{"id":"n-1e0f3b745edf2f86","revision":0,"first_index":0,"fingerprint":"f6-579059b7d2bdb7eb","name":"derivative of Triangle of coefficients of polynomials v(n,x) jointly generated with A208508; see the Formula section.","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"],["5"],["14","2"],["30","14"],["55","54","3"],["91","154","27"],["140","364","132","4"],["204","756","468","44"],["285","1428","1350","260","5"],["385","2508","3366","1100","65"],["506","4158","7524","3740","450","6"],["650","6578","15444","10868","2210","90"],["819","10010","29601","28028","8645","714","7"]],"degrees":[-1,-1,0,0,1,1,2,2,3,3,4,4,5,5,6],"row_sums":["0","0","1","5","16","44","112","272","640","1472","3328","7424","16384","35840","77824"],"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":[["-12","-2"],["14","-1"],["-4","1"]]},"terms":[{"coeff":{"coeffs":[["0","-16","0"],["2","21","-3"],["-4","-3","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","8","-8","0"],["0","-8","10","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":172272,"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-94f9892e0eb7737f"]},{"id":172273,"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-94f9892e0eb7737f"]},{"id":172274,"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-94f9892e0eb7737f"]}],"region_exact":{},"incoming_edges":[{"id":"e-94f9892e0eb7737f","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-1e0f3b745edf2f86","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":"7ea520c956498ba6f9c16798b70574ca43ad39be63c8e43321c1ffc61f5e8d3b","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-d4c5acf24bc40277","role":"sequence","name":"Triangle of coefficients of polynomials v(n,x) jointly generated with A208508; see the Formula section.","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:   5;\n  4:  14,   2;\n  5:  30,  14;\n  6:  55,  54,   3;\n  7:  91, 154,  27;\n  8: 140, 364, 132,  4;\n  9: 204, 756, 468, 44;\n","recurrence_sentence":"The row polynomials satisfy -(4*n^2-14*n+12-(n^2-n-2)*x)*P(n,x) = -(4*n^2-2*n+(3*n^2-21*n+16)*x-(n^2-3*n)*x^2)*P(n-1,x) - ((8*n-8)*x-(10*n-8)*x^2+2*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[(-(4*n^2 - 2*n + (3*n^2 - 21*n + 16)*x - (n^2 - 3*n)*x^2)*P[n - 1, x] - ((8*n - 8)*x - (10*n - 8)*x^2 + 2*n*x^3)*D[P[n - 1, x], x])/(-4*n^2 + 14*n - 12 + (n^2 - n - 2)*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] = (-(4*n^2 - 2*n + (3*n^2 - 21*n + 16)*x - (n^2 - 3*n)*x^2)*v[i - 1] - ((8*n - 8)*x - (10*n - 8)*x^2 + 2*n*x^3)*deriv(v[i - 1], x))/(-4*n^2 + 14*n - 12 + (n^2 - n - 2)*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-d4c5acf24bc40277","steps":["D"],"edges":["e-94f9892e0eb7737f"],"text":"A208509 \u2192 D"},"headline":"A208509 \u2192 D","chain":null}