{"id":"n-1b0fc3e5a40ede1a","revision":1,"first_index":0,"fingerprint":"f6-9e57d05d9d9a1ce7","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).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["2","1"],["4","6","1"],["8","24","12","1"],["16","80","80","20","1"],["32","240","400","200","30","1"],["64","672","1680","1400","420","42","1"],["128","1792","6272","7840","3920","784","56","1"],["256","4608","21504","37632","28224","9408","1344","72","1"],["512","11520","69120","161280","169344","84672","20160","2160","90","1"],["1024","28160","211200","633600","887040","620928","221760","39600","3300","110","1"],["2048","67584","619520","2323200","4181760","3902976","1951488","522720","72600","4840","132","1"],["4096","159744","1757184","8053760","18120960","21745152","14496768","5436288","1132560","125840","6864","156","1"],["8192","372736","4845568","26650624","73289216","109933824","94228992","47114496","13741728","2290288","208208","9464","182","1"],["16384","860160","13045760","84797440","279831552","513024512","549669120","353358720","137417280","32064032","4372368","331240","12740","210","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","11","45","197","903","4279","20793","103049","518859","2646723","13648869","71039373","372693519","1968801519"],"nonn":true,"definitions":[{"position":0,"source":"derived:reversal","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["2"],["1"]]},"terms":[{"coeff":{"coeffs":[["4","0"],["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":165281,"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-0ad5e2ffb4ffd4ba"]},{"id":165282,"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-0ad5e2ffb4ffd4ba"]},{"id":165283,"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-0ad5e2ffb4ffd4ba"]}],"region_exact":{},"incoming_edges":[{"id":"e-0ad5e2ffb4ffd4ba","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-1b0fc3e5a40ede1a","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":"dd80eeaedcfb41f09f9ba20d0d596acbf40b430de5bbaf25b80c06d8eeec52de","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-8ce61d302cec295b","role":"sequence","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).","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]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-35aedd47ad51ab4e","kind":"fingerprint_related","node_a_id":"n-1b0fc3e5a40ede1a","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-6a30765140629bd5","other":{"id":"n-6a30765140629bd5","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.","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"],["4","2"],["8","12","2"]],"card_rows":[["2"],["4","2"],["8","12","2"],["16","48","24","2"]],"row_sums_preview":["2","6","22","90","394","1806","8558","41586"],"regions":["(-inf, 0)"],"region_exact":{},"oeis_ids":["A114655"],"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}},{"id":"l-653b8f9b5c0b1e40","kind":"fingerprint_related","node_a_id":"n-c3c79310b753c17e","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-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-7af882ad23e1fc73","kind":"fingerprint_related","node_a_id":"n-0d75f6272d2a2a84","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-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-f83adb07cf90df43","kind":"fingerprint_related","node_a_id":"n-8ce61d302cec295b","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-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-5cbca8bdabac4251","kind":"oeis_match","node_a_id":"n-1b0fc3e5a40ede1a","node_b_id":null,"fingerprint":null,"oeis_id":"A114687","in_catalog":1,"target":"reversed_triangle","row_alignment":"tabl","rows_matched":9,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":9,"target":"reversed_triangle","terms_compared":54},"in_catalog":true,"oeis_id":"A114687","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-c74be215faeea33e","kind":"oeis_match","node_a_id":"n-1b0fc3e5a40ede1a","node_b_id":null,"fingerprint":null,"oeis_id":"A114656","in_catalog":1,"target":"triangle","row_alignment":"tabl","rows_matched":9,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":9,"target":"triangle","terms_compared":54},"in_catalog":true,"oeis_id":"A114656","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-e7aed48f1ee2c712","kind":"oeis_match","node_a_id":"n-1b0fc3e5a40ede1a","node_b_id":null,"fingerprint":null,"oeis_id":"A001003","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":"A001003","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A114687","relation":"match_reversed_triangle"},{"oeis_id":"A001003","relation":"match_row_sums"},{"oeis_id":"A114656","relation":"match_triangle"}],"bfiles":[{"node_id":"n-1b0fc3e5a40ede1a","oeis_id":"A114656","target":"triangle","oeis_offset":1,"first_row":0,"rows_used":44,"terms_matched":54,"oeis_terms_agreeing":54,"oeis_terms_available":54,"term_count":990,"raw_size":19564},{"node_id":"n-1b0fc3e5a40ede1a","oeis_id":"A114687","target":"reversed_triangle","oeis_offset":1,"first_row":0,"rows_used":44,"terms_matched":54,"oeis_terms_agreeing":54,"oeis_terms_available":54,"term_count":990,"raw_size":19564}],"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,     1;\n  2:   4,     6,     1;\n  3:   8,    24,    12,      1;\n  4:  16,    80,    80,     20,      1;\n  5:  32,   240,   400,    200,     30,     1;\n  6:  64,   672,  1680,   1400,    420,    42,     1;\n  7: 128,  1792,  6272,   7840,   3920,   784,    56,    1;\n  8: 256,  4608, 21504,  37632,  28224,  9408,  1344,   72,  1;\n  9: 512, 11520, 69120, 161280, 169344, 84672, 20160, 2160, 90, 1;\n","recurrence_sentence":"The row polynomials satisfy (n+2)*P(n,x) = (2*n+4+3*n*x)*P(n-1,x) + (4*x-2*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n + 4 + 3*n*x)*P[n - 1, x] + (4*x - 2*x^2)*D[P[n - 1, x], x])/(n + 2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((2*n + 4 + 3*n*x)*v[i - 1] + (4*x - 2*x^2)*deriv(v[i - 1], x))/(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":"A114656","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,     1;\n   3:   4,     6,     1;\n   4:   8,    24,    12,      1;\n   5:  16,    80,    80,     20,      1;\n   6:  32,   240,   400,    200,     30,     1;\n   7:  64,   672,  1680,   1400,    420,    42,     1;\n   8: 128,  1792,  6272,   7840,   3920,   784,    56,    1;\n   9: 256,  4608, 21504,  37632,  28224,  9408,  1344,   72,  1;\n  10: 512, 11520, 69120, 161280, 169344, 84672, 20160, 2160, 90, 1;\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) = 1.","mathematica":"P[1, x_] := 1;\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] = 1; 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":0,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":true}},{"oeis_id":"A114687","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: 1,  2;\n   3: 1,  6,    4;\n   4: 1, 12,   24,     8;\n   5: 1, 20,   80,    80,    16;\n   6: 1, 30,  200,   400,   240,     32;\n   7: 1, 42,  420,  1400,  1680,    672,     64;\n   8: 1, 56,  784,  3920,  7840,   6272,   1792,   128;\n   9: 1, 72, 1344,  9408, 28224,  37632,  21504,  4608,   256;\n  10: 1, 90, 2160, 20160, 84672, 169344, 161280, 69120, 11520, 512;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where (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) = 1.","mathematica":"P[1, x_] := 1;\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[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] = ((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 = 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-8ce61d302cec295b","steps":["reverse"],"edges":["e-0ad5e2ffb4ffd4ba"],"text":"A114687 \u2192 reverse"},"headline":"A114687 \u2192 reverse","chain":null}