{"id":"n-2526cbe057b697bc","revision":0,"first_index":0,"fingerprint":"f6-8e10c2d09761267e","name":"derivative of theta_plus_one of Interpolation operator: Triangle with an even number of zeros in each line followed by 1 or 2 ones.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[[],["2"],["0","6"],["0","6","12"],["0","0","0","20"],["0","0","0","20","30"],["0","0","0","0","0","42"],["0","0","0","0","0","42","56"],["0","0","0","0","0","0","0","72"],["0","0","0","0","0","0","0","72","90"],["0","0","0","0","0","0","0","0","0","110"],["0","0","0","0","0","0","0","0","0","110","132"],["0","0","0","0","0","0","0","0","0","0","0","156"],["0","0","0","0","0","0","0","0","0","0","0","156","182"],["0","0","0","0","0","0","0","0","0","0","0","0","0","210"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","2","6","18","20","50","42","98","72","162","110","242","156","338","210"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["2"],["0","6"]],"recurrence":{"denominator":{"coeffs":[["2"],["-3"],["1"]]},"terms":[{"coeff":{"coeffs":[["6","-6"],["-11","-3"],["6","7"],["-1","-2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2","-2"],["0","-3","-2"],["0","1","2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":171015,"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-b99ffe0e49f77b90"]},{"id":171016,"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-b99ffe0e49f77b90"]},{"id":171017,"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-b99ffe0e49f77b90"]}],"region_exact":{},"incoming_edges":[{"id":"e-b99ffe0e49f77b90","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-2526cbe057b697bc","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":"4a66dd6bd445e3e88da8e6c111ea7097c3c9a0564a808d0d0eba4e88c1a91545","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-9f865b2dcffd9592","role":"sequence","name":"theta_plus_one of Interpolation operator: Triangle with an even number of zeros in each line followed by 1 or 2 ones.","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: 0, 6;\n  3: 0, 6, 12;\n  4: 0, 0,  0, 20;\n  5: 0, 0,  0, 20, 30;\n  6: 0, 0,  0,  0,  0, 42;\n  7: 0, 0,  0,  0,  0, 42, 56;\n  8: 0, 0,  0,  0,  0,  0,  0, 72;\n  9: 0, 0,  0,  0,  0,  0,  0, 72, 90;\n","recurrence_sentence":"The row polynomials satisfy (n^2-3*n+2)*P(n,x) = -(n^3-6*n^2+11*n-6+(2*n^3-7*n^2+3*n+6)*x)*P(n-1,x) + ((n^2-3*n+2)*x+(2*n^2-2*n-2)*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 2, P(2,x) = 6*x.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 2;\nP[2, x_] := 6*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^3 - 6*n^2 + 11*n - 6 + (2*n^3 - 7*n^2 + 3*n + 6)*x)*P[n - 1, x] + ((n^2 - 3*n + 2)*x + (2*n^2 - 2*n - 2)*x^2)*D[P[n - 1, x], x])/(n^2 - 3*n + 2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; v[3] = 6*x; for(n = 3, N, my(i = n + 1); v[i] = (-(n^3 - 6*n^2 + 11*n - 6 + (2*n^3 - 7*n^2 + 3*n + 6)*x)*v[i - 1] + ((n^2 - 3*n + 2)*x + (2*n^2 - 2*n - 2)*x^2)*deriv(v[i - 1], x))/(n^2 - 3*n + 2)); 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-ea2139b8dbf6bf8d","steps":["\u03b8+1","D"],"edges":["e-246a0576b6d0a23b","e-b99ffe0e49f77b90"],"text":"A133080 \u2192 \u03b8+1 \u2192 D"},"headline":"A133080 \u2192 \u03b8+1 \u2192 D","chain":null}