(3*n+3)*x)*P(n-1,x) - (3*n+(2*n-4)*x+3*n*x^2)*P(n-2,x) + (n-1-(n-1)*x-(n-1)*x^2+(n-1)*x^3)*P(n-3,x) for n >= 3, with P(0,x) = 1, P(1,x) = 2 + 2*x, P(2,x) = 3 + 9*x + 3*x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 2*x;\nP[2, x_] := 3 + 9*x + 3*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((3*n + 3 + (3*n + 3)*x)*P[n - 1, x] - (3*n + (2*n - 4)*x + 3*n*x^2)*P[n - 2, x] + (n - 1 - (n - 1)*x - (n - 1)*x^2 + (n - 1)*x^3)*P[n - 3, x])/(n + 2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 2*x; v[3] = 3 + 9*x + 3*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((3*n + 3 + (3*n + 3)*x)*v[i - 1] - (3*n + (2*n - 4)*x + 3*n*x^2)*v[i - 2] + (n - 1 - (n - 1)*x - (n - 1)*x^2 + (n - 1)*x^3)*v[i - 3])/(n + 2)); 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":"A132812","target":"triangle","first_row":0,"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:  2,   2;\n   3:  3,   9,    3;\n   4:  4,  24,   24,     4;\n   5:  5,  50,  100,    50,     5;\n   6:  6,  90,  300,   300,    90,     6;\n   7:  7, 147,  735,  1225,   735,   147,     7;\n   8:  8, 224, 1568,  3920,  3920,  1568,   224,    8;\n   9:  9, 324, 3024, 10584, 15876, 10584,  3024,  324,   9;\n  10: 10, 450, 5400, 25200, 52920, 52920, 25200, 5400, 450, 10;\n","recurrence_sentence":"The row polynomials satisfy (n+1)*P(n,x) = (3*n+3*n*x)*P(n-1,x) - (3*n-3+(2*n-6)*x+(3*n-3)*x^2)*P(n-2,x) + (n-2-(n-2)*x-(n-2)*x^2+(n-2)*x^3)*P(n-3,x) for n >= 4, with P(1,x) = 1, P(2,x) = 2 + 2*x, P(3,x) = 3 + 9*x + 3*x^2.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 2 + 2*x;\nP[3, x_] := 3 + 9*x + 3*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((3*n + 3*n*x)*P[n - 1, x] - (3*n - 3 + (2*n - 6)*x + (3*n - 3)*x^2)*P[n - 2, x] + (n - 2 - (n - 2)*x - (n - 2)*x^2 + (n - 2)*x^3)*P[n - 3, x])/(n + 1)]];\nTable[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 2*x; v[3] = 3 + 9*x + 3*x^2; for(n = 4, N, my(i = n); v[i] = ((3*n + 3*n*x)*v[i - 1] - (3*n - 3 + (2*n - 6)*x + (3*n - 3)*x^2)*v[i - 2] + (n - 2 - (n - 2)*x - (n - 2)*x^2 + (n - 2)*x^3)*v[i - 3])/(n + 1)); 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":0,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":true}},{"oeis_id":"A132812","target":"reversed_triangle","first_row":0,"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:  2,   2;\n   3:  3,   9,    3;\n   4:  4,  24,   24,     4;\n   5:  5,  50,  100,    50,     5;\n   6:  6,  90,  300,   300,    90,     6;\n   7:  7, 147,  735,  1225,   735,   147,     7;\n   8:  8, 224, 1568,  3920,  3920,  1568,   224,    8;\n   9:  9, 324, 3024, 10584, 15876, 10584,  3024,  324,   9;\n  10: 10, 450, 5400, 25200, 52920, 52920, 25200, 5400, 450, 10;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where (n+1)*P(n,x) = (3*n+3*n*x)*P(n-1,x) - (3*n-3+(2*n-6)*x+(3*n-3)*x^2)*P(n-2,x) + (n-2-(n-2)*x-(n-2)*x^2+(n-2)*x^3)*P(n-3,x) for n >= 4, with P(1,x) = 1, P(2,x) = 2 + 2*x, P(3,x) = 3 + 9*x + 3*x^2.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 2 + 2*x;\nP[3, x_] := 3 + 9*x + 3*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((3*n + 3*n*x)*P[n - 1, x] - (3*n - 3 + (2*n - 6)*x + (3*n - 3)*x^2)*P[n - 2, x] + (n - 2 - (n - 2)*x - (n - 2)*x^2 + (n - 2)*x^3)*P[n - 3, x])/(n + 1)]];\nTable[Reverse[PadRight[CoefficientList[P[n, x], x], n]], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 2*x; v[3] = 3 + 9*x + 3*x^2; for(n = 4, N, my(i = n); v[i] = ((3*n + 3*n*x)*v[i - 1] - (3*n - 3 + (2*n - 6)*x + (3*n - 3)*x^2)*v[i - 2] + (n - 2 - (n - 2)*x - (n - 2)*x^2 + (n - 2)*x^3)*v[i - 3])/(n + 1)); 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":0,"offset":1,"reversed":true,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-984320118b653f6b","steps":[],"edges":[],"text":"A132812"},"headline":"A132812: Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k)^2/(n-k+1).","chain":null}