{"id":"n-7213f12fdfdd9709","revision":0,"first_index":0,"fingerprint":"f6-ea55a274263d0861","name":"basis_map_A013610 of Triangle read by rows: T(n,k) is k-th entry of the gamma-vector of the n-dimensional cyclohedron, 0 <= k <= floor(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":false,"rows":[["1"],["1"],["3","6"],["7","18"],["19","72","54"],["51","240","270"],["141","810","1350","540"],["393","2646","5670","3780"],["1107","8568","22680","22680","5670"],["3139","27432","86184","113400","51030"],["8953","87210","317520","521640","357210","61236"],["25653","275550","1140480","2245320","2058210","673596"],["73789","866448","4018410","9230760","10665270","5388768","673596"],["212941","2713464","13938210","36602280","51081030","35026992","8756748"],["616227","8469006","47729682","141080940","231080850","201405204","78810732","7505784"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","1","9","25","145","561","2841","12489","60705","281185","1353769","6418809","30917041","148331665","716698425"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013610","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["1"]],"recurrence":{"denominator":{"coeffs":[["0"],["1"]]},"terms":[{"coeff":{"coeffs":[["-3","-12"],["3","12"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-1"],["2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":201741,"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-65e4455e79974bb5"]},{"id":201742,"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-65e4455e79974bb5"]},{"id":201743,"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-65e4455e79974bb5"]},{"id":201744,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-65e4455e79974bb5"]}],"region_exact":{},"incoming_edges":[{"id":"e-65e4455e79974bb5","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-7213f12fdfdd9709","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":"910a37a2ae8e09b25706506863942806c0bdf97c07e1155f89e9f499bf0fd05e","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-8bfdd58279be42d7","role":"sequence","name":"Triangle read by rows: T(n,k) is k-th entry of the gamma-vector of the n-dimensional cyclohedron, 0 <= k <= floor(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-5bb6668f9d21fdca","kind":"fingerprint_related","node_a_id":"n-7213f12fdfdd9709","node_b_id":"n-2a11fc404a876fb1","fingerprint":"f6-ea55a274263d0861","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-ea55a274263d0861"},"other_id":"n-2a11fc404a876fb1","other":{"id":"n-2a11fc404a876fb1","name":"basis_map_A013610 of gamma_polynomial of Triangle T(n, k) of numbers of square lattice walks that start and end at origin after 2*n steps and contain exactly k steps to the east, possibly touching origin at intermediate stages.","first_index":0,"fingerprint":"f6-ea55a274263d0861","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2"],["18","36"]],"card_rows":[["1"],["2"],["18","36"],["140","360"]],"row_sums_preview":["1","2","54","500","10150","141372","2625084","42862248"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-fdc03fad975441d0","steps":["\u03b3","basis A013610"],"edges":["e-510754e51c3ada00","e-df1ab89ac4c01da9"],"text":"A069466 \u2192 \u03b3 \u2192 basis A013610"},"headline":"A069466 \u2192 \u03b3 \u2192 basis A013610","chain":null}},{"id":"l-f0bcc93002b616ad","kind":"fingerprint_related","node_a_id":"n-a25c84ed241587e6","node_b_id":"n-7213f12fdfdd9709","fingerprint":"f6-ea55a274263d0861","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-ea55a274263d0861"},"other_id":"n-a25c84ed241587e6","other":{"id":"n-a25c84ed241587e6","name":"basis_map_A013610 of T(n,k) = binomial(n,2*k)*binomial(2*k,k) for 0 <= k <= n, triangle read by rows.","first_index":1,"fingerprint":"f6-ea55a274263d0861","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1"],["3","6"]],"card_rows":[["1"],["1"],["3","6"],["7","18"]],"row_sums_preview":["1","1","9","25","145","561","2841","12489"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4b168caf0bc63f8b","steps":["basis A013610"],"edges":["e-6d32a50c7e7f7f1a"],"text":"A089627 \u2192 basis A013610"},"headline":"A089627 \u2192 basis A013610","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,     6;\n  3:    7,    18;\n  4:   19,    72,    54;\n  5:   51,   240,   270;\n  6:  141,   810,  1350,    540;\n  7:  393,  2646,  5670,   3780;\n  8: 1107,  8568, 22680,  22680,  5670;\n  9: 3139, 27432, 86184, 113400, 51030;\n","recurrence_sentence":"The row polynomials satisfy n*P(n,x) = (3*n-3+(12*n-12)*x)*P(n-2,x) + (2*n-1)*P(n-1,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[Together[((3*n - 3 + (12*n - 12)*x)*P[n - 2, x] + (2*n - 1)*P[n - 1, x])/(n)]];\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*n - 3 + (12*n - 12)*x)*v[i - 2] + (2*n - 1)*v[i - 1])/(n)); 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-8bfdd58279be42d7","steps":["basis A013610"],"edges":["e-65e4455e79974bb5"],"text":"A391403 \u2192 basis A013610"},"headline":"A391403 \u2192 basis A013610","chain":null}
