{"id":"n-ff283d31b1efde95","revision":1,"first_index":0,"fingerprint":"f6-f60f31b5a675bca5","name":"basis_map_A008277 of Triangle, read by rows, where T(n,k) = n!/(k!*(n-4*k)!*4^k) for n>=4*k>=0.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"irregular","degree_slope":null,"degree_intercept":null,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1"],["1"],["1"],["7","6"],["31","30"],["91","90"],["211","210"],["1681","4200","1260"],["12097","34776","11340"],["57961","171360","56700"],["209881","625680","207900"],["1874071","10605870","8108100","1247400"],["17842111","118382550","98918820","16216200"],["117303187","805951146","684864180","113513400"]],"degrees":[0,0,0,0,1,1,1,1,2,2,2,2,3,3,3],"row_sums":["1","1","1","1","13","61","181","421","7141","58213","286021","1043461","21835441","251359681","1721631913"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A008277","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["1"],["1"],["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-6","-6"],["11","11"],["-6","-6"],["1","1"]]},"deriv_order":0,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["0","-6"],["0","11"],["0","-6"],["0","1"]]},"deriv_order":1,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162279,"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-510f97d2d65daac7"]},{"id":162280,"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-510f97d2d65daac7"]},{"id":162281,"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-510f97d2d65daac7"]}],"region_exact":{},"incoming_edges":[{"id":"e-510f97d2d65daac7","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-ff283d31b1efde95","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":"85d0ff144313b167cb8f901564af2dc6c05748d4f7f78a359a66914c0ab2f2e0","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-fa138429dacf7586","role":"sequence","name":"Triangle, read by rows, where T(n,k) = n!/(k!*(n-4*k)!*4^k) for n>=4*k>=0.","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:     1;\n  2:     1;\n  3:     1;\n  4:     7,     6;\n  5:    31,    30;\n  6:    91,    90;\n  7:   211,   210;\n  8:  1681,  4200,  1260;\n  9: 12097, 34776, 11340;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (n^3-6*n^2+11*n-6+(n^3-6*n^2+11*n-6)*x)*P(n-4,x) + (n^3-6*n^2+11*n-6)*x*d/dx P(n-4,x) + P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1, P(2,x) = 1, P(3,x) = 1.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[2, x_] := 1;\nP[3, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(n^3 - 6*n^2 + 11*n - 6 + (n^3 - 6*n^2 + 11*n - 6)*x)*P[n - 4, x] + (n^3 - 6*n^2 + 11*n - 6)*x*D[P[n - 4, x], x] + P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; v[3] = 1; v[4] = 1; for(n = 4, N, my(i = n + 1); v[i] = (n^3 - 6*n^2 + 11*n - 6 + (n^3 - 6*n^2 + 11*n - 6)*x)*v[i - 4] + (n^3 - 6*n^2 + 11*n - 6)*x*deriv(v[i - 4], x) + v[i - 1]); 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-fa138429dacf7586","steps":["basis A008277"],"edges":["e-510f97d2d65daac7"],"text":"A118933 \u2192 basis A008277"},"headline":"A118933 \u2192 basis A008277","chain":null}