{"id":"n-db86dad16b91ae0d","revision":0,"first_index":0,"fingerprint":"f6-3d61edfdb7d83d14","name":"derivative of basis_map_A013609 of Even row Pascal-square read by antidiagonals.","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":[[],["4"],["20","24"],["76","176","96"],["260","888","960","320"],["840","3792","6144","4160","960"],["2616","14688","31776","32704","15680","2688"],["7940","53328","144480","200320","147840","53760","7168"],["23644","184872","602112","1051840","1054080","596736","172032","18432"],["69380","619024","2355840","4969280","6308800","4884480","2214912","522240","46080"],["201220","2017416","8786592","21730240","33445440","32968320","20579328","7704576","1520640","112640"],["578064","6433824","31561728","89566592","162126720","194460672","155251712","80492544","25445376","4280320","270336"],["1647600","20157120","109982400","352297600","733539200","1037702400","1011512320","673587200","296524800","80537600","11714560","638976"],["4664836","62222880","373788480","1334099200","3141734400","5124026880","5906503680","4817479680","2734617600","1039667200","246005760","31309824","1490944"],["13132580","189678264","1244006400","4895708800","12867456000","23779929600","31669693440","30621542400","21352780800","10506496000","3496796160","729071616","82001920","3440640"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","4","44","348","2428","15896","100152","614836","3703748","21990036","129066412","750467888","4329841776","24817611364","141451734620"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["4"]],"recurrence":{"denominator":{"coeffs":[["2"],["-6"],["4"]]},"terms":[{"coeff":{"coeffs":[["-2","-4"],["-16","-28"],["16","24"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","0"],["-5","-18","-16"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":180071,"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-75de35b230d3b348"]},{"id":180072,"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-75de35b230d3b348"]},{"id":180073,"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-75de35b230d3b348"]}],"region_exact":{},"incoming_edges":[{"id":"e-75de35b230d3b348","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-db86dad16b91ae0d","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":"48356519ee696bbf9d60f6b9c0356df3dac2a9fc2b822712a863609d9ed2ff6d","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-7c7a913d07444057","role":"sequence","name":"basis_map_A013609 of Even row Pascal-square read by antidiagonals.","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:     4;\n  2:    20,     24;\n  3:    76,    176,      96;\n  4:   260,    888,     960,     320;\n  5:   840,   3792,    6144,    4160,     960;\n  6:  2616,  14688,   31776,   32704,   15680,    2688;\n  7:  7940,  53328,  144480,  200320,  147840,   53760,    7168;\n  8: 23644, 184872,  602112, 1051840, 1054080,  596736,  172032,  18432;\n  9: 69380, 619024, 2355840, 4969280, 6308800, 4884480, 2214912, 522240, 46080;\n","recurrence_sentence":"The row polynomials satisfy (4*n^2-6*n+2)*P(n,x) = (16*n^2-16*n-2+(24*n^2-28*n-4)*x)*P(n-1,x) - (5*n+18*n*x+16*n*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 4;\nP[n_, x_] := P[n, x] = Expand[Together[((16*n^2 - 16*n - 2 + (24*n^2 - 28*n - 4)*x)*P[n - 1, x] - (5*n + 18*n*x + 16*n*x^2)*D[P[n - 1, x], x])/(4*n^2 - 6*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] = 4; for(n = 2, N, my(i = n + 1); v[i] = ((16*n^2 - 16*n - 2 + (24*n^2 - 28*n - 4)*x)*v[i - 1] - (5*n + 18*n*x + 16*n*x^2)*deriv(v[i - 1], x))/(4*n^2 - 6*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-79ebafb266da3b41","steps":["basis A013609","D"],"edges":["e-1a56a7670c2d3c41","e-75de35b230d3b348"],"text":"A172431 \u2192 basis A013609 \u2192 D"},"headline":"A172431 \u2192 basis A013609 \u2192 D","chain":null}