{"id":"n-57a09c7aaa187100","revision":1,"first_index":0,"fingerprint":"f6-ec1b6da9b0cf6ba2","name":"gamma_polynomial of Triangle T(n, k) = (m*(n-k) + 1)*T(n-1, k-1) + (m*(k-1) + 1)*T(n-1, k) + j*T(n-2, k-1), where T(n, 1) = T(n, n) = 1, m = 1, and j = 4, read by rows.","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","6"],["1","20"],["1","50","64"],["1","112","472"],["1","238","2288","1200"],["1","492","9216","15840"],["1","1002","33520","127808","36480"],["1","2024","114552","816256","756992"],["1","4070","376000","4544624","9193728","1659904"],["1","8164","1201216","23148704","85590656","49762304"],["1","16354","3766880","110915840","677618816","848892672","106164224"],["1","32736","11660376","508738304","4811678976","10856669184","4337769472"],["1","65502","35766736","2259907504","31624394496","115967280128","98899972096","9100195840"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","1","7","21","115","585","3727","25549","198811","1689825","15778327","159711045","1747374787","20526549049","257887582303"],"nonn":true,"definitions":[{"position":0,"source":"derived:gamma_polynomial","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","-2","0"],["0","2","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1","-4"],["0","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","4","0"],["0","0","0"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162543,"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-94133232adc31a68"]},{"id":162544,"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-94133232adc31a68"]},{"id":162545,"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-94133232adc31a68"]}],"region_exact":{},"incoming_edges":[{"id":"e-94133232adc31a68","operation":{"index_shift":0,"operation":"gamma_polynomial","scale":"1"},"operation_detail":"scale 1, shift 0","conclusion_id":"n-57a09c7aaa187100","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":"4ef13180f5589c9060591a8f4f262598d2688c495a6f79a3894af68a72653842","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-9189beb5ff99695e","role":"sequence","name":"Triangle T(n, k) = (m*(n-k) + 1)*T(n-1, k-1) + (m*(k-1) + 1)*T(n-1, k) + j*T(n-2, k-1), where T(n, 1) = T(n, n) = 1, m = 1, and j = 4, read by rows.","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,    6;\n  3: 1,   20;\n  4: 1,   50,     64;\n  5: 1,  112,    472;\n  6: 1,  238,   2288,   1200;\n  7: 1,  492,   9216,  15840;\n  8: 1, 1002,  33520, 127808,  36480;\n  9: 1, 2024, 114552, 816256, 756992;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+(2*n-2)*x)*P(n-1,x) + (x-4*x^2)*d/dx P(n-1,x) + 4*x*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(1 + (2*n - 2)*x)*P[n - 1, x] + (x - 4*x^2)*D[P[n - 1, x], x] + 4*x*P[n - 2, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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