{"id":"n-af70f5feba9b6c29","revision":0,"first_index":0,"fingerprint":"f6-c7214c48a77bf119","name":"derivative of reversal of Number triangle T(n,k)=C(n,k)C(2,n-k).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"irregular","degree_slope":null,"degree_intercept":null,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[[],["2"],["4","2"],["6","6"],["8","12"],["10","20"],["12","30"],["14","42"],["16","56"],["18","72"],["20","90"],["22","110"],["24","132"],["26","156"],["28","182"]],"degrees":[-1,0,1,1,1,1,1,1,1,1,1,1,1,1,1],"row_sums":["0","2","6","12","20","30","42","56","72","90","110","132","156","182","210"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["2"],["4","2"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["-3"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["3"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":171269,"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-448920caad879473"]},{"id":171270,"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-448920caad879473"]},{"id":171271,"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-448920caad879473"]}],"region_exact":{},"incoming_edges":[{"id":"e-448920caad879473","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-af70f5feba9b6c29","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":"36b0fd23fc4f52a72b7edefd2a45fd2a9f6e6af3086c55d19d382792642fe8d6","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-88b9d8361c4b11db","role":"sequence","name":"reversal of Number triangle T(n,k)=C(n,k)C(2,n-k).","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:  2;\n  2:  4,  2;\n  3:  6,  6;\n  4:  8, 12;\n  5: 10, 20;\n  6: 12, 30;\n  7: 14, 42;\n  8: 16, 56;\n  9: 18, 72;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = P(n-3,x) - 3*P(n-2,x) + 3*P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 2, P(2,x) = 4 + 2*x.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 2;\nP[2, x_] := 4 + 2*x;\nP[n_, x_] := P[n, x] = Expand[P[n - 3, x] - 3*P[n - 2, x] + 3*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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