{"id":"n-e1d160f3b707da31","revision":1,"first_index":0,"fingerprint":"f6-5659ab0d98136444","name":"Riordan array ((1+x)^3,x).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":3,"palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":true,"rows":[["1","3","3","1"],["0","1","3","3","1"],["0","0","1","3","3","1"],["0","0","0","1","3","3","1"],["0","0","0","0","1","3","3","1"],["0","0","0","0","0","1","3","3","1"],["0","0","0","0","0","0","1","3","3","1"],["0","0","0","0","0","0","0","1","3","3","1"],["0","0","0","0","0","0","0","0","1","3","3","1"],["0","0","0","0","0","0","0","0","0","1","3","3","1"],["0","0","0","0","0","0","0","0","0","0","1","3","3","1"],["0","0","0","0","0","0","0","0","0","0","0","1","3","3","1"],["0","0","0","0","0","0","0","0","0","0","0","0","1","3","3","1"],["0","0","0","0","0","0","0","0","0","0","0","0","0","1","3","3","1"],["0","0","0","0","0","0","0","0","0","0","0","0","0","0","1","3","3","1"]],"degrees":[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],"row_sums":["8","8","8","8","8","8","8","8","8","8","8","8","8","8","8"],"nonn":true,"definitions":[{"position":0,"source":"oeis_catalog:A122431","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1","3","3","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":159152,"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-73f8c6c9e8d957af"]},{"id":159153,"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-73f8c6c9e8d957af"]},{"id":159154,"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-73f8c6c9e8d957af"]},{"id":159155,"position":3,"claim":"basis_map_preserver","from_index":null,"region_key":null,"provenance":"edges","claim_data":{"claim":"basis_map_preserver","input_interval":"(-inf, +inf)","output_interval":"(-inf, +inf)","statement":"every real-rooted input has a real-rooted image (an affine substitution times g)"},"label":"Basis map preserves real-rootedness","description":"Basis map u^k -> P_k sends roots in (-inf, +inf) to roots in (-inf, +inf): every real-rooted input has a real-rooted image (an affine substitution times g)","region_exact":null,"edges":["e-468701cdcc848e53"]},{"id":159156,"position":4,"claim":"basis_map_preserver","from_index":null,"region_key":null,"provenance":"edges","claim_data":{"claim":"basis_map_preserver","input_interval":"(-inf, 0]","output_interval":"(-inf, 0]","statement":"an input root r <= 0 gives the root (r - a)/d <= 0; a larger root gives a positive root"},"label":"Basis map preserves real-rootedness","description":"Basis map u^k -> P_k sends roots in (-inf, 0] to roots in (-inf, 0]: an input root r <= 0 gives the root (r - a)/d <= 0; a larger root gives a positive root","region_exact":null,"edges":["e-468701cdcc848e53"]}],"region_exact":{},"incoming_edges":[{"id":"e-468701cdcc848e53","operation":{"operation":"basis_map_theorem"},"operation_detail":null,"conclusion_id":"n-e1d160f3b707da31","premise_count":0,"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":"1149ca7f69460456aabccfa46290ea5106f3afbf6693793e6881df63a5b7a848","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"basis_map_preserver","input_interval":"(-inf, +inf)","output_interval":"(-inf, +inf)","statement":"every real-rooted input has a real-rooted image (an affine substitution times g)"},{"claim":"basis_map_preserver","input_interval":"(-inf, 0]","output_interval":"(-inf, 0]","statement":"an input root r <= 0 gives the root (r - a)/d <= 0; a larger root gives a positive root"}],"premises":[]},{"id":"e-73f8c6c9e8d957af","operation":{"operation":"parent_provers","route":"periodic_product"},"operation_detail":"periodic_product","conclusion_id":"n-e1d160f3b707da31","premise_count":0,"theorem":"polytool parent provers, route periodic_product: every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign","lean_route":null,"lean_declarations":[],"certificate_hash":"a51cc6eabfb7a52d6d64f4a8125f35d5374473604eb49cd8b9cc0f0fa4b42596","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[]}],"used_by":[{"id":"e-02645aa9ee0a8704","operation":"basis_map","operation_detail":"A013610","conclusion_id":"n-0482a458bbc39c21","conclusion_name":"basis_map_A013610 of Riordan array ((1+x)^3,x)."},{"id":"e-09b769bb10417be2","operation":"basis_map","operation_detail":"A007318","conclusion_id":"n-9aa9e3dd2403bca2","conclusion_name":"shift_x_plus_one of Riordan array ((1+x)^3,x)."},{"id":"e-22aa1241157a3ec8","operation":"basis_map","operation_detail":"A008277","conclusion_id":"n-0ddc9ddd9e0aa41f","conclusion_name":"basis_map_A008277 of Riordan array ((1+x)^3,x)."},{"id":"e-4464410b9667d6e5","operation":"basis_map","operation_detail":"A013611","conclusion_id":"n-c77d53cf93e5da02","conclusion_name":"basis_map_A013611 of Riordan array ((1+x)^3,x)."},{"id":"e-5fbd3e16dd967779","operation":"basis_map","operation_detail":"A013609","conclusion_id":"n-48e70e0cceeb235f","conclusion_name":"basis_map_A013609 of Riordan array ((1+x)^3,x)."},{"id":"e-112d4d02aeda07ce","operation":"preserver_image","operation_detail":"mobius, shift 0","conclusion_id":"n-7f9ce95b8e46d30e","conclusion_name":"mobius_x_over_x_plus_one of Riordan array ((1+x)^3,x)."},{"id":"e-56c62e2300be82c0","operation":"preserver_image","operation_detail":"affine, shift 0","conclusion_id":"n-9aa9e3dd2403bca2","conclusion_name":"shift_x_plus_one of Riordan array ((1+x)^3,x)."},{"id":"e-797abd8656cf91d6","operation":"preserver_image","operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-9436aa07118c53db","conclusion_name":"one_plus_derivative of Riordan array ((1+x)^3,x)."},{"id":"e-a87ec3100784b141","operation":"preserver_image","operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-a537a025418ac17d","conclusion_name":"theta_plus_one_squared of Riordan array ((1+x)^3,x)."},{"id":"e-b524f7ca945d9ab3","operation":"preserver_image","operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-59d89a30eaddd429","conclusion_name":"derivative of Riordan array ((1+x)^3,x)."},{"id":"e-c8839bd61c0ac6a5","operation":"preserver_image","operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-8ce0c552efe4fe6e","conclusion_name":"theta_plus_one of Riordan array ((1+x)^3,x)."},{"id":"e-d15d6831373644cf","operation":"preserver_image","operation_detail":"reversal, shift 0","conclusion_id":"n-1bc415e3a319469d","conclusion_name":"reversal of Riordan array ((1+x)^3,x)."},{"id":"e-267d8a8f48506a34","operation":"veronese_section","operation_detail":"r=2, j=1, shift 0","conclusion_id":"n-ab55e05a7226d6fc","conclusion_name":"veronese_section_2_1 of Riordan array ((1+x)^3,x)."},{"id":"e-74f6ac036b23734d","operation":"veronese_section","operation_detail":"r=2, j=0, shift 0","conclusion_id":"n-07c6db3457191f00","conclusion_name":"veronese_section_2_0 of Riordan array ((1+x)^3,x)."}],"used_by_total":14,"links":[{"id":"l-cbef84bdbf931141","kind":"fingerprint_related","node_a_id":"n-e1d160f3b707da31","node_b_id":"n-1bc415e3a319469d","fingerprint":"f6-5659ab0d98136444","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-5659ab0d98136444"},"other_id":"n-1bc415e3a319469d","other":{"id":"n-1bc415e3a319469d","name":"reversal of Riordan array ((1+x)^3,x).","first_index":0,"fingerprint":"f6-5659ab0d98136444","polytool_family":null,"degree_pattern":"affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1","3","3","1"],["1","3","3","1"],["1","3","3","1"]],"card_rows":[["1","3","3","1"],["1","3","3","1"],["1","3","3","1"],["1","3","3","1"]],"row_sums_preview":["8","8","8","8","8","8","8","8"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-e1d160f3b707da31","steps":["reverse"],"edges":["e-d15d6831373644cf"],"text":"A122431 \u2192 reverse"},"headline":"A122431 \u2192 reverse","chain":null}}],"oeis":[{"oeis_id":"A122431","relation":"tail"}],"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, 3, 3, 1;\n  1: 0, 1, 3, 3, 1;\n  2: 0, 0, 1, 3, 3, 1;\n  3: 0, 0, 0, 1, 3, 3, 1;\n  4: 0, 0, 0, 0, 1, 3, 3, 1;\n  5: 0, 0, 0, 0, 0, 1, 3, 3, 1;\n  6: 0, 0, 0, 0, 0, 0, 1, 3, 3, 1;\n  7: 0, 0, 0, 0, 0, 0, 0, 1, 3, 3, 1;\n  8: 0, 0, 0, 0, 0, 0, 0, 0, 1, 3, 3, 1;\n  9: 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 3, 3, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = x*P(n-1,x) for n >= 1, with P(0,x) = 1 + 3*x + 3*x^2 + x^3.","mathematica":"P[0, x_] := 1 + 3*x + 3*x^2 + x^3;\nP[n_, x_] := P[n, x] = Expand[x*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1 + 3*x + 3*x^2 + x^3; for(n = 1, N, my(i = n + 1); v[i] = x*v[i - 1]); 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-e1d160f3b707da31","steps":[],"edges":[],"text":"A122431"},"headline":"Riordan array ((1+x)^3,x).","chain":null}