{"id":"n-78ba05b4a3dd925d","revision":2,"first_index":1,"fingerprint":"f6-fd15deff59829444","name":"Triangle read by rows: T(n,k) is the number of lattice paths from (0,0) to (n,n) using steps E=(1,0), N=(0,1) and D=(1,1) (i.e., bilateral Schroeder paths), having k D=(1,1) steps.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["2","1"],["6","6","1"],["20","30","12","1"],["70","140","90","20","1"],["252","630","560","210","30","1"],["924","2772","3150","1680","420","42","1"],["3432","12012","16632","11550","4200","756","56","1"],["12870","51480","84084","72072","34650","9240","1260","72","1"],["48620","218790","411840","420420","252252","90090","18480","1980","90","1"],["184756","923780","1969110","2333760","1681680","756756","210210","34320","2970","110","1"],["705432","3879876","9237800","12471030","10501920","5717712","2018016","450450","60060","4290","132","1"],["2704156","16224936","42678636","64664600","62355150","39907296","17153136","4900896","900900","100100","6006","156","1"],["10400600","67603900","194699232","327202876","355655300","261891630","133024320","46558512","11027016","1701700","160160","8190","182","1"],["40116600","280816200","878850700","1622493600","1963217256","1636014380","960269310","399072960","116396280","23279256","3063060","247520","10920","210","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","13","63","321","1683","8989","48639","265729","1462563","8097453","45046719","251595969","1409933619","7923848253"],"nonn":true,"definitions":[{"position":0,"source":"oeis_catalog:A104684","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["2","1"]],"recurrence":{"denominator":{"coeffs":[["1"],["-1"]]},"terms":[{"coeff":{"coeffs":[["6","3"],["-4","-2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-2"],["0","0","1"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"}},{"position":1,"source":"derived:reversal","source_kind":"derived","equality_edge_id":"e-0431973470657681","recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["-6","-5"],["4","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-2","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":159024,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, +inf)","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)","region_exact":null,"edges":["e-0e64edb308434359"]},{"id":159025,"position":1,"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-74c20febd3423e2e"]},{"id":159026,"position":2,"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-5047c71cb920acb6","e-74c20febd3423e2e"]},{"id":159027,"position":3,"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-5047c71cb920acb6","e-74c20febd3423e2e"]},{"id":159028,"position":4,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"kind":"strict","orientation":"previous_interlaces_next"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (strict, previous interlaces next) for every m >= 2","region_exact":null,"edges":["e-74c20febd3423e2e"]},{"id":159029,"position":5,"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-5047c71cb920acb6"]}],"region_exact":{},"incoming_edges":[{"id":"e-0431973470657681","operation":{"operation":"definition_equality"},"operation_detail":null,"conclusion_id":"n-78ba05b4a3dd925d","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":"4f5777a256d8c2b0c85e0d15cf0a4eacdf350db9365aa4c4fa0a5ce4ff5345b4","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[],"premises":[]},{"id":"e-5047c71cb920acb6","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-78ba05b4a3dd925d","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":"c54eef74f9d1bd1377c66f97246a16999011765bd730f6106b654c539a1bf5f6","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-4689fb1386d440ad","role":"sequence","name":"T(n,k) = binomial(n,k)*binomial(n+k,k), 0 <= k <= n, triangle read by rows.","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-0e64edb308434359","operation":{"operation":"theorem_check","theorem":"three_term_recurrence"},"operation_detail":"three_term_recurrence","conclusion_id":"n-78ba05b4a3dd925d","premise_count":0,"theorem":"three-term recurrence Q_n = (a_n y + b_n) Q_(n-1) - c_n Q_(n-2) with a_n != 0 and beta_n = c_n / (a_n a_(n-1)) > 0: simple real roots and strict interlacing of consecutive rows (zero separation for orthogonal polynomials: G. Szeg\u0151, Orthogonal Polynomials, \u00a73.3; Favard, C. R. Acad. Sci. Paris 200 (1935)); root regions by a positive-ratio (chain-sequence) induction","lean_route":null,"lean_declarations":[],"certificate_hash":"e9fc74af52e5717c2ec1077937293ac29978755aebf62d6ca0334776e3f7bd29","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{}}],"premises":[]},{"id":"e-74c20febd3423e2e","operation":{"operation":"theorem_check","theorem":"three_term_recurrence"},"operation_detail":"three_term_recurrence","conclusion_id":"n-78ba05b4a3dd925d","premise_count":0,"theorem":"three-term recurrence Q_n = (a_n y + b_n) Q_(n-1) - c_n Q_(n-2) with a_n != 0 and beta_n = c_n / (a_n a_(n-1)) > 0: simple real roots and strict interlacing of consecutive rows (zero separation for orthogonal polynomials: G. Szeg\u0151, Orthogonal Polynomials, \u00a73.3; Favard, C. R. Acad. Sci. Paris 200 (1935)); root regions by a positive-ratio (chain-sequence) induction","lean_route":null,"lean_declarations":[],"certificate_hash":"d390c6eb52e0c99623771bc02bb92b3509c6126904026d1601c9598b1dc873dd","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":"strict","orientation":"previous_interlaces_next"}],"premises":[]}],"used_by":[{"id":"e-478032ea05246a28","operation":"basis_map","operation_detail":"A013610","conclusion_id":"n-914a474e9f3b0e2a","conclusion_name":"basis_map_A013610 of Triangle read by rows: T(n,k) is the number of lattice paths from (0,0) to (n,n) using steps E=(1,0), N=(0,1) and D=(1,1) (i.e., bilateral Schroeder paths), having k D=(1,1) steps."},{"id":"e-7b1b09c6fd205866","operation":"basis_map","operation_detail":"A007318","conclusion_id":"n-118f91023ad73cd8","conclusion_name":"basis_map_A007318 of Triangle read by rows: T(n,k) is the number of lattice paths from (0,0) to (n,n) using steps E=(1,0), N=(0,1) and D=(1,1) (i.e., bilateral Schroeder paths), having k D=(1,1) steps."},{"id":"e-7f59bcf02d573532","operation":"basis_map","operation_detail":"A013609","conclusion_id":"n-60c7f085d1907563","conclusion_name":"basis_map_A013609 of Triangle read by rows: T(n,k) is the number of lattice paths from (0,0) to (n,n) using steps E=(1,0), N=(0,1) and D=(1,1) (i.e., bilateral Schroeder paths), having k D=(1,1) steps."},{"id":"e-907fec0ae54b396c","operation":"basis_map","operation_detail":"A008277","conclusion_id":"n-989609c84742d817","conclusion_name":"basis_map_A008277 of Triangle read by rows: T(n,k) is the number of lattice paths from (0,0) to (n,n) using steps E=(1,0), N=(0,1) and D=(1,1) (i.e., bilateral Schroeder paths), having k D=(1,1) steps."},{"id":"e-d797656a7302c65e","operation":"basis_map","operation_detail":"A013611","conclusion_id":"n-6373ae06480cbab5","conclusion_name":"basis_map_A013611 of Triangle read by rows: T(n,k) is the number of lattice paths from (0,0) to (n,n) using steps E=(1,0), N=(0,1) and D=(1,1) (i.e., bilateral Schroeder paths), having k D=(1,1) steps."},{"id":"e-00d47151fcd2720e","operation":"preserver_image","operation_detail":"reversal, shift 0","conclusion_id":"n-4689fb1386d440ad","conclusion_name":"T(n,k) = binomial(n,k)*binomial(n+k,k), 0 <= k <= n, triangle read by rows."},{"id":"e-22fed0d4659f5966","operation":"preserver_image","operation_detail":"mobius, shift 0","conclusion_id":"n-3f2e898dea3fb1c5","conclusion_name":"mobius_x_over_x_plus_one of Triangle read by rows: T(n,k) is the number of lattice paths from (0,0) to (n,n) using steps E=(1,0), N=(0,1) and D=(1,1) (i.e., bilateral Schroeder paths), having k D=(1,1) steps."},{"id":"e-33e8cdf9e54d9cd4","operation":"preserver_image","operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-8ed5cfff0540233a","conclusion_name":"theta_plus_one of Triangle read by rows: T(n,k) is the number of lattice paths from (0,0) to (n,n) using steps E=(1,0), N=(0,1) and D=(1,1) (i.e., bilateral Schroeder paths), having k D=(1,1) steps."},{"id":"e-62274b400b6ee467","operation":"preserver_image","operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-814eff484ba2be47","conclusion_name":"one_plus_derivative of Triangle read by rows: T(n,k) is the number of lattice paths from (0,0) to (n,n) using steps E=(1,0), N=(0,1) and D=(1,1) (i.e., bilateral Schroeder paths), having k D=(1,1) steps."},{"id":"e-8470b587e8da7dfc","operation":"preserver_image","operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-f1857dc0e9cd7932","conclusion_name":"theta_plus_one_squared of Triangle read by rows: T(n,k) is the number of lattice paths from (0,0) to (n,n) using steps E=(1,0), N=(0,1) and D=(1,1) (i.e., bilateral Schroeder paths), having k D=(1,1) steps."},{"id":"e-da18ebef7a232130","operation":"preserver_image","operation_detail":"affine, shift 0","conclusion_id":"n-118f91023ad73cd8","conclusion_name":"basis_map_A007318 of Triangle read by rows: T(n,k) is the number of lattice paths from (0,0) to (n,n) using steps E=(1,0), N=(0,1) and D=(1,1) (i.e., bilateral Schroeder paths), having k D=(1,1) steps."},{"id":"e-efbf4c1b988cfbc1","operation":"preserver_image","operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-1ff7b1d8c09df87e","conclusion_name":"derivative of Triangle read by rows: T(n,k) is the number of lattice paths from (0,0) to (n,n) using steps E=(1,0), N=(0,1) and D=(1,1) (i.e., bilateral Schroeder paths), having k D=(1,1) steps."}],"used_by_total":12,"links":[{"id":"l-18f6804afb046ffd","kind":"fingerprint_related","node_a_id":"n-78ba05b4a3dd925d","node_b_id":"n-57c367574a30fd7b","fingerprint":"f6-fd15deff59829444","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-fd15deff59829444"},"other_id":"n-57c367574a30fd7b","other":{"id":"n-57c367574a30fd7b","name":"basis_map_A007318 of Square the entries of Pascal's triangle.","first_index":0,"fingerprint":"f6-fd15deff59829444","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","1"],["6","6","1"]],"card_rows":[["1"],["2","1"],["6","6","1"],["20","30","12","1"]],"row_sums_preview":["1","3","13","63","321","1683","8989","48639"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-b13531ad79e7c629","steps":["x \u21a6 x+1"],"edges":["e-40b4efc61cad5541"],"text":"A008459 \u2192 x \u21a6 x+1"},"headline":"A008459 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-34572875e9b4626f","kind":"fingerprint_related","node_a_id":"n-4689fb1386d440ad","node_b_id":"n-78ba05b4a3dd925d","fingerprint":"f6-fd15deff59829444","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-fd15deff59829444"},"other_id":"n-4689fb1386d440ad","other":{"id":"n-4689fb1386d440ad","name":"T(n,k) = binomial(n,k)*binomial(n+k,k), 0 <= k <= n, triangle read by rows.","first_index":1,"fingerprint":"f6-fd15deff59829444","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","2"],["1","6","6"]],"card_rows":[["1"],["1","2"],["1","6","6"],["1","12","30","20"]],"row_sums_preview":["1","3","13","63","321","1683","8989","48639"],"regions":["(-inf, +inf)","(-inf, 0)","(-1, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":["A063007"],"derivation":{"root":"n-4689fb1386d440ad","steps":[],"edges":[],"text":"A063007"},"headline":"A063007: T(n,k) = binomial(n,k)*binomial(n+k,k), 0 <= k <= n, triangle read by rows.","chain":null}},{"id":"l-362a29052958ebe3","kind":"fingerprint_related","node_a_id":"n-78ba05b4a3dd925d","node_b_id":"n-55a66d3aeb8cbce1","fingerprint":"f6-fd15deff59829444","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-fd15deff59829444"},"other_id":"n-55a66d3aeb8cbce1","other":{"id":"n-55a66d3aeb8cbce1","name":"mobius_x_over_x_plus_one of Square the entries of Pascal's triangle.","first_index":0,"fingerprint":"f6-fd15deff59829444","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","2"],["1","6","6"]],"card_rows":[["1"],["1","2"],["1","6","6"],["1","12","30","20"]],"row_sums_preview":["1","3","13","63","321","1683","8989","48639"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-b13531ad79e7c629","steps":["x \u21a6 x/(x+1)"],"edges":["e-2ce1a392fcdf3dd9"],"text":"A008459 \u2192 x \u21a6 x/(x+1)"},"headline":"A008459 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-d4e46f3d8f80f72f","kind":"fingerprint_related","node_a_id":"n-78ba05b4a3dd925d","node_b_id":"n-ed80a7592b428c53","fingerprint":"f6-fd15deff59829444","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-fd15deff59829444"},"other_id":"n-ed80a7592b428c53","other":{"id":"n-ed80a7592b428c53","name":"basis_map_A007318 of Triangle T(n, k) of numbers of square lattice walks that start and end at origin after 2*n steps and contain exactly k steps to the east, possibly touching origin at intermediate stages.","first_index":0,"fingerprint":"f6-fd15deff59829444","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["4","2"],["36","36","6"]],"card_rows":[["1"],["4","2"],["36","36","6"],["400","600","240","20"]],"row_sums_preview":["1","6","78","1260","22470","424116","8305836","166929048"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-fdc03fad975441d0","steps":["basis A007318"],"edges":["e-2cd9cd27e716475b"],"text":"A069466 \u2192 basis A007318"},"headline":"A069466 \u2192 basis A007318","chain":null}},{"id":"l-1b54d37e1ad33d02","kind":"oeis_match","node_a_id":"n-78ba05b4a3dd925d","node_b_id":null,"fingerprint":null,"oeis_id":"A063007","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":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":10,"target":"reversed_triangle","terms_compared":55},"in_catalog":true,"oeis_id":"A063007","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-276374ccc25cc7b1","kind":"oeis_match","node_a_id":"n-78ba05b4a3dd925d","node_b_id":null,"fingerprint":null,"oeis_id":"A001850","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":"A001850","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-94fa52f883b01723","kind":"oeis_match","node_a_id":"n-78ba05b4a3dd925d","node_b_id":null,"fingerprint":null,"oeis_id":"A104684","in_catalog":1,"target":"triangle","row_alignment":"tabl","rows_matched":9,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":9,"target":"triangle","terms_compared":50},"in_catalog":true,"oeis_id":"A104684","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A104684","relation":"identification"},{"oeis_id":"A063007","relation":"match_reversed_triangle"},{"oeis_id":"A001850","relation":"match_row_sums"},{"oeis_id":"A104684","relation":"match_triangle"}],"bfiles":[{"node_id":"n-78ba05b4a3dd925d","oeis_id":"A001850","target":"row_sums","oeis_offset":0,"first_row":1,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":24,"oeis_terms_available":24,"term_count":200,"raw_size":15989},{"node_id":"n-78ba05b4a3dd925d","oeis_id":"A063007","target":"reversed_triangle","oeis_offset":0,"first_row":1,"rows_used":44,"terms_matched":55,"oeis_terms_agreeing":55,"oeis_terms_available":55,"term_count":990,"raw_size":20689},{"node_id":"n-78ba05b4a3dd925d","oeis_id":"A104684","target":"triangle","oeis_offset":0,"first_row":1,"rows_used":44,"terms_matched":50,"oeis_terms_agreeing":50,"oeis_terms_available":50,"term_count":990,"raw_size":20689}],"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:     2,      1;\n   3:     6,      6,      1;\n   4:    20,     30,     12,      1;\n   5:    70,    140,     90,     20,      1;\n   6:   252,    630,    560,    210,     30,     1;\n   7:   924,   2772,   3150,   1680,    420,    42,     1;\n   8:  3432,  12012,  16632,  11550,   4200,   756,    56,    1;\n   9: 12870,  51480,  84084,  72072,  34650,  9240,  1260,   72,  1;\n  10: 48620, 218790, 411840, 420420, 252252, 90090, 18480, 1980, 90, 1;\n","recurrence_sentence":"The row polynomials satisfy -(n-1)*P(n,x) = -(4*n-6+(2*n-3)*x)*P(n-1,x) + (n-2)*x^2*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 2 + x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(4*n - 6 + (2*n - 3)*x)*P[n - 1, x] + (n - 2)*x^2*P[n - 2, x])/(-n + 1)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + x; for(n = 3, N, my(i = n); v[i] = (-(4*n - 6 + (2*n - 3)*x)*v[i - 1] + (n - 2)*x^2*v[i - 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}},{"oeis_id":"A063007","target":"reversed_triangle","first_row":1,"oeis_offset":0,"reversed":true,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0: 1;\n  1: 1,  2;\n  2: 1,  6,    6;\n  3: 1, 12,   30,    20;\n  4: 1, 20,   90,   140,    70;\n  5: 1, 30,  210,   560,   630,    252;\n  6: 1, 42,  420,  1680,  3150,   2772,    924;\n  7: 1, 56,  756,  4200, 11550,  16632,  12012,   3432;\n  8: 1, 72, 1260,  9240, 34650,  72072,  84084,  51480,  12870;\n  9: 1, 90, 1980, 18480, 90090, 252252, 420420, 411840, 218790, 48620;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where -n*P(n,x) = -(4*n-2+(2*n-1)*x)*P(n-1,x) + (n-1)*x^2*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(4*n - 2 + (2*n - 1)*x)*P[n - 1, x] + (n - 1)*x^2*P[n - 2, x])/(-n)]];\nTable[Reverse[PadRight[CoefficientList[P[n, x], x], n + 1]], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = (-(4*n - 2 + (2*n - 1)*x)*v[i - 1] + (n - 1)*x^2*v[i - 2])/(-n)); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c = Vecrev(c); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":1,"offset":0,"reversed":true,"trim_zeros":"none","regular_triangle":true}},{"oeis_id":"A104684","target":"triangle","first_row":1,"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:     2,      1;\n  2:     6,      6,      1;\n  3:    20,     30,     12,      1;\n  4:    70,    140,     90,     20,      1;\n  5:   252,    630,    560,    210,     30,     1;\n  6:   924,   2772,   3150,   1680,    420,    42,     1;\n  7:  3432,  12012,  16632,  11550,   4200,   756,    56,    1;\n  8: 12870,  51480,  84084,  72072,  34650,  9240,  1260,   72,  1;\n  9: 48620, 218790, 411840, 420420, 252252, 90090, 18480, 1980, 90, 1;\n","recurrence_sentence":"The row polynomials satisfy -n*P(n,x) = -(4*n-2+(2*n-1)*x)*P(n-1,x) + (n-1)*x^2*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(4*n - 2 + (2*n - 1)*x)*P[n - 1, x] + (n - 1)*x^2*P[n - 2, x])/(-n)]];\nTable[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = (-(4*n - 2 + (2*n - 1)*x)*v[i - 1] + (n - 1)*x^2*v[i - 2])/(-n)); 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":1,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-78ba05b4a3dd925d","steps":[],"edges":[],"text":"A104684"},"headline":"A104684: Triangle read by rows: T(n,k) is the number of lattice paths from (0,0) to (n,n) using steps E=(1,0), N=(0,1) and D=(1,1) (i.e., bilateral Schroeder paths), having k D=(1,1) steps.","chain":null}