{"id":"n-4bf8d7c41abf0ae8","revision":0,"first_index":0,"fingerprint":"f6-e945b9f95949136c","name":"derivative of Triangle read by rows; T(n,k) is the number of achiral rows of n colors using exactly k colors.","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","4"],["1","12","18"],["1","12","18"],["1","28","108","96"],["1","28","108","96"],["1","60","450","960","600"],["1","60","450","960","600"],["1","124","1620","6240","9000","4320"],["1","124","1620","6240","9000","4320"],["1","252","5418","33600","84000","90720","35280"],["1","252","5418","33600","84000","90720","35280"]],"degrees":[-1,0,0,1,1,2,2,3,3,4,4,5,5,6,6],"row_sums":["0","1","1","5","5","31","31","233","233","2071","2071","21305","21305","249271","249271"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["1"],["1"],["1","4"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","0","-2"]]},"deriv_order":0,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["0","0","-1","-1"]]},"deriv_order":1,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["1","4"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","1","1"]]},"deriv_order":1,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":171399,"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-6ad7c47b1d9ca509"]},{"id":171400,"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-6ad7c47b1d9ca509"]},{"id":171401,"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-6ad7c47b1d9ca509"]}],"region_exact":{},"incoming_edges":[{"id":"e-6ad7c47b1d9ca509","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"0\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-4bf8d7c41abf0ae8","premise_count":1,"theorem":"Laguerre's theorem on differential operators: if phi is a real polynomial with only real roots, then phi(D) maps every real-rooted polynomial to a real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"e4fff54e0d20debb60756d606b26311f8c62f5c02d804f12e928bb2a7565cdba","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-9e68379063b83f9d","role":"sequence","name":"Triangle read by rows; T(n,k) is the number of achiral rows of n colors using exactly k colors.","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: 0;\n  1: 1;\n  2: 1;\n  3: 1,  4;\n  4: 1,  4;\n  5: 1, 12,  18;\n  6: 1, 12,  18;\n  7: 1, 28, 108,  96;\n  8: 1, 28, 108,  96;\n  9: 1, 60, 450, 960, 600;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -2*x^2*P(n-4,x) - (x^2+x^3)*d/dx P(n-4,x) + (1+4*x)*P(n-2,x) + (x+x^2)*d/dx P(n-2,x) for n >= 4, with P(0,x) = 0, P(1,x) = 1, P(2,x) = 1, P(3,x) = 1 + 4*x.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[2, x_] := 1;\nP[3, x_] := 1 + 4*x;\nP[n_, x_] := P[n, x] = Expand[-2*x^2*P[n - 4, x] - (x^2 + x^3)*D[P[n - 4, x], x] + (1 + 4*x)*P[n - 2, x] + (x + x^2)*D[P[n - 2, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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