{"id":"n-711cbd693420f053","revision":0,"first_index":0,"fingerprint":"f6-afdf9d915441c1e3","name":"one_plus_derivative of basis_map_A013611 of A convolution triangle of numbers based on A001906 (even-indexed Fibonacci numbers).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["8","4"],["47","64","16"],["240","568","384","64"],["1137","3904","4560","2048","256"],["5144","23308","39424","30464","10240","1024"],["22559","127360","284256","317440","183040","49152","4096"],["96736","654960","1822464","2683520","2232320","1026048","229376","16384"],["407841","3222912","10763616","19757056","21651200","14336000","5476352","1048576","65536"],["1696808","15337236","59834880","131846912","180045824","156914688","86245376","28180480","4718592","262144"],["6984655","71095232","317508336","817555456","1340120320","1454702592","1051574272","493879296","140967936","20971520","1048576"],["28500688","322662056","1623929984","4788635840","9173155840","11949973504","10754064384","6638075904","2721054720","689438720","92274688","4194304"],["115448337","1439128256","8061397680","26800435200","58803750400","89516998656","96651362304","74236035072","39969423360","14533263360","3310354432","402653184","16777216"],["464754744","6325975580","39041264640","144519528960","357578813440","623508977664","786568052736","724619624448","485255282688","231623884800","75665244160","15653142528","1744830464","67108864"],["1860988735","27465393920","185188713920","755570708480","2082077155840","4094399709184","5916068487168","6366780456960","5112306794496","3033899466752","1300404109312","385607532544","73064775680","7516192768","268435456"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","12","127","1256","11905","109604","987903","8761808","76729089","665082940","5716408191","48785960632","413857027457","3492636485716","29342478921215"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0","0","0"],["10","8","-8"],["0","0","0"],["8","0","0"]]},"terms":[{"coeff":{"coeffs":[["24","20","-16","-16"],["56","52","-16","-16"],["48","48","0","0"],["16","16","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["3","-4","-40","-48","-16"],["12","32","16","0","0"],["12","32","16","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":183025,"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-2a76a9e79102a2c9"]},{"id":183026,"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-2a76a9e79102a2c9"]},{"id":183027,"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-2a76a9e79102a2c9"]},{"id":183028,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-2a76a9e79102a2c9"]}],"region_exact":{},"incoming_edges":[{"id":"e-2a76a9e79102a2c9","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-711cbd693420f053","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":"c2885bd33e846bbcc6dd18c3c69c96e371d8571d2fd39d40fecfaef2c67479db","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-1f449c58b1930aa5","role":"sequence","name":"basis_map_A013611 of A convolution triangle of numbers based on A001906 (even-indexed Fibonacci numbers).","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:       1;\n  1:       8,        4;\n  2:      47,       64,       16;\n  3:     240,      568,      384,        64;\n  4:    1137,     3904,     4560,      2048,       256;\n  5:    5144,    23308,    39424,     30464,     10240,      1024;\n  6:   22559,   127360,   284256,    317440,    183040,     49152,     4096;\n  7:   96736,   654960,  1822464,   2683520,   2232320,   1026048,   229376,    16384;\n  8:  407841,  3222912, 10763616,  19757056,  21651200,  14336000,  5476352,  1048576,   65536;\n  9: 1696808, 15337236, 59834880, 131846912, 180045824, 156914688, 86245376, 28180480, 4718592, 262144;\n","recurrence_sentence":"The row polynomials satisfy (8*n^3+10*n+8*n*x-8*n*x^2)*P(n,x) = (16*n^3+48*n^2+56*n+24+(16*n^3+48*n^2+52*n+20)*x-(16*n+16)*x^2-(16*n+16)*x^3)*P(n-1,x) + (12*n^2+12*n+3+(32*n^2+32*n-4)*x+(16*n^2+16*n-40)*x^2-48*x^3-16*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((16*n^3 + 48*n^2 + 56*n + 24 + (16*n^3 + 48*n^2 + 52*n + 20)*x - (16*n + 16)*x^2 - (16*n + 16)*x^3)*P[n - 1, x] + (12*n^2 + 12*n + 3 + (32*n^2 + 32*n - 4)*x + (16*n^2 + 16*n - 40)*x^2 - 48*x^3 - 16*x^4)*D[P[n - 1, x], x])/(8*n^3 + 10*n + 8*n*x - 8*n*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((16*n^3 + 48*n^2 + 56*n + 24 + (16*n^3 + 48*n^2 + 52*n + 20)*x - (16*n + 16)*x^2 - (16*n + 16)*x^3)*v[i - 1] + (12*n^2 + 12*n + 3 + (32*n^2 + 32*n - 4)*x + (16*n^2 + 16*n - 40)*x^2 - 48*x^3 - 16*x^4)*deriv(v[i - 1], x))/(8*n^3 + 10*n + 8*n*x - 8*n*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-f287bed89e07609f","steps":["basis A013611","D+1"],"edges":["e-362ae2b737f320ab","e-2a76a9e79102a2c9"],"text":"A125662 \u2192 basis A013611 \u2192 D+1"},"headline":"A125662 \u2192 basis A013611 \u2192 D+1","chain":null}
