{"id":"n-77b7f4b7ebead213","revision":1,"first_index":0,"fingerprint":"f6-c00a555c8a33253e","name":"basis_map_A008277 of 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.","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"],["5","6","1"],["14","31","12","1"],["42","155","105","20","1"],["132","771","815","265","30","1"],["429","3850","5971","2940","560","42","1"],["1430","19363","42420","29631","8400","1050","56","1"],["4862","98187","296269","281862","111195","20454","1806","72","1"],["16796","502063","2049855","2583715","1359792","343749","44310","2910","90","1"],["58786","2587937","14114221","23107535","15763000","5245086","920997","87780","4455","110","1"],["208012","13440857","96985141","203226606","175987845","74834782","17133732","2209977","162030","6545","132","1"],["742900","70296473","666247231","1766825632","1912144091","1017047460","294632338","49205442","4857567","282425","9295","156","1"],["2674440","370019079","4580736797","15238750891","20363140798","13330267951","4780603256","1002776632","127483356","9939930","469469","12831","182","1"],["9694845","1959106674","31543746599","130720107700","213621372328","169959091302","74218383239","19125240420","3036494890","303579276","19165146","749840","17290","210","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","12","58","323","2014","13793","102351","814708","6903281","61889908","584195660","5782291011","59806875613","644516749760"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A008277","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["2","1"]],"recurrence":{"denominator":{"coeffs":[["2"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","1","1"],["0","-1","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","1","2"],["0","-1","-2"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","1"],["0","0","-1"]]},"deriv_order":2,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["2","1"],["4","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1"],["0","2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162267,"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-da5c511e60ab53ca"]},{"id":162268,"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-da5c511e60ab53ca"]},{"id":162269,"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-da5c511e60ab53ca"]}],"region_exact":{},"incoming_edges":[{"id":"e-da5c511e60ab53ca","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-77b7f4b7ebead213","premise_count":1,"theorem":"ray basis maps: for L f = (a + b t) f + kappa (t - h) f', b > 0, kappa >= 0, and g real-rooted with every root <= h, of multiplicity m at h, T(u^k) = L^k g sends every real polynomial with all roots in S to one with all roots in (-inf, h] if and only if S lies in (-inf, a + b h + kappa m] (RealRooted rayBasisMap_maps_realRootedIn_iff; the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"dc1af7d2c8e9cfed1caeae7f8af68cfb92c25174ca72207d33b71fce3dc79ccf","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-c2eaf5212a4dec13","role":"sequence","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.","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-0fd07b079c65aaa3","kind":"fingerprint_related","node_a_id":"n-a91ed9a3e111e514","node_b_id":"n-77b7f4b7ebead213","fingerprint":"f6-c00a555c8a33253e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-c00a555c8a33253e"},"other_id":"n-a91ed9a3e111e514","other":{"id":"n-a91ed9a3e111e514","name":"basis_map_A008277 of narayana","first_index":1,"fingerprint":"f6-c00a555c8a33253e","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","1"],["5","6","1"]],"card_rows":[["1"],["2","1"],["5","6","1"],["14","31","12","1"]],"row_sums_preview":["1","3","12","58","323","2014","13793","102351"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2a05d8340c329b21","steps":["basis A008277"],"edges":["e-9b44e0c74e7ea693"],"text":"narayana \u2192 basis A008277"},"headline":"narayana \u2192 basis A008277","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:     2,      1;\n  2:     5,      6,       1;\n  3:    14,     31,      12,       1;\n  4:    42,    155,     105,      20,       1;\n  5:   132,    771,     815,     265,      30,      1;\n  6:   429,   3850,    5971,    2940,     560,     42,     1;\n  7:  1430,  19363,   42420,   29631,    8400,   1050,    56,    1;\n  8:  4862,  98187,  296269,  281862,  111195,  20454,  1806,   72,  1;\n  9: 16796, 502063, 2049855, 2583715, 1359792, 343749, 44310, 2910, 90, 1;\n","recurrence_sentence":"The row polynomials satisfy (n+2)*P(n,x) = -((n-1)*x+(n-1)*x^2)*P(n-2,x) - ((n-1)*x+(2*n-2)*x^2)*d/dx P(n-2,x) - (n-1)*x^2*d^2/dx^2 P(n-2,x) + (4*n+2+(2*n+1)*x)*P(n-1,x) + (2*n+1)*x*d/dx P(n-1,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[(-((n - 1)*x + (n - 1)*x^2)*P[n - 2, x] - ((n - 1)*x + (2*n - 2)*x^2)*D[P[n - 2, x], x] - (n - 1)*x^2*D[P[n - 2, x], {x, 2}] + (4*n + 2 + (2*n + 1)*x)*P[n - 1, x] + (2*n + 1)*x*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; v[2] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = (-((n - 1)*x + (n - 1)*x^2)*v[i - 2] - ((n - 1)*x + (2*n - 2)*x^2)*deriv(v[i - 2], x) - (n - 1)*x^2*deriv(deriv(v[i - 2], x), x) + (4*n + 2 + (2*n + 1)*x)*v[i - 1] + (2*n + 1)*x*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}}],"derivation":{"root":"n-c2eaf5212a4dec13","steps":["basis A008277"],"edges":["e-da5c511e60ab53ca"],"text":"A001263 \u2192 basis A008277"},"headline":"A001263 \u2192 basis A008277","chain":null}