{"id":"n-6a30765140629bd5","revision":1,"first_index":1,"fingerprint":"f6-9e57d05d9d9a1ce7","name":"Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k weak ascents (1<=k<=n). A Schroeder path of length 2n is a lattice path from (0,0) to (2n,0) consisting of U=(1,1), D=(1,-1) and H=(2,0) steps and never going below the x-axis. A weak ascent in a Schroeder path is a maximal sequence of consecutive U and H 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":[["2"],["4","2"],["8","12","2"],["16","48","24","2"],["32","160","160","40","2"],["64","480","800","400","60","2"],["128","1344","3360","2800","840","84","2"],["256","3584","12544","15680","7840","1568","112","2"],["512","9216","43008","75264","56448","18816","2688","144","2"],["1024","23040","138240","322560","338688","169344","40320","4320","180","2"],["2048","56320","422400","1267200","1774080","1241856","443520","79200","6600","220","2"],["4096","135168","1239040","4646400","8363520","7805952","3902976","1045440","145200","9680","264","2"],["8192","319488","3514368","16107520","36241920","43490304","28993536","10872576","2265120","251680","13728","312","2"],["16384","745472","9691136","53301248","146578432","219867648","188457984","94228992","27483456","4580576","416416","18928","364","2"],["32768","1720320","26091520","169594880","559663104","1026049024","1099338240","706717440","274834560","64128064","8744736","662480","25480","420","2"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["2","6","22","90","394","1806","8558","41586","206098","1037718","5293446","27297738","142078746","745387038","3937603038"],"nonn":true,"definitions":[{"position":0,"source":"oeis_catalog:A114655","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["2"]],"recurrence":{"denominator":{"coeffs":[["1"],["1"]]},"terms":[{"coeff":{"coeffs":[["2","-3"],["2","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","4","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":159087,"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-784ea0037cdd0731"]},{"id":159088,"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-784ea0037cdd0731"]},{"id":159089,"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-784ea0037cdd0731"]},{"id":159090,"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-784ea0037cdd0731"]}],"region_exact":{},"incoming_edges":[{"id":"e-784ea0037cdd0731","operation":{"operation":"theorem_check","theorem":"liu_wang"},"operation_detail":"liu_wang","conclusion_id":"n-6a30765140629bd5","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":"4038d0d63986a4e8828d41ed5d118cff35811796571252181fa2ffd045042e27","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-798bfa3a83d1e4b4","operation":"obreschkoff_combination","operation_detail":null,"conclusion_id":"n-68458fe983bd99b5","conclusion_name":"obreschkoff_combination of Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k weak ascents (1<=k<=n). A Schroeder path of length 2n is a lattice path from (0,0) to (2n,0) consisting of U=(1,1), D=(1,-1) and H=(2,0) steps and never going below the x-axis. A weak ascent in a Schroeder path is a maximal sequence of consecutive U and H steps."},{"id":"e-798bfa3a83d1e4b4","operation":"obreschkoff_combination","operation_detail":null,"conclusion_id":"n-68458fe983bd99b5","conclusion_name":"obreschkoff_combination of Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k weak ascents (1<=k<=n). A Schroeder path of length 2n is a lattice path from (0,0) to (2n,0) consisting of U=(1,1), D=(1,-1) and H=(2,0) steps and never going below the x-axis. A weak ascent in a Schroeder path is a maximal sequence of consecutive U and H steps."},{"id":"e-460d39e35063c243","operation":"preserver_image","operation_detail":"mobius, shift 0","conclusion_id":"n-4a5cb5ea4703a9a6","conclusion_name":"mobius_x_over_x_plus_one of Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k weak ascents (1<=k<=n). A Schroeder path of length 2n is a lattice path from (0,0) to (2n,0) consisting of U=(1,1), D=(1,-1) and H=(2,0) steps and never going below the x-axis. A weak ascent in a Schroeder path is a maximal sequence of consecutive U and H steps."},{"id":"e-5ebd2cd7e542e718","operation":"preserver_image","operation_detail":"affine, shift 0","conclusion_id":"n-25065a706ad43d32","conclusion_name":"shift_x_plus_one of Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k weak ascents (1<=k<=n). A Schroeder path of length 2n is a lattice path from (0,0) to (2n,0) consisting of U=(1,1), D=(1,-1) and H=(2,0) steps and never going below the x-axis. A weak ascent in a Schroeder path is a maximal sequence of consecutive U and H steps."},{"id":"e-b0d131c104e489dd","operation":"preserver_image","operation_detail":"reversal, shift 0","conclusion_id":"n-435b4d5bded8f11f","conclusion_name":"reversal of Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k weak ascents (1<=k<=n). A Schroeder path of length 2n is a lattice path from (0,0) to (2n,0) consisting of U=(1,1), D=(1,-1) and H=(2,0) steps and never going below the x-axis. A weak ascent in a Schroeder path is a maximal sequence of consecutive U and H steps."},{"id":"e-b19e39c8911e8bee","operation":"preserver_image","operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-ffe0c414befb86c3","conclusion_name":"theta_plus_one_squared of Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k weak ascents (1<=k<=n). A Schroeder path of length 2n is a lattice path from (0,0) to (2n,0) consisting of U=(1,1), D=(1,-1) and H=(2,0) steps and never going below the x-axis. A weak ascent in a Schroeder path is a maximal sequence of consecutive U and H steps."},{"id":"e-c818d14e04074107","operation":"preserver_image","operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-7f2a74b39253adec","conclusion_name":"one_plus_derivative of Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k weak ascents (1<=k<=n). A Schroeder path of length 2n is a lattice path from (0,0) to (2n,0) consisting of U=(1,1), D=(1,-1) and H=(2,0) steps and never going below the x-axis. A weak ascent in a Schroeder path is a maximal sequence of consecutive U and H steps."},{"id":"e-d11937e4427f3a96","operation":"preserver_image","operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-936054a41761ef4d","conclusion_name":"derivative of Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k weak ascents (1<=k<=n). A Schroeder path of length 2n is a lattice path from (0,0) to (2n,0) consisting of U=(1,1), D=(1,-1) and H=(2,0) steps and never going below the x-axis. A weak ascent in a Schroeder path is a maximal sequence of consecutive U and H steps."},{"id":"e-f3cda92ee6452a8a","operation":"preserver_image","operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-7f47b2246e50a89b","conclusion_name":"theta_plus_one of Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k weak ascents (1<=k<=n). A Schroeder path of length 2n is a lattice path from (0,0) to (2n,0) consisting of U=(1,1), D=(1,-1) and H=(2,0) steps and never going below the x-axis. A weak ascent in a Schroeder path is a maximal sequence of consecutive U and H steps."}],"used_by_total":9,"links":[{"id":"l-23402b409fb77b06","kind":"fingerprint_related","node_a_id":"n-6a30765140629bd5","node_b_id":"n-8ce61d302cec295b","fingerprint":"f6-9e57d05d9d9a1ce7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-9e57d05d9d9a1ce7"},"other_id":"n-8ce61d302cec295b","other":{"id":"n-8ce61d302cec295b","name":"Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also called marked Dyck paths) of semilength n and having k double rises (0 <= k <= n-1).","first_index":0,"fingerprint":"f6-9e57d05d9d9a1ce7","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","2"],["1","6","4"]],"card_rows":[["1"],["1","2"],["1","6","4"],["1","12","24","8"]],"row_sums_preview":["1","3","11","45","197","903","4279","20793"],"regions":["(-inf, +inf)","(-inf, 0)"],"region_exact":{},"oeis_ids":["A114687"],"derivation":{"root":"n-8ce61d302cec295b","steps":[],"edges":[],"text":"A114687"},"headline":"A114687: Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also called marked Dyck paths) of semilength n and having k double rises (0 <= k <= n-1).","chain":null}},{"id":"l-270ee1dff4805936","kind":"fingerprint_related","node_a_id":"n-6a30765140629bd5","node_b_id":"n-0d75f6272d2a2a84","fingerprint":"f6-9e57d05d9d9a1ce7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-9e57d05d9d9a1ce7"},"other_id":"n-0d75f6272d2a2a84","other":{"id":"n-0d75f6272d2a2a84","name":"Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also called marked Dyck paths) of semilength n and having k peaks (1 <= k <= n).","first_index":1,"fingerprint":"f6-9e57d05d9d9a1ce7","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","1"],["4","6","1"]],"card_rows":[["1"],["2","1"],["4","6","1"],["8","24","12","1"]],"row_sums_preview":["1","3","11","45","197","903","4279","20793"],"regions":["(-inf, +inf)","(-inf, 0)"],"region_exact":{},"oeis_ids":["A114656"],"derivation":{"root":"n-0d75f6272d2a2a84","steps":[],"edges":[],"text":"A114656"},"headline":"A114656: Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also called marked Dyck paths) of semilength n and having k peaks (1 <= k <= n).","chain":null}},{"id":"l-3f37579f19fc2ca9","kind":"fingerprint_related","node_a_id":"n-6a30765140629bd5","node_b_id":"n-435b4d5bded8f11f","fingerprint":"f6-9e57d05d9d9a1ce7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-9e57d05d9d9a1ce7"},"other_id":"n-435b4d5bded8f11f","other":{"id":"n-435b4d5bded8f11f","name":"reversal of Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k weak ascents (1<=k<=n). A Schroeder path of length 2n is a lattice path from (0,0) to (2n,0) consisting of U=(1,1), D=(1,-1) and H=(2,0) steps and never going below the x-axis. A weak ascent in a Schroeder path is a maximal sequence of consecutive U and H steps.","first_index":1,"fingerprint":"f6-9e57d05d9d9a1ce7","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["2"],["2","4"],["2","12","8"]],"card_rows":[["2"],["2","4"],["2","12","8"],["2","24","48","16"]],"row_sums_preview":["2","6","22","90","394","1806","8558","41586"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-6a30765140629bd5","steps":["reverse"],"edges":["e-b0d131c104e489dd"],"text":"A114655 \u2192 reverse"},"headline":"A114655 \u2192 reverse","chain":null}},{"id":"l-5a9e530e24a22f07","kind":"fingerprint_related","node_a_id":"n-6a30765140629bd5","node_b_id":"n-1b0fc3e5a40ede1a","fingerprint":"f6-9e57d05d9d9a1ce7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-9e57d05d9d9a1ce7"},"other_id":"n-1b0fc3e5a40ede1a","other":{"id":"n-1b0fc3e5a40ede1a","name":"reversal of Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also called marked Dyck paths) of semilength n and having k double rises (0 <= k <= n-1).","first_index":0,"fingerprint":"f6-9e57d05d9d9a1ce7","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","1"],["4","6","1"]],"card_rows":[["1"],["2","1"],["4","6","1"],["8","24","12","1"]],"row_sums_preview":["1","3","11","45","197","903","4279","20793"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8ce61d302cec295b","steps":["reverse"],"edges":["e-0ad5e2ffb4ffd4ba"],"text":"A114687 \u2192 reverse"},"headline":"A114687 \u2192 reverse","chain":null}},{"id":"l-ebb22178ad365807","kind":"fingerprint_related","node_a_id":"n-6a30765140629bd5","node_b_id":"n-c3c79310b753c17e","fingerprint":"f6-9e57d05d9d9a1ce7","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-9e57d05d9d9a1ce7"},"other_id":"n-c3c79310b753c17e","other":{"id":"n-c3c79310b753c17e","name":"reversal of Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also called marked Dyck paths) of semilength n and having k peaks (1 <= k <= n).","first_index":1,"fingerprint":"f6-9e57d05d9d9a1ce7","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","2"],["1","6","4"]],"card_rows":[["1"],["1","2"],["1","6","4"],["1","12","24","8"]],"row_sums_preview":["1","3","11","45","197","903","4279","20793"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-0d75f6272d2a2a84","steps":["reverse"],"edges":["e-fbde760708cf0a27"],"text":"A114656 \u2192 reverse"},"headline":"A114656 \u2192 reverse","chain":null}},{"id":"l-064c8195c2a9d2a9","kind":"oeis_match","node_a_id":"n-6a30765140629bd5","node_b_id":null,"fingerprint":null,"oeis_id":"A155069","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":"A155069","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-9ceba7b04a918872","kind":"oeis_match","node_a_id":"n-6a30765140629bd5","node_b_id":null,"fingerprint":null,"oeis_id":"A114655","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":1,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":9,"target":"triangle","terms_compared":51},"in_catalog":true,"oeis_id":"A114655","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-e33bdc3ec199c172","kind":"oeis_match","node_a_id":"n-6a30765140629bd5","node_b_id":null,"fingerprint":null,"oeis_id":"A025240","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":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A025240","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-fdb8f9caa49a65fc","kind":"oeis_match","node_a_id":"n-6a30765140629bd5","node_b_id":null,"fingerprint":null,"oeis_id":"A006318","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":"A006318","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A114655","relation":"identification"},{"oeis_id":"A006318","relation":"match_row_sums"},{"oeis_id":"A025240","relation":"match_row_sums"},{"oeis_id":"A155069","relation":"match_row_sums"},{"oeis_id":"A114655","relation":"match_triangle"}],"bfiles":[{"node_id":"n-6a30765140629bd5","oeis_id":"A114655","target":"triangle","oeis_offset":1,"first_row":1,"rows_used":44,"terms_matched":51,"oeis_terms_agreeing":51,"oeis_terms_available":51,"term_count":990,"raw_size":19853}],"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:    2;\n   2:    4,     2;\n   3:    8,    12,      2;\n   4:   16,    48,     24,      2;\n   5:   32,   160,    160,     40,      2;\n   6:   64,   480,    800,    400,     60,      2;\n   7:  128,  1344,   3360,   2800,    840,     84,     2;\n   8:  256,  3584,  12544,  15680,   7840,   1568,   112,    2;\n   9:  512,  9216,  43008,  75264,  56448,  18816,  2688,  144,   2;\n  10: 1024, 23040, 138240, 322560, 338688, 169344, 40320, 4320, 180, 2;\n","recurrence_sentence":"The row polynomials satisfy (n+1)*P(n,x) = (2*n+2+(3*n-3)*x)*P(n-1,x) + (4*x-2*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 2.","mathematica":"P[1, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n + 2 + (3*n - 3)*x)*P[n - 1, x] + (4*x - 2*x^2)*D[P[n - 1, x], x])/(n + 1)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 2; for(n = 2, N, my(i = n); v[i] = ((2*n + 2 + (3*n - 3)*x)*v[i - 1] + (4*x - 2*x^2)*deriv(v[i - 1], x))/(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":"A114655","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:    2;\n   2:    4,     2;\n   3:    8,    12,      2;\n   4:   16,    48,     24,      2;\n   5:   32,   160,    160,     40,      2;\n   6:   64,   480,    800,    400,     60,      2;\n   7:  128,  1344,   3360,   2800,    840,     84,     2;\n   8:  256,  3584,  12544,  15680,   7840,   1568,   112,    2;\n   9:  512,  9216,  43008,  75264,  56448,  18816,  2688,  144,   2;\n  10: 1024, 23040, 138240, 322560, 338688, 169344, 40320, 4320, 180, 2;\n","recurrence_sentence":"The row polynomials satisfy (n+1)*P(n,x) = (2*n+2+(3*n-3)*x)*P(n-1,x) + (4*x-2*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 2.","mathematica":"P[1, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n + 2 + (3*n - 3)*x)*P[n - 1, x] + (4*x - 2*x^2)*D[P[n - 1, x], x])/(n + 1)]];\nTable[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 2; for(n = 2, N, my(i = n); v[i] = ((2*n + 2 + (3*n - 3)*x)*v[i - 1] + (4*x - 2*x^2)*deriv(v[i - 1], x))/(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":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-6a30765140629bd5","steps":[],"edges":[],"text":"A114655"},"headline":"A114655: Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k weak ascents (1<=k<=n). A Schroeder path of length 2n is a lattice path from (0,0) to (2n,0) consisting of U=(1,1), D=(1,-1) and H=(2,0) steps and never going below the x-axis. A weak ascent in a Schroeder path is a maximal sequence of consecutive U and H steps.","chain":null}