{"id":"n-2114ace3c07775a2","revision":1,"first_index":0,"fingerprint":"f6-b047158a8b919e42","name":"basis_map_A013609 of Triangle T(n,k) read by rows: number of size-k antisymmetric subsets of {1,2,...,n}.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["1"],["1"],["3","4"],["3","4"],["9","24","16"],["9","24","16"],["27","108","144","64"],["27","108","144","64"],["81","432","864","768","256"],["81","432","864","768","256"],["243","1620","4320","5760","3840","1024"],["243","1620","4320","5760","3840","1024"],["729","5832","19440","34560","34560","18432","4096"],["729","5832","19440","34560","34560","18432","4096"],["2187","20412","81648","181440","241920","193536","86016","16384"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","1","7","7","49","49","343","343","2401","2401","16807","16807","117649","117649","823543"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013609","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["3","4"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":166294,"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-1a0f394398e61cd4"]},{"id":166295,"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-1a0f394398e61cd4"]},{"id":166296,"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-1a0f394398e61cd4"]}],"region_exact":{},"incoming_edges":[{"id":"e-1a0f394398e61cd4","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-2114ace3c07775a2","premise_count":1,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"4fffb1a0b101b1a4e68e866670be0c6f85820077acdcc5de8c684bb6d43dff45","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-ba3bb18490b15e2e","role":"sequence","name":"Triangle T(n,k) read by rows: number of size-k antisymmetric subsets of {1,2,...,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":[{"id":"l-025912f0bdc70a42","kind":"fingerprint_related","node_a_id":"n-2114ace3c07775a2","node_b_id":"n-612921ba5e07601f","fingerprint":"f6-b047158a8b919e42","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b047158a8b919e42"},"other_id":"n-612921ba5e07601f","other":{"id":"n-612921ba5e07601f","name":"basis_map_A013609 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).","first_index":0,"fingerprint":"f6-b047158a8b919e42","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3"],["9","12"]],"card_rows":[["1"],["3"],["9","12"],["27","36"]],"row_sums_preview":["1","3","21","63","441","1323","9261","27783"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7c641d81e21514ce","steps":["basis A013609"],"edges":["e-8f465536c6a75fc0"],"text":"A274498 \u2192 basis A013609"},"headline":"A274498 \u2192 basis A013609","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:  1;\n  2:  3,   4;\n  3:  3,   4;\n  4:  9,  24,  16;\n  5:  9,  24,  16;\n  6: 27, 108, 144,  64;\n  7: 27, 108, 144,  64;\n  8: 81, 432, 864, 768, 256;\n  9: 81, 432, 864, 768, 256;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3+4*x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(3 + 4*x)*P[n - 2, 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; for(n = 2, N, my(i = n + 1); v[i] = (3 + 4*x)*v[i - 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-ba3bb18490b15e2e","steps":["basis A013609"],"edges":["e-1a0f394398e61cd4"],"text":"A188440 \u2192 basis A013609"},"headline":"A188440 \u2192 basis A013609","chain":null}
