{"id":"n-e216abf6fe8d06ac","revision":1,"first_index":0,"fingerprint":"f6-1f4ee20ba4318b4a","name":"gamma_polynomial of Triangle read by rows: T(n,k) (1 <= k <= n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*k-m+1)*T(n-1,k), where m = 6.","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"],["1","12"],["1","108"],["1","792","1296"],["1","5592","35856"],["1","39204","667440","430272"],["1","274500","10558512","24193728"],["1","1921584","153730656","839787264","290324736"],["1","13451184","2136852576","23335029504","27413012736"],["1","94158396","28908982944","571576715136","1525386380544","328956152832"],["1","659108892","384855984288","12941404359552","65570341840128","46806913870848"],["1","4613762376","5074311556080","278214585511680","2415742807576320","3811546636240896","561682966450176"],["1","32296336776","66519701714160","5773211034105600","80425020346248960","234113600487131136","112259389028438016"],["1","226074357588","869019238738512","116875137433135680","2495575235521094400","12083022835876002816","12581687011588927488","1347112668341256192"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","1","13","109","2089","41449","1136917","35026741","1285764241","52898346001","2454922389853","125704175163709","7072265921097529","432637772893974649","28625142134073512677"],"nonn":true,"definitions":[{"position":0,"source":"derived:gamma_polynomial","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","-12","0"],["0","12","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","6","-24"],["0","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162639,"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-de18761c144e2c93"]},{"id":162640,"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-de18761c144e2c93"]},{"id":162641,"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-de18761c144e2c93"]}],"region_exact":{},"incoming_edges":[{"id":"e-de18761c144e2c93","operation":{"index_shift":0,"operation":"gamma_polynomial","scale":"1"},"operation_detail":"scale 1, shift 0","conclusion_id":"n-e216abf6fe8d06ac","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":"18ef1bea5704a13af781d41d6dc07204817ba1e9e9058f88db16fd6ab799d42a","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-3bc26d3108ee6bfd","role":"sequence","name":"Triangle read by rows: T(n,k) (1 <= k <= n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*k-m+1)*T(n-1,k), where m = 6.","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":[],"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: 1,       12;\n  3: 1,      108;\n  4: 1,      792,       1296;\n  5: 1,     5592,      35856;\n  6: 1,    39204,     667440,      430272;\n  7: 1,   274500,   10558512,    24193728;\n  8: 1,  1921584,  153730656,   839787264,   290324736;\n  9: 1, 13451184, 2136852576, 23335029504, 27413012736;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+(12*n-12)*x)*P(n-1,x) + (6*x-24*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(1 + (12*n - 12)*x)*P[n - 1, x] + (6*x - 24*x^2)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (1 + (12*n - 12)*x)*v[i - 1] + (6*x - 24*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 - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-3bc26d3108ee6bfd","steps":["\u03b3"],"edges":["e-de18761c144e2c93"],"text":"A142461 \u2192 \u03b3"},"headline":"A142461 \u2192 \u03b3","chain":null}