{"id":"n-702e0082e91994b4","revision":1,"first_index":1,"fingerprint":"f6-69966eafa97dc573","name":"veronese_section_2_1 of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":-2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[[],["1"],["1"],["0","1"],["0","3"],["0","3","1"],["0","1","5"],["0","0","10","1"],["0","0","10","7"],["0","0","5","21","1"],["0","0","1","35","9"],["0","0","0","35","36","1"],["0","0","0","21","84","11"],["0","0","0","7","126","55","1"],["0","0","0","1","126","165","13"]],"degrees":[-1,0,0,1,1,2,2,3,3,4,4,5,5,6,6],"row_sums":["0","1","1","1","3","4","6","11","17","27","45","72","116","189","305"],"nonn":true,"definitions":[{"position":0,"source":"derived:veronese_section_2_1","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["0"],["1"],["1"],["0","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","2"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["0","1"]]},"deriv_order":0,"offset":4,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162977,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-7b39f86f218b1f84"]},{"id":162978,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-7b39f86f218b1f84"]},{"id":162979,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-7b39f86f218b1f84"]}],"region_exact":{},"incoming_edges":[{"id":"e-7b39f86f218b1f84","operation":{"index_shift":0,"modulus":2,"operation":"veronese_section","residue":1},"operation_detail":"r=2, j=1, shift 0","conclusion_id":"n-702e0082e91994b4","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":"bcd65a3e20fa100b420fe60e3814754ef3e3d929c3e5f58e7abc697971f131ae","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-392f49b496f79fbf","role":"sequence","name":"Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-85062e87ffa08e62","kind":"fingerprint_related","node_a_id":"n-702e0082e91994b4","node_b_id":"n-579c11d996f64808","fingerprint":"f6-69966eafa97dc573","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-69966eafa97dc573"},"other_id":"n-579c11d996f64808","other":{"id":"n-579c11d996f64808","name":"veronese_section_2_0 of Triangle read by rows: T(n,k) = binomial(k,n-k).","first_index":0,"fingerprint":"f6-69966eafa97dc573","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1"],["0","1"]],"card_rows":[["1"],["1"],["0","1"],["0","3"]],"row_sums_preview":["1","1","1","3","4","6","11","17"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-21854f4257effb01","steps":["Veronese 2"],"edges":["e-3ba5b4221c802eaa"],"text":"A030528 \u2192 Veronese 2"},"headline":"A030528 \u2192 Veronese 2","chain":null}},{"id":"l-9a25592ead4e0f97","kind":"oeis_match","node_a_id":"n-702e0082e91994b4","node_b_id":null,"fingerprint":null,"oeis_id":"A093040","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":2,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"target":"row_sums","terms_compared":14},"in_catalog":false,"oeis_id":"A093040","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A093040","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-702e0082e91994b4","oeis_id":"A093040","target":"row_sums","oeis_offset":0,"first_row":2,"rows_used":200,"terms_matched":14,"oeis_terms_agreeing":39,"oeis_terms_available":39,"term_count":200,"raw_size":5104}],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 0;\n   2: 1;\n   3: 1;\n   4: 0, 1;\n   5: 0, 3;\n   6: 0, 3,  1;\n   7: 0, 1,  5;\n   8: 0, 0, 10,  1;\n   9: 0, 0, 10,  7;\n  10: 0, 0,  5, 21, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = x*P(n-2,x) + 2*x*P(n-3,x) + x*P(n-4,x) for n >= 5, with P(1,x) = 0, P(2,x) = 1, P(3,x) = 1, P(4,x) = x.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 1;\nP[3, x_] := 1;\nP[4, x_] := x;\nP[n_, x_] := P[n, x] = Expand[x*P[n - 2, x] + 2*x*P[n - 3, x] + x*P[n - 4, x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; v[3] = 1; v[4] = x; for(n = 5, N, my(i = n); v[i] = x*v[i - 2] + 2*x*v[i - 3] + 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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-392f49b496f79fbf","steps":["Veronese 2 (j=1)"],"edges":["e-7b39f86f218b1f84"],"text":"A026729 \u2192 Veronese 2 (j=1)"},"headline":"A026729 \u2192 Veronese 2 (j=1)","chain":null}