{"id":"n-28e5dd0a4ca0936e","revision":1,"first_index":0,"fingerprint":"f6-39ed4fb7856d5a32","name":"basis_map_A013611 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":"half_affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["1"],["3"],["9","24"],["27","72"],["81","432","576"],["243","1296","1728"],["729","5832","15552","13824"],["2187","17496","46656","41472"],["6561","69984","279936","497664","331776"],["19683","209952","839808","1492992","995328"],["59049","787320","4199040","11197440","14929920","7962624"],["177147","2361960","12597120","33592320","44789760","23887872"],["531441","8503056","56687040","201553920","403107840","429981696","191102976"],["1594323","25509168","170061120","604661760","1209323520","1289945088","573308928"],["4782969","89282088","714256704","3174474240","8465264640","13544423424","12039487488","4586471424"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","3","33","99","1089","3267","35937","107811","1185921","3557763","39135393","117406179","1291467969","3874403907","42618442977"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013611","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["3"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["9","24"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":167834,"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-5fc7194d1c12a028"]},{"id":167835,"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-5fc7194d1c12a028"]},{"id":167836,"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-5fc7194d1c12a028"]}],"region_exact":{},"incoming_edges":[{"id":"e-5fc7194d1c12a028","operation":{"map":"A013611","operation":"basis_map"},"operation_detail":"A013611","conclusion_id":"n-28e5dd0a4ca0936e","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":"3101a8e5ee4362103c08e1230e75c17ff842199af0c80d314f05ff1ec767fce4","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-5b14745f91ed3c80","kind":"fingerprint_related","node_a_id":"n-8a70fbd3ff1954d6","node_b_id":"n-28e5dd0a4ca0936e","fingerprint":"f6-39ed4fb7856d5a32","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-39ed4fb7856d5a32"},"other_id":"n-8a70fbd3ff1954d6","other":{"id":"n-8a70fbd3ff1954d6","name":"basis_map_A013611 of Triangle T(n,k) read by rows: number of size-k antisymmetric subsets of {1,2,...,n}.","first_index":0,"fingerprint":"f6-39ed4fb7856d5a32","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1"],["3","8"]],"card_rows":[["1"],["1"],["3","8"],["3","8"]],"row_sums_preview":["1","1","11","11","121","121","1331","1331"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ba3bb18490b15e2e","steps":["basis A013611"],"edges":["e-d1c49253f9d90c32"],"text":"A188440 \u2192 basis A013611"},"headline":"A188440 \u2192 basis A013611","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:     3;\n  2:     9,     24;\n  3:    27,     72;\n  4:    81,    432,    576;\n  5:   243,   1296,   1728;\n  6:   729,   5832,  15552,   13824;\n  7:  2187,  17496,  46656,   41472;\n  8:  6561,  69984, 279936,  497664, 331776;\n  9: 19683, 209952, 839808, 1492992, 995328;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (9+24*x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3;\nP[n_, x_] := P[n, x] = Expand[(9 + 24*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] = 3; for(n = 2, N, my(i = n + 1); v[i] = (9 + 24*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-7c641d81e21514ce","steps":["basis A013611"],"edges":["e-5fc7194d1c12a028"],"text":"A274498 \u2192 basis A013611"},"headline":"A274498 \u2192 basis A013611","chain":null}