{"id":"n-a73bf794859ac224","revision":0,"first_index":0,"fingerprint":"f6-099df3ed915090b2","name":"basis_map_A008277 of Triangle T(n,k) = number of k-part order-consecutive partitions of n (1<=k<=n).","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"],["41","152","104","20","1"],["122","721","780","260","30","1"],["365","3342","5377","2756","545","42","1"],["1094","15247","35112","25893","7742","1015","56","1"],["3281","68816","221072","224920","94858","18592","1736","72","1"],["9842","308353","1357176","1851448","1054092","287562","39816","2784","90","1"],["29525","1374966","8184001","14668120","10941130","3971380","758082","78120","4245","110","1"],["88574","6111007","48718692","113003033","108081292","50615862","12722600","1794474","143022","6215","132","1"],["265721","27100328","287300216","852590220","1028941339","608053632","195042200","35942280","3898851","247544","8800","156","1"],["797162","120002065","1682504772","6331578396","9523207434","6984172923","2794890904","651871376","91823446","7898891","408980","12116","182","1"],["2391485","530841534","9802271809","46450896828","86231262075","77478643326","38027502723","10970951304","1943412991","216025810","15098083","649740","16289","210","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","12","58","318","1914","12428","86160","633348","4911164","40009680","341284904","3039391288","28189168648","271669964208"],"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":{"terms":[{"coeff":{"coeffs":[["-3","-4","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","-4","-2"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","-1"]]},"deriv_order":2,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["4","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":196279,"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-b2a33f5e9ab45e0c"]},{"id":196280,"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-b2a33f5e9ab45e0c"]},{"id":196281,"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-b2a33f5e9ab45e0c"]}],"region_exact":{},"incoming_edges":[{"id":"e-b2a33f5e9ab45e0c","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-a73bf794859ac224","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":"0686d45fca88e359a6936576052a59932f012387c563d55c763a82b34f5b6b74","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-a6f3b35778192ffb","role":"sequence","name":"Triangle T(n,k) = number of k-part order-consecutive partitions of n (1<=k<=n).","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":[],"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:   41,    152,     104,      20,       1;\n  5:  122,    721,     780,     260,      30,      1;\n  6:  365,   3342,    5377,    2756,     545,     42,     1;\n  7: 1094,  15247,   35112,   25893,    7742,   1015,    56,    1;\n  8: 3281,  68816,  221072,  224920,   94858,  18592,  1736,   72,  1;\n  9: 9842, 308353, 1357176, 1851448, 1054092, 287562, 39816, 2784, 90, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(3+4*x+x^2)*P(n-2,x) - (4*x+2*x^2)*d/dx P(n-2,x) - x^2*d^2/dx^2 P(n-2,x) + (4+2*x)*P(n-1,x) + 2*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[-(3 + 4*x + x^2)*P[n - 2, x] - (4*x + 2*x^2)*D[P[n - 2, x], x] - x^2*D[P[n - 2, x], {x, 2}] + (4 + 2*x)*P[n - 1, x] + 2*x*D[P[n - 1, x], x]];\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] = -(3 + 4*x + x^2)*v[i - 2] - (4*x + 2*x^2)*deriv(v[i - 2], x) - x^2*deriv(deriv(v[i - 2], x), x) + (4 + 2*x)*v[i - 1] + 2*x*deriv(v[i - 1], x)); 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-a6f3b35778192ffb","steps":["basis A008277"],"edges":["e-b2a33f5e9ab45e0c"],"text":"A056241 \u2192 basis A008277"},"headline":"A056241 \u2192 basis A008277","chain":null}
