{"id":"n-70c7c76d08f033c7","revision":0,"first_index":1,"fingerprint":"f6-2a56fc2b007d406c","name":"shift_x_plus_one of 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"],["1","1"],["3","3"],["4","5","1"],["6","11","5"],["11","23","13","1"],["17","41","31","7"],["27","77","74","25","1"],["45","143","160","71","9"],["72","254","331","189","41","1"],["116","454","677","467","139","11"],["189","806","1342","1081","416","61","1"],["305","1410","2604","2415","1146","243","13"]],"degrees":[-1,0,0,1,1,2,2,3,3,4,4,5,5,6,6],"row_sums":["0","1","1","2","6","10","22","48","96","204","428","888","1864","3896","8136"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["0"],["1"],["1"],["1","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","1"]]},"deriv_order":0,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["2","2"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["1","1"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":171122,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-5dbcb37b55d2a09c"]},{"id":171123,"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-5dbcb37b55d2a09c"]},{"id":171124,"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-5dbcb37b55d2a09c"]}],"region_exact":{},"incoming_edges":[{"id":"e-5dbcb37b55d2a09c","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-70c7c76d08f033c7","premise_count":1,"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"730987d2400cd57cf627980b09f576ae485ceae72e769f5364631b5c6a6cac4b","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"-1"}}}],"premises":[{"position":0,"node_id":"n-702e0082e91994b4","role":"sequence","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.","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-dfc33e3026892f88","kind":"fingerprint_related","node_a_id":"n-70c7c76d08f033c7","node_b_id":"n-a82559089062a78c","fingerprint":"f6-2a56fc2b007d406c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2a56fc2b007d406c"},"other_id":"n-a82559089062a78c","other":{"id":"n-a82559089062a78c","name":"shift_x_plus_one of veronese_section_2_0 of Triangle read by rows: T(n,k) = binomial(k,n-k).","first_index":0,"fingerprint":"f6-2a56fc2b007d406c","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1"],["1","1"]],"card_rows":[["1"],["1"],["1","1"],["3","3"]],"row_sums_preview":["1","1","2","6","10","22","48","96"],"regions":["(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-21854f4257effb01","steps":["Veronese 2","x \u21a6 x+1"],"edges":["e-3ba5b4221c802eaa","e-6d965e704a735b5a"],"text":"A030528 \u2192 Veronese 2 \u2192 x \u21a6 x+1"},"headline":"A030528 \u2192 Veronese 2 \u2192 x \u21a6 x+1","chain":null}}],"oeis":[],"bfiles":[],"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:  1,  1;\n   5:  3,  3;\n   6:  4,  5,  1;\n   7:  6, 11,  5;\n   8: 11, 23, 13,  1;\n   9: 17, 41, 31,  7;\n  10: 27, 77, 74, 25, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+x)*P(n-4,x) + (2+2*x)*P(n-3,x) + (1+x)*P(n-2,x) for n >= 5, with P(1,x) = 0, P(2,x) = 1, P(3,x) = 1, P(4,x) = 1 + x.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 1;\nP[3, x_] := 1;\nP[4, x_] := 1 + x;\nP[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 4, x] + (2 + 2*x)*P[n - 3, x] + (1 + x)*P[n - 2, 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] = 1 + x; for(n = 5, N, my(i = n); v[i] = (1 + x)*v[i - 4] + (2 + 2*x)*v[i - 3] + (1 + 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, 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)","x \u21a6 x+1"],"edges":["e-12539302e0d545d2","e-5dbcb37b55d2a09c"],"text":"A026729 \u2192 Veronese 2 (j=1) \u2192 x \u21a6 x+1"},"headline":"A026729 \u2192 Veronese 2 (j=1) \u2192 x \u21a6 x+1","chain":null}
