{"id":"n-5adb183b9026eb81","revision":0,"first_index":0,"fingerprint":"f6-63d66b489de6a076","name":"mobius_x_over_x_plus_one of Triangular array of generalized Narayana numbers T(n,k) = 2*binomial(n+1,k)* binomial(n-2,k-1)/(n+1) for n >= 1 and 0 <= k <= n-1, read by rows.","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"],["0","2"],["0","2","5"],["0","2","12","14"],["0","2","21","56","42"],["0","2","32","144","240","132"],["0","2","45","300","825","990","429"],["0","2","60","550","2200","4290","4004","1430"],["0","2","77","924","5005","14014","21021","16016","4862"],["0","2","96","1456","10192","38220","81536","99008","63648","16796"],["0","2","117","2184","19110","91728","259896","445536","453492","251940","58786"],["0","2","140","3150","33600","199920","719712","1627920","2325600","2034900","994840","208012"],["0","2","165","4400","56100","403920","1790712","5116320","9593100","11724900","8989090","3922512","742900"],["0","2","192","5984","89760","767448","4093056","14325696","33767712","53934540","57530176","39225120","15452320","2674440"],["0","2","221","7956","138567","1385670","8729721","36581688","105172353","210344706","292145425","276210220","169492635","60843510","9694845"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","2","7","28","121","550","2591","12536","61921","310954","1582791","8147796","42344121","221866446","1170747519"],"nonn":true,"definitions":[{"position":0,"source":"derived:mobius_x_over_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0","0","0"],["2","4","2"],["1","2","1"]]},"terms":[{"coeff":{"coeffs":[["-1","-1","1","1"],["0","3","6","3"],["1","4","5","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1","3","3","1"],["0","2","6","6","2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":173766,"position":0,"claim":"real_rooted","from_index":0,"region_key":"[-1, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in [-1, 0]","region_exact":null,"edges":["e-9729c9503f3cbc35"]},{"id":173767,"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-9729c9503f3cbc35"]},{"id":173768,"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-9729c9503f3cbc35"]}],"region_exact":{},"incoming_edges":[{"id":"e-9729c9503f3cbc35","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"mobius\",\"a\":\"1\",\"b\":\"0\",\"c\":\"1\",\"e\":\"1\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":1}}"},"operation_detail":"mobius, shift 0","conclusion_id":"n-5adb183b9026eb81","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":"fTransform","lean_declarations":["RealRooted.splits_fPolynomial"],"certificate_hash":"19643bf5e9979bc54df5f6ce33d5d7ef700bdccf61a4cfe90a61a6a3cab2da11","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-8a65e165240c40e0","role":"sequence","name":"Triangular array of generalized Narayana numbers T(n,k) = 2*binomial(n+1,k)* binomial(n-2,k-1)/(n+1) for n >= 1 and 0 <= k <= n-1, read by rows.","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-0b0bae4f5d5c24f7","kind":"fingerprint_related","node_a_id":"n-5adb183b9026eb81","node_b_id":"n-0248ad0b5ffb0432","fingerprint":"f6-63d66b489de6a076","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-63d66b489de6a076"},"other_id":"n-0248ad0b5ffb0432","other":{"id":"n-0248ad0b5ffb0432","name":"shift_x_plus_one of reversal of Triangle read by rows: T(n,k) = number of vertices of degree k in an origami flip graph OFG(A2n).","first_index":0,"fingerprint":"f6-63d66b489de6a076","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["8"],["30","12"],["112","96","16"]],"card_rows":[["8"],["30","12"],["112","96","16"],["420","560","210","20"]],"row_sums_preview":["8","42","224","1210","6600","36274","200576","1114578"],"regions":["(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7a4827a398e28fe4","steps":["reverse","x \u21a6 x+1"],"edges":["e-8b10d4d15ed56acc","e-6b54788a23565e9e"],"text":"A352880 \u2192 reverse \u2192 x \u21a6 x+1"},"headline":"A352880 \u2192 reverse \u2192 x \u21a6 x+1","chain":null}},{"id":"l-0c2b5d93a49a9204","kind":"fingerprint_related","node_a_id":"n-a9fe45b73042a3d2","node_b_id":"n-5adb183b9026eb81","fingerprint":"f6-63d66b489de6a076","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-63d66b489de6a076"},"other_id":"n-a9fe45b73042a3d2","other":{"id":"n-a9fe45b73042a3d2","name":"mobius_x_over_x_plus_one of Triangle read by rows: T(n,k) = number of vertices of degree k in an origami flip graph OFG(A2n).","first_index":0,"fingerprint":"f6-63d66b489de6a076","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["8"],["12","30"],["16","96","112"]],"card_rows":[["8"],["12","30"],["16","96","112"],["20","210","560","420"]],"row_sums_preview":["8","42","224","1210","6600","36274","200576","1114578"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7a4827a398e28fe4","steps":["x \u21a6 x/(x+1)"],"edges":["e-95ef58c5a4cf27ac"],"text":"A352880 \u2192 x \u21a6 x/(x+1)"},"headline":"A352880 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-255c965a8ec587bc","kind":"fingerprint_related","node_a_id":"n-5adb183b9026eb81","node_b_id":"n-d7c9d6946ca46996","fingerprint":"f6-63d66b489de6a076","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-63d66b489de6a076"},"other_id":"n-d7c9d6946ca46996","other":{"id":"n-d7c9d6946ca46996","name":"mobius_x_over_x_plus_one of reversal of Triangle of Dyck paths counted by number of long interior inclines.","first_index":0,"fingerprint":"f6-63d66b489de6a076","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["2"],["2","5"],["2","12","14"]],"card_rows":[["2"],["2","5"],["2","12","14"],["2","21","56","42"]],"row_sums_preview":["2","7","28","121","550","2591","12536","61921"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c963abfc0266d66c","steps":["reverse","x \u21a6 x/(x+1)"],"edges":["e-7ac651410315272d","e-991d808aa6d1f73f"],"text":"A108838 \u2192 reverse \u2192 x \u21a6 x/(x+1)"},"headline":"A108838 \u2192 reverse \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-68b9d9931326f043","kind":"fingerprint_related","node_a_id":"n-5adb183b9026eb81","node_b_id":"n-a9b81a684a479992","fingerprint":"f6-63d66b489de6a076","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-63d66b489de6a076"},"other_id":"n-a9b81a684a479992","other":{"id":"n-a9b81a684a479992","name":"shift_x_plus_one of derivative of narayana","first_index":1,"fingerprint":"f6-63d66b489de6a076","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["1"],["5","2"]],"card_rows":[["1"],["5","2"],["21","18","3"],["84","112","42","4"]],"row_sums_preview":["0","1","7","42","242","1375","7773","43876"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2a05d8340c329b21","steps":["basis A007318","D"],"edges":["e-58b8cdf8af6ac600","e-5ab75c3f6302f8ca"],"text":"narayana \u2192 basis A007318 \u2192 D"},"headline":"narayana \u2192 basis A007318 \u2192 D","chain":null}},{"id":"l-69b36114002e1832","kind":"fingerprint_related","node_a_id":"n-5adb183b9026eb81","node_b_id":"n-bdd7c1342151940b","fingerprint":"f6-63d66b489de6a076","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-63d66b489de6a076"},"other_id":"n-bdd7c1342151940b","other":{"id":"n-bdd7c1342151940b","name":"basis_map_A007318 of reversal of Triangle read by rows: T(n,k) = number of vertices of degree k in an origami flip graph OFG(A2n).","first_index":0,"fingerprint":"f6-63d66b489de6a076","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["8"],["30","12"],["112","96","16"]],"card_rows":[["8"],["30","12"],["112","96","16"],["420","560","210","20"]],"row_sums_preview":["8","42","224","1210","6600","36274","200576","1114578"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7a4827a398e28fe4","steps":["reverse","basis A007318"],"edges":["e-8b10d4d15ed56acc","e-0e8c1a5f2917a4f6"],"text":"A352880 \u2192 reverse \u2192 basis A007318"},"headline":"A352880 \u2192 reverse \u2192 basis A007318","chain":null}},{"id":"l-6afb28ef64d435aa","kind":"fingerprint_related","node_a_id":"n-5adb183b9026eb81","node_b_id":"n-4b52bcea0ddc9e78","fingerprint":"f6-63d66b489de6a076","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-63d66b489de6a076"},"other_id":"n-4b52bcea0ddc9e78","other":{"id":"n-4b52bcea0ddc9e78","name":"shift_x_plus_one of derivative of Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k)^2/(n-k+1).","first_index":0,"fingerprint":"f6-63d66b489de6a076","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["2"],["15","6"]],"card_rows":[["2"],["15","6"],["84","72","12"],["420","560","210","20"]],"row_sums_preview":["0","2","21","168","1210","8250","54411","351008"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-984320118b653f6b","steps":["basis A007318","D"],"edges":["e-2345ffa617ccabd7","e-abb34fcab799982f"],"text":"A132812 \u2192 basis A007318 \u2192 D"},"headline":"A132812 \u2192 basis A007318 \u2192 D","chain":null}},{"id":"l-a0c41fa225312112","kind":"fingerprint_related","node_a_id":"n-3e16b3cbabac7254","node_b_id":"n-5adb183b9026eb81","fingerprint":"f6-63d66b489de6a076","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-63d66b489de6a076"},"other_id":"n-3e16b3cbabac7254","other":{"id":"n-3e16b3cbabac7254","name":"derivative of Triangle read by rows: T(n,k) is the number of Schroeder paths of semilength n containing exactly k peaks but no peaks at level one (n >= 1; 0 <= k <= n-1).","first_index":0,"fingerprint":"f6-63d66b489de6a076","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["1"],["5","2"]],"card_rows":[["1"],["5","2"],["21","18","3"],["84","112","42","4"]],"row_sums_preview":["0","1","7","42","242","1375","7773","43876"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-57b4b14604f4c0d7","steps":["D"],"edges":["e-a4f714b61c97080d"],"text":"A126216 \u2192 D"},"headline":"A126216 \u2192 D","chain":null}},{"id":"l-ce8dd67083fbe035","kind":"fingerprint_related","node_a_id":"n-5adb183b9026eb81","node_b_id":"n-212d4d84971f6f98","fingerprint":"f6-63d66b489de6a076","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-63d66b489de6a076"},"other_id":"n-212d4d84971f6f98","other":{"id":"n-212d4d84971f6f98","name":"reversal of basis_map_A007318 of Triangle of Dyck paths counted by number of long interior inclines.","first_index":0,"fingerprint":"f6-63d66b489de6a076","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["2"],["2","5"],["2","12","14"]],"card_rows":[["2"],["2","5"],["2","12","14"],["2","21","56","42"]],"row_sums_preview":["2","7","28","121","550","2591","12536","61921"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c963abfc0266d66c","steps":["x \u21a6 x+1","reverse"],"edges":["e-454321f2b34b0823","e-8e698c4d3cc74184"],"text":"A108838 \u2192 x \u21a6 x+1 \u2192 reverse"},"headline":"A108838 \u2192 x \u21a6 x+1 \u2192 reverse","chain":null}},{"id":"l-f4051bf82d063922","kind":"fingerprint_related","node_a_id":"n-3b95e3dd4c91b92c","node_b_id":"n-5adb183b9026eb81","fingerprint":"f6-63d66b489de6a076","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-63d66b489de6a076"},"other_id":"n-3b95e3dd4c91b92c","other":{"id":"n-3b95e3dd4c91b92c","name":"basis_map_A007318 of Triangle of Dyck paths counted by number of long interior inclines.","first_index":0,"fingerprint":"f6-63d66b489de6a076","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["2"],["5","2"],["14","12","2"]],"card_rows":[["2"],["5","2"],["14","12","2"],["42","56","21","2"]],"row_sums_preview":["2","7","28","121","550","2591","12536","61921"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c963abfc0266d66c","steps":["x \u21a6 x+1"],"edges":["e-454321f2b34b0823"],"text":"A108838 \u2192 x \u21a6 x+1"},"headline":"A108838 \u2192 x \u21a6 x+1","chain":null}}],"oeis":[],"bfiles":[],"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: 0, 2;\n  2: 0, 2,  5;\n  3: 0, 2, 12,   14;\n  4: 0, 2, 21,   56,    42;\n  5: 0, 2, 32,  144,   240,   132;\n  6: 0, 2, 45,  300,   825,   990,   429;\n  7: 0, 2, 60,  550,  2200,  4290,  4004,  1430;\n  8: 0, 2, 77,  924,  5005, 14014, 21021, 16016,  4862;\n  9: 0, 2, 96, 1456, 10192, 38220, 81536, 99008, 63648, 16796;\n","recurrence_sentence":"The row polynomials satisfy (n^2+2*n+(2*n^2+4*n)*x+(n^2+2*n)*x^2)*P(n,x) = (n^2-1+(4*n^2+3*n-1)*x+(5*n^2+6*n+1)*x^2+(2*n^2+3*n+1)*x^3)*P(n-1,x) + ((2*n+1)*x+(6*n+3)*x^2+(6*n+3)*x^3+(2*n+1)*x^4)*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[((n^2 - 1 + (4*n^2 + 3*n - 1)*x + (5*n^2 + 6*n + 1)*x^2 + (2*n^2 + 3*n + 1)*x^3)*P[n - 1, x] + ((2*n + 1)*x + (6*n + 3)*x^2 + (6*n + 3)*x^3 + (2*n + 1)*x^4)*D[P[n - 1, x], x])/(n^2 + 2*n + (2*n^2 + 4*n)*x + (n^2 + 2*n)*x^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] = ((n^2 - 1 + (4*n^2 + 3*n - 1)*x + (5*n^2 + 6*n + 1)*x^2 + (2*n^2 + 3*n + 1)*x^3)*v[i - 1] + ((2*n + 1)*x + (6*n + 3)*x^2 + (6*n + 3)*x^3 + (2*n + 1)*x^4)*deriv(v[i - 1], x))/(n^2 + 2*n + (2*n^2 + 4*n)*x + (n^2 + 2*n)*x^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}}],"derivation":{"root":"n-8a65e165240c40e0","steps":["x \u21a6 x/(x+1)"],"edges":["e-9729c9503f3cbc35"],"text":"A281260 \u2192 x \u21a6 x/(x+1)"},"headline":"A281260 \u2192 x \u21a6 x/(x+1)","chain":null}