{"id":"n-e491b6b31274204e","revision":0,"first_index":0,"fingerprint":"f6-c0cba542a6350c02","name":"derivative of basis_map_A013609 of Pascal-(1,4,1) array.","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":false,"rows":[[],["2"],["16","8"],["72","112","24"],["320","800","480","64"],["1280","4480","4800","1664","160"],["4992","22400","34560","21504","5120","384"],["18816","103168","209280","193024","81280","14592","896"],["69632","451072","1135104","1398784","880640","274944","39424","2048"],["253440","1894400","5698560","8828928","7513600","3502080","860160","102400","4608"],["911360","7720960","27033600","50556928","54609920","34590720","12615680","2539520","258048","10240"],["3244032","30720000","122757120","269303808","353955840","286384128","142069760","42188800","7170048","634880","22528"],["11452416","119881728","538460160","1356546048","2105180160","2084880384","1325678592","534609920","133152768","19537920","1531904","49152"],["40148992","460357632","2296233984","6535610368","11710586880","13765705728","10757275648","5567021056","1877557248","401244160","51724288","3637248","106496"],["139919360","1744076800","9565962240","30364270592","61746380800","84186071040","78396129280","49981030400","21625012224","6236569600","1164247040","133693440","8519680","229376"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","2","24","208","1664","12384","88960","621056","4251648","28658176","190846976","1258450944","8230961152","53467209728","345292111872"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["2"]],"recurrence":{"denominator":{"coeffs":[["-5","0","2"],["5","-5","-2"],["0","5","0"]]},"terms":[{"coeff":{"coeffs":[["-70","-170","-72","-8"],["105","225","48","0"],["-25","-45","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-25","-110","-145","-52","-4"],["25","100","105","10","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":180280,"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-be28eb177b1686fb"]},{"id":180281,"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-be28eb177b1686fb"]},{"id":180282,"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-be28eb177b1686fb"]}],"region_exact":{},"incoming_edges":[{"id":"e-be28eb177b1686fb","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-e491b6b31274204e","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":"0940d41e9c91123a9331107eea977946a1491452047e8097106ca8410fec894b","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-bf2f08bd0370af8d","role":"sequence","name":"basis_map_A013609 of Pascal-(1,4,1) array.","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:     16,       8;\n  3:     72,     112,      24;\n  4:    320,     800,     480,      64;\n  5:   1280,    4480,    4800,    1664,     160;\n  6:   4992,   22400,   34560,   21504,    5120,     384;\n  7:  18816,  103168,  209280,  193024,   81280,   14592,    896;\n  8:  69632,  451072, 1135104, 1398784,  880640,  274944,  39424,   2048;\n  9: 253440, 1894400, 5698560, 8828928, 7513600, 3502080, 860160, 102400, 4608;\n","recurrence_sentence":"The row polynomials satisfy (5*n-5+(5*n^2-5*n)*x-(2*n-2)*x^2)*P(n,x) = -(25*n^2-105*n+70+(45*n^2-225*n+170)*x-(48*n-72)*x^2+8*x^3)*P(n-1,x) + (25*n-25+(100*n-110)*x+(105*n-145)*x^2+(10*n-52)*x^3-4*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(25*n^2 - 105*n + 70 + (45*n^2 - 225*n + 170)*x - (48*n - 72)*x^2 + 8*x^3)*P[n - 1, x] + (25*n - 25 + (100*n - 110)*x + (105*n - 145)*x^2 + (10*n - 52)*x^3 - 4*x^4)*D[P[n - 1, x], x])/(5*n - 5 + (5*n^2 - 5*n)*x - (2*n - 2)*x^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; for(n = 2, N, my(i = n + 1); v[i] = (-(25*n^2 - 105*n + 70 + (45*n^2 - 225*n + 170)*x - (48*n - 72)*x^2 + 8*x^3)*v[i - 1] + (25*n - 25 + (100*n - 110)*x + (105*n - 145)*x^2 + (10*n - 52)*x^3 - 4*x^4)*deriv(v[i - 1], x))/(5*n - 5 + (5*n^2 - 5*n)*x - (2*n - 2)*x^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-ed9434bfd889c409","steps":["basis A013609","D"],"edges":["e-175e767cc5e2f6ad","e-be28eb177b1686fb"],"text":"A081579 \u2192 basis A013609 \u2192 D"},"headline":"A081579 \u2192 basis A013609 \u2192 D","chain":null}