{"id":"n-6f0ac3e1273260af","revision":0,"first_index":0,"fingerprint":"f6-980b18e5562a0336","name":"derivative of veronese_section_2_1 of Triangular array binomial(2*n-k, k), k=0..n, n >= 0.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":-3,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[[],[],[],["1"],["10"],["35","2"],["84","42"],["165","252","3"],["286","924","108"],["455","2574","990","4"],["680","6006","5148","220"],["969","12376","19305","2860","5"],["1330","23256","58344","20020","390"],["1771","40698","151164","97240","6825","6"],["2300","67298","348840","369512","61880","630"]],"degrees":[-1,-1,-1,0,0,1,1,2,2,3,3,4,4,5,5],"row_sums":["0","0","0","1","10","37","126","420","1318","4023","12054","35515","103340","297704","850460"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["0"],["0"],["1"],["10"],["35","2"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","0","12","1"]]},"deriv_order":0,"offset":6,"sign":"none"},{"coeff":{"coeffs":[["0","0","0","4"]]},"deriv_order":1,"offset":6,"sign":"none"},{"coeff":{"coeffs":[["0","-24","-15"]]},"deriv_order":0,"offset":5,"sign":"none"},{"coeff":{"coeffs":[["0","0","-12","-4"]]},"deriv_order":1,"offset":5,"sign":"none"},{"coeff":{"coeffs":[["12","3","-3"]]},"deriv_order":0,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["0","12"]]},"deriv_order":1,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["27","10"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["-4","-4","8"]]},"deriv_order":1,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["-11","3"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["8","12"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["5"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-4","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":172761,"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-a610d38b236f0ca3"]},{"id":172762,"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-a610d38b236f0ca3"]},{"id":172763,"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-a610d38b236f0ca3"]}],"region_exact":{},"incoming_edges":[{"id":"e-a610d38b236f0ca3","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-6f0ac3e1273260af","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":"9bc94b321a873a75927f6e79e5b37a2d82bfa2490bfcdb70578644ed9935ed93","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-3d82b1c51e4bade1","role":"sequence","name":"veronese_section_2_1 of Triangular array binomial(2*n-k, k), k=0..n, n >= 0.","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:   0;\n  3:   1;\n  4:  10;\n  5:  35,    2;\n  6:  84,   42;\n  7: 165,  252,   3;\n  8: 286,  924, 108;\n  9: 455, 2574, 990, 4;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (12*x^2+x^3)*P(n-6,x) + 4*x^3*d/dx P(n-6,x) - (24*x+15*x^2)*P(n-5,x) - (12*x^2+4*x^3)*d/dx P(n-5,x) + (12+3*x-3*x^2)*P(n-4,x) + 12*x*d/dx P(n-4,x) + (27+10*x)*P(n-3,x) - (4+4*x-8*x^2)*d/dx P(n-3,x) - (11-3*x)*P(n-2,x) + (8+12*x)*d/dx P(n-2,x) + 5*P(n-1,x) - (4+4*x)*d/dx P(n-1,x) for n >= 6, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 0, P(3,x) = 1, P(4,x) = 10, P(5,x) = 35 + 2*x.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 0;\nP[2, x_] := 0;\nP[3, x_] := 1;\nP[4, x_] := 10;\nP[5, x_] := 35 + 2*x;\nP[n_, x_] := P[n, x] = Expand[(12*x^2 + x^3)*P[n - 6, x] + 4*x^3*D[P[n - 6, x], x] - (24*x + 15*x^2)*P[n - 5, x] - (12*x^2 + 4*x^3)*D[P[n - 5, x], x] + (12 + 3*x - 3*x^2)*P[n - 4, x] + 12*x*D[P[n - 4, x], x] + (27 + 10*x)*P[n - 3, x] - (4 + 4*x - 8*x^2)*D[P[n - 3, x], x] - (11 - 3*x)*P[n - 2, x] + (8 + 12*x)*D[P[n - 2, x], x] + 5*P[n - 1, x] - (4 + 4*x)*D[P[n - 1, x], 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] = 0; v[4] = 1; v[5] = 10; v[6] = 35 + 2*x; for(n = 6, N, my(i = n + 1); v[i] = (12*x^2 + x^3)*v[i - 6] + 4*x^3*deriv(v[i - 6], x) - (24*x + 15*x^2)*v[i - 5] - (12*x^2 + 4*x^3)*deriv(v[i - 5], x) + (12 + 3*x - 3*x^2)*v[i - 4] + 12*x*deriv(v[i - 4], x) + (27 + 10*x)*v[i - 3] - (4 + 4*x - 8*x^2)*deriv(v[i - 3], x) - (11 - 3*x)*v[i - 2] + (8 + 12*x)*deriv(v[i - 2], x) + 5*v[i - 1] - (4 + 4*x)*deriv(v[i - 1], 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-6758fe7729845be8","steps":["Veronese 2 (j=1)","D"],"edges":["e-33a38ab7b1aab5c6","e-a610d38b236f0ca3"],"text":"A054142 \u2192 Veronese 2 (j=1) \u2192 D"},"headline":"A054142 \u2192 Veronese 2 (j=1) \u2192 D","chain":null}