{"id":"n-db2d48c86b57e427","revision":0,"first_index":0,"fingerprint":"f6-657752d90232a0f4","name":"derivative of basis_map_A013611 of Riordan array (1-x,x/(1-x)).","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":true,"rows":[["8","32"],["20","128","192"],["48","416","1152","1024"],["112","1216","4800","8192","5120"],["256","3328","16896","41984","51200","24576"],["576","8704","53760","174080","312320","294912","114688"],["1280","22016","159744","634880","1495040","2088960","1605632","524288"],["2816","54272","451584","2121728","6164480","11354112","12959744","8388608","2359296"],["6144","131072","1228800","6651904","22937600","52297728","78905344","76021760","42467328","10485760"],["13312","311296","3244032","19857408","79134720","214695936","401866752","512753664","427032576","209715200","46137344"],["28672","729088","8355840","57016320","257556480","809238528","1805647872","2862612480","3161456640","2317352960","1015021568","201326592"],["61440","1687552","21086208","158597120","800194560","2854748160","7387742208","13979615232","19204669440","18685624320","12226396160","4831838208","872415232"],["131072","3866624","52297728","429654016","2393374720","9550430208","28097642496","61731766272","101317607424","122725335040","106669539328","63015223296","22682796032","3758096384"],["278528","8781824","127795200","1138229248","6935019520","30589059072","100763959296","251909898240","480428163072","695751147520","753330552832","591497527296","318431559680","105226698752","16106127360"],["589824","19791872","308281344","2958032896","19561185280","94466211840","344276860928","964455038976","2094450868224","3526738575360","4567966154752","4470255648768","3200019070976","1582158577664","483183820800","68719476736"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["40","340","2640","19440","138240","959040","6531840","43856640","291133440","1914762240","12496343040","81024675840","522427760640","3352244797440","21419538186240"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["8","32"]],"recurrence":{"denominator":{"coeffs":[["3","4"],["1","4"]]},"terms":[{"coeff":{"coeffs":[["8","32","32"],["2","12","16"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":184374,"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-c7e19c83e65c2de8"]},{"id":184375,"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-c7e19c83e65c2de8"]},{"id":184376,"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-c7e19c83e65c2de8"]},{"id":184377,"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-c7e19c83e65c2de8"]}],"region_exact":{},"incoming_edges":[{"id":"e-c7e19c83e65c2de8","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-db2d48c86b57e427","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":"acec6246f11587dc316f46f5ec74d1c5615cdc3f9db30439144605ea4e37cdbf","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-84c761192e0db3c0","role":"sequence","name":"basis_map_A013611 of Riordan array (1-x,x/(1-x)).","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:     8,     32;\n  1:    20,    128,     192;\n  2:    48,    416,    1152,     1024;\n  3:   112,   1216,    4800,     8192,     5120;\n  4:   256,   3328,   16896,    41984,    51200,     24576;\n  5:   576,   8704,   53760,   174080,   312320,    294912,    114688;\n  6:  1280,  22016,  159744,   634880,  1495040,   2088960,   1605632,    524288;\n  7:  2816,  54272,  451584,  2121728,  6164480,  11354112,  12959744,   8388608,   2359296;\n  8:  6144, 131072, 1228800,  6651904, 22937600,  52297728,  78905344,  76021760,  42467328,  10485760;\n  9: 13312, 311296, 3244032, 19857408, 79134720, 214695936, 401866752, 512753664, 427032576, 209715200, 46137344;\n","recurrence_sentence":"The row polynomials satisfy (n+3+(4*n+4)*x)*P(n,x) = (2*n+8+(12*n+32)*x+(16*n+32)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 8 + 32*x.","mathematica":"P[0, x_] := 8 + 32*x;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n + 8 + (12*n + 32)*x + (16*n + 32)*x^2)*P[n - 1, x])/(n + 3 + (4*n + 4)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 8 + 32*x; for(n = 1, N, my(i = n + 1); v[i] = ((2*n + 8 + (12*n + 32)*x + (16*n + 32)*x^2)*v[i - 1])/(n + 3 + (4*n + 4)*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-7717563017cb9e5f","steps":["basis A013611","D"],"edges":["e-d3bf3843e93b142d","e-c7e19c83e65c2de8"],"text":"A159854 \u2192 basis A013611 \u2192 D"},"headline":"A159854 \u2192 basis A013611 \u2192 D","chain":null}