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DELTA (0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":-2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[[],[],["1"],["3"],["7","2"],["15","8"],["30","24","3"],["58","62","15"],["109","146","54","4"],["201","324","162","24"],["365","688","435","100","5"],["655","1414","1083","340","35"],["1164","2832","2550","1020","165","6"],["2052","5556","5754","2804","625","48"],["3593","10716","12552","7224","2065","252","7"]],"degrees":[-1,-1,0,0,1,1,2,2,3,3,4,4,5,5,6],"row_sums":["0","0","1","3","9","23","57","135","313","711","1593","3527","7737","16839","36409"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["0"],["1"]],"recurrence":{"denominator":{"coeffs":[["-14","-4"],["9","0"],["-1","1"]]},"terms":[{"coeff":{"coeffs":[["-30","-24","0"],["23","10","-6"],["-3","1","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-15","-27","-12","0"],["5","4","-5","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":171775,"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-ff4a6837ba6ee6c3"]},{"id":171776,"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-ff4a6837ba6ee6c3"]},{"id":171777,"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-ff4a6837ba6ee6c3"]}],"region_exact":{},"incoming_edges":[{"id":"e-ff4a6837ba6ee6c3","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-91dbf3ed4f4cdb7e","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":"43c0166e4673603b32f16a719fea88c3b71dd0297b56de753eaa45084926de3b","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-72cbbb2f152c8f9c","role":"sequence","name":"Triangle T(n,k), read by rows, given by (2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.","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:   0;\n  2:   1;\n  3:   3;\n  4:   7,   2;\n  5:  15,   8;\n  6:  30,  24,   3;\n  7:  58,  62,  15;\n  8: 109, 146,  54,  4;\n  9: 201, 324, 162, 24;\n","recurrence_sentence":"The row polynomials satisfy -(n^2-9*n+14-(n^2-4)*x)*P(n,x) = -(3*n^2-23*n+30-(n^2+10*n-24)*x-(2*n^2-6*n)*x^2)*P(n-1,x) + (5*n-15+(4*n-27)*x-(5*n+12)*x^2-4*n*x^3)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 0;\nP[2, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(3*n^2 - 23*n + 30 - (n^2 + 10*n - 24)*x - (2*n^2 - 6*n)*x^2)*P[n - 1, x] + (5*n - 15 + (4*n - 27)*x - (5*n + 12)*x^2 - 4*n*x^3)*D[P[n - 1, x], x])/(-n^2 + 9*n - 14 + (n^2 - 4)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 0; v[3] = 1; for(n = 3, N, my(i = n + 1); v[i] = (-(3*n^2 - 23*n + 30 - (n^2 + 10*n - 24)*x - (2*n^2 - 6*n)*x^2)*v[i - 1] + (5*n - 15 + (4*n - 27)*x - (5*n + 12)*x^2 - 4*n*x^3)*deriv(v[i - 1], x))/(-n^2 + 9*n - 14 + (n^2 - 4)*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-72cbbb2f152c8f9c","steps":["D"],"edges":["e-ff4a6837ba6ee6c3"],"text":"A209599 \u2192 D"},"headline":"A209599 \u2192 D","chain":null}