{"id":"n-a52cc17175b71643","revision":1,"first_index":0,"fingerprint":"f5-7e0c00f93492814d","name":"veronese_section_2_1 of Triangle read by rows: T(n,k) is the number of ternary words of length n having degree of asymmetry equal to k (n>=0; 0<=k<=n/2).","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":[[],[],["6"],["18"],["36"],["108"],["162","216"],["486","648"],["648","2592"],["1944","7776"],["2430","19440","7776"],["7290","58320","23328"],["8748","116640","139968"],["26244","349920","419904"],["30618","612360","1469664","279936"]],"degrees":[-1,-1,0,0,0,0,1,1,1,1,2,2,2,2,3],"row_sums":["0","0","6","18","36","108","378","1134","3240","9720","29646","88938","265356","796068","2392578"],"nonn":true,"definitions":[{"position":0,"source":"derived:veronese_section_2_1","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["0"],["6"],["18"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["6","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-9","36"]]},"deriv_order":0,"offset":4,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":168350,"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-2c5dd3537a50c97c"]},{"id":168351,"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-2c5dd3537a50c97c"]},{"id":168352,"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-2c5dd3537a50c97c"]}],"region_exact":{},"incoming_edges":[{"id":"e-2c5dd3537a50c97c","operation":{"index_shift":0,"modulus":2,"operation":"veronese_section","residue":1},"operation_detail":"r=2, j=1, shift 0","conclusion_id":"n-a52cc17175b71643","premise_count":1,"theorem":"Veronese sections of Polya frequency sequences (Aissen--Schoenberg--Whitney--Edrei): if P is real-rooted with coefficients of one sign, equivalently nonzero with all roots in (-inf, 0], every section V_(r,j)(P) is zero or real-rooted with roots in (-inf, 0]","lean_route":null,"lean_declarations":[],"certificate_hash":"9e686468a285f308485a2ef636d88b950aba3f264ed8cab5303c528987bd6675","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-7c641d81e21514ce","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of ternary words of length n having degree of asymmetry equal to k (n>=0; 0<=k<=n/2).","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-6976d5a793799254","kind":"fingerprint_related","node_a_id":"n-737bf0cfa519ec4d","node_b_id":"n-a52cc17175b71643","fingerprint":"f5-7e0c00f93492814d","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f5-7e0c00f93492814d"},"other_id":"n-737bf0cfa519ec4d","other":{"id":"n-737bf0cfa519ec4d","name":"veronese_section_2_1 of Triangle T(n,k) read by rows: number of size-k antisymmetric subsets of {1,2,...,n}.","first_index":0,"fingerprint":"f5-7e0c00f93492814d","polytool_family":null,"degree_pattern":"irregular","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],[],["2"]],"card_rows":[["2"],["2"],["4"],["4"]],"row_sums_preview":["0","0","2","2","4","4","14","14"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ba3bb18490b15e2e","steps":["Veronese 2 (j=1)"],"edges":["e-4266cbd95404e739"],"text":"A188440 \u2192 Veronese 2 (j=1)"},"headline":"A188440 \u2192 Veronese 2 (j=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:    0;\n  1:    0;\n  2:    6;\n  3:   18;\n  4:   36;\n  5:  108;\n  6:  162,  216;\n  7:  486,  648;\n  8:  648, 2592;\n  9: 1944, 7776;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = 6*P(n-2,x) - (9-36*x)*P(n-4,x) for n >= 4, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 6, P(3,x) = 18.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 0;\nP[2, x_] := 6;\nP[3, x_] := 18;\nP[n_, x_] := P[n, x] = Expand[6*P[n - 2, x] - (9 - 36*x)*P[n - 4, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 0; v[3] = 6; v[4] = 18; for(n = 4, N, my(i = n + 1); v[i] = 6*v[i - 2] - (9 - 36*x)*v[i - 4]); 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-7c641d81e21514ce","steps":["Veronese 2 (j=1)"],"edges":["e-2c5dd3537a50c97c"],"text":"A274498 \u2192 Veronese 2 (j=1)"},"headline":"A274498 \u2192 Veronese 2 (j=1)","chain":null}