{"id":"n-5d8bbcaae52422ed","revision":1,"first_index":1,"fingerprint":"f6-f394a67368746b91","name":"gamma_polynomial of Triangle read by rows: Eulerian numbers of type B, T(n,k) (1 <= k <= n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (2*n - 2*k + 1)*T(n-1, k-1) + (2*k - 1)*T(n-1, k).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1"],["1","4"],["1","20"],["1","72","80"],["1","232","976"],["1","716","7664","3904"],["1","2172","49776","88640"],["1","6544","292320","1217792","354560"],["1","19664","1618656","13201664","12933376"],["1","59028","8643872","124784768","274820352","51733504"],["1","177124","45108256","1080946304","4469939456","2767631360"],["1","531416","231917744","8829655296","61848381184","84083218432","11070525440"],["1","1594296","1180845360","69228954880","768547157760","1914489501696","816582578176"],["1","4782940","5974375824","527113117120","8855335156480","36430327673856","33589447536640","3266330312704"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","1","5","21","153","1209","12285","140589","1871217","27773361","460041525","8363802501","166064229513","3570030632169","82674532955565"],"nonn":true,"definitions":[{"position":0,"source":"derived:gamma_polynomial","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","-8","0"],["0","4","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2","-8"],["0","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162555,"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-af71d95d59cd25b7"]},{"id":162556,"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-af71d95d59cd25b7"]},{"id":162557,"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-af71d95d59cd25b7"]}],"region_exact":{},"incoming_edges":[{"id":"e-af71d95d59cd25b7","operation":{"index_shift":0,"operation":"gamma_polynomial","scale":"1"},"operation_detail":"scale 1, shift 0","conclusion_id":"n-5d8bbcaae52422ed","premise_count":1,"theorem":"gamma criterion (RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff): if P = sum_j gamma_j x^j (1 + x)^(d - 2j) with deg Gamma <= d/2, then P is nonzero and real-rooted with roots in (-inf, 0] if and only if Gamma(z) = sum_j gamma_j z^j is; Gamma(c z) with c != 0 has the roots of Gamma divided by c","lean_route":null,"lean_declarations":[],"certificate_hash":"9a35c4f0ba471dbc2066fecc55c49595afe78f7c7d7b14608ff61924b773276e","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-1be5eefbe5bc79ec","role":"sequence","name":"Triangle read by rows: Eulerian numbers of type B, T(n,k) (1 <= k <= n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (2*n - 2*k + 1)*T(n-1, k-1) + (2*k - 1)*T(n-1, k).","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-691aa1eb31cfc8cd","kind":"fingerprint_related","node_a_id":"n-5d8bbcaae52422ed","node_b_id":"n-d7237f25d7d0f8de","fingerprint":"f6-f394a67368746b91","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f394a67368746b91"},"other_id":"n-d7237f25d7d0f8de","other":{"id":"n-d7237f25d7d0f8de","name":"gamma_polynomial of Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) =  4*x + 2.","first_index":0,"fingerprint":"f6-f394a67368746b91","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2"],["4","16"]],"card_rows":[["1"],["2"],["4","16"],["8","160"]],"row_sums_preview":["1","2","20","168","2448","38688","786240","17995392"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2c43e713cd32b015","steps":["\u03b3"],"edges":["e-b43c1d901148031a"],"text":"A257612 \u2192 \u03b3"},"headline":"A257612 \u2192 \u03b3","chain":null}},{"id":"l-6e8672446f7b48b3","kind":"fingerprint_related","node_a_id":"n-5d8bbcaae52422ed","node_b_id":"n-a7592a0a48b88a6a","fingerprint":"f6-f394a67368746b91","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f394a67368746b91"},"other_id":"n-a7592a0a48b88a6a","other":{"id":"n-a7592a0a48b88a6a","name":"gamma_polynomial of Triangle read by rows: T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(n) = 6*n + 3.","first_index":0,"fingerprint":"f6-f394a67368746b91","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3"],["9","36"]],"card_rows":[["1"],["3"],["9","36"],["27","540"]],"row_sums_preview":["1","3","45","567","12393","293787","8955765","307468143"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-444025a5f2f6e0e5","steps":["\u03b3"],"edges":["e-1933fd2398425d03"],"text":"A257625 \u2192 \u03b3"},"headline":"A257625 \u2192 \u03b3","chain":null}},{"id":"l-c2e393d403b313cd","kind":"fingerprint_related","node_a_id":"n-5d8bbcaae52422ed","node_b_id":"n-7b489708d21ffcfe","fingerprint":"f6-f394a67368746b91","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f394a67368746b91"},"other_id":"n-7b489708d21ffcfe","other":{"id":"n-7b489708d21ffcfe","name":"gamma_polynomial of type_b_eulerian","first_index":0,"fingerprint":"f6-f394a67368746b91","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1"],["1","4"]],"card_rows":[["1"],["1"],["1","4"],["1","20"]],"row_sums_preview":["1","1","5","21","153","1209","12285","140589"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d8312bb744732845","steps":["\u03b3"],"edges":["e-fcc9671b64420c56"],"text":"type_b_eulerian \u2192 \u03b3"},"headline":"type_b_eulerian \u2192 \u03b3","chain":null}},{"id":"l-4978e0bac22f61c4","kind":"oeis_match","node_a_id":"n-5d8bbcaae52422ed","node_b_id":null,"fingerprint":null,"oeis_id":"A333273","in_catalog":0,"target":"triangle","row_alignment":"node_lengths","rows_matched":11,"oeis_nonn":1,"details":{"alignment":{"first_row":2,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":0,"row_alignment":"node_lengths","rows_matched":11,"target":"triangle","terms_compared":41},"in_catalog":false,"oeis_id":"A333273","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-e1c1cd3322715781","kind":"oeis_match","node_a_id":"n-5d8bbcaae52422ed","node_b_id":null,"fingerprint":null,"oeis_id":"A182825","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A182825","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A182825","relation":"match_row_sums"},{"oeis_id":"A333273","relation":"match_triangle"}],"bfiles":[{"node_id":"n-5d8bbcaae52422ed","oeis_id":"A182825","target":"row_sums","oeis_offset":0,"first_row":1,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":21,"oeis_terms_available":21,"term_count":200,"raw_size":38394},{"node_id":"n-5d8bbcaae52422ed","oeis_id":"A333273","target":"triangle","oeis_offset":1,"first_row":2,"rows_used":61,"terms_matched":41,"oeis_terms_agreeing":41,"oeis_terms_available":41,"term_count":991,"raw_size":52402}],"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: 1;\n   2: 1;\n   3: 1,     4;\n   4: 1,    20;\n   5: 1,    72,      80;\n   6: 1,   232,     976;\n   7: 1,   716,    7664,     3904;\n   8: 1,  2172,   49776,    88640;\n   9: 1,  6544,  292320,  1217792,   354560;\n  10: 1, 19664, 1618656, 13201664, 12933376;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+(4*n-8)*x)*P(n-1,x) + (2*x-8*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(1 + (4*n - 8)*x)*P[n - 1, x] + (2*x - 8*x^2)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (1 + (4*n - 8)*x)*v[i - 1] + (2*x - 8*x^2)*deriv(v[i - 1], x)); 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}},{"oeis_id":"A333273","target":"triangle","first_row":2,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 1;\n   2: 1,     4;\n   3: 1,    20;\n   4: 1,    72,      80;\n   5: 1,   232,     976;\n   6: 1,   716,    7664,      3904;\n   7: 1,  2172,   49776,     88640;\n   8: 1,  6544,  292320,   1217792,    354560;\n   9: 1, 19664, 1618656,  13201664,  12933376;\n  10: 1, 59028, 8643872, 124784768, 274820352, 51733504;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+(4*n-4)*x)*P(n-1,x) + (2*x-8*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(1 + (4*n - 4)*x)*P[n - 1, x] + (2*x - 8*x^2)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (1 + (4*n - 4)*x)*v[i - 1] + (2*x - 8*x^2)*deriv(v[i - 1], x)); 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":2,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-1be5eefbe5bc79ec","steps":["\u03b3"],"edges":["e-af71d95d59cd25b7"],"text":"A060187 \u2192 \u03b3"},"headline":"A060187 \u2192 \u03b3","chain":null}
