{"id":"n-d5de2886ab0383d8","revision":0,"first_index":1,"fingerprint":"f6-f49167bc79a7af36","name":"derivative of basis_map_A013611 of Coefficients of a special case of Poisson-Charlier polynomials.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[[],["4"],["20","32"],["100","352","192"],["576","3424","5376","1024"],["3844","33728","98112","96256","5120"],["29252","347008","1530432","2973696","2155520","24576"],["250100","3780192","22693632","67593216","99819520","58466304","114688"],["2374944","43884480","336001536","1363782656","3093176320","3714662400","1844183040","524288"],["24817476","544079232","5089291392","26320902144","81254661120","149660073984","152203591680","65893564416","2359296"],["283182868","7201752288","79837445760","503789876224","1978626984960","4951225098240","7706162659328","6817737867264","2623756566528","10485760"],["3505127300","101613880160","1305270189888","9749103208448","46650185815040","148272141262848","312945486708736","422820573085696","331722205691904","115093746483200","46137344"],["46791847840","1524941039392","22306348080384","192831334782976","1090797840793600","4218181526347776","11290903982833664","20652060327280640","24697087727763456","17431622141870080","5512550208765952","201326592"],["670315041348","24281509760320","398919679851456","3923128601079808","25657825456445440","117159217809309696","381074272749076480","882765032836825088","1427006640254091264","1532743792040345600","984266444986384384","286194337912455168","872415232"],["10258917877636","409199335653440","7466370577396416","82398602020060160","612505071135631360","3230462830986928128","12394288213231583232","34849905233257562112","71262433151512215552","103332648697158696960","100836625014167961600","59446290213123391488","16007371420925427712","3758096384"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","4","52","644","10400","237060","7060484","252717652","10398589664","480993340740","24668631919220","1388663877590564","85109913372732352","5641214565023081284","402062814280108482180"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["0"],["4"],["20","32"]],"recurrence":{"denominator":{"coeffs":[["-16"],["8"]]},"terms":[{"coeff":{"coeffs":[["264","3456","14976","21504"],["-232","-3168","-14208","-20992"],["72","960","4224","6144"],["-8","-96","-384","-512"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["24","408","2592","7296","7680"],["-23","-400","-2592","-7424","-7936"],["8","136","864","2432","2560"],["-1","-16","-96","-256","-256"]]},"deriv_order":1,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["-120","-896","-1536"],["104","832","1536"],["-24","-192","-384"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-8","-136","-736","-1280"],["13","192","944","1536"],["-6","-84","-384","-576"],["1","12","48","64"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["40","256"],["-48","-256"],["16","64"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-8","-72","-160"],["5","48","112"],["-1","-8","-16"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":188904,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-97ced682499ca77e"]},{"id":188905,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-97ced682499ca77e"]},{"id":188906,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-97ced682499ca77e"]}],"region_exact":{},"incoming_edges":[{"id":"e-97ced682499ca77e","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-d5de2886ab0383d8","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":"e61a200bae2e6f9e755fc3eb74af6afcb263fcd816557d587014936ab1903617","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-d300994ed88c0a37","role":"sequence","name":"basis_map_A013611 of Coefficients of a special case of Poisson-Charlier polynomials.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:        0;\n   2:        4;\n   3:       20,        32;\n   4:      100,       352,        192;\n   5:      576,      3424,       5376,        1024;\n   6:     3844,     33728,      98112,       96256,        5120;\n   7:    29252,    347008,    1530432,     2973696,     2155520,        24576;\n   8:   250100,   3780192,   22693632,    67593216,    99819520,     58466304,       114688;\n   9:  2374944,  43884480,  336001536,  1363782656,  3093176320,   3714662400,   1844183040,      524288;\n  10: 24817476, 544079232, 5089291392, 26320902144, 81254661120, 149660073984, 152203591680, 65893564416, 2359296;\n","recurrence_sentence":"The row polynomials satisfy (8*n-16)*P(n,x) = -(8*n^3-72*n^2+232*n-264+(96*n^3-960*n^2+3168*n-3456)*x+(384*n^3-4224*n^2+14208*n-14976)*x^2+(512*n^3-6144*n^2+20992*n-21504)*x^3)*P(n-3,x) - (n^3-8*n^2+23*n-24+(16*n^3-136*n^2+400*n-408)*x+(96*n^3-864*n^2+2592*n-2592)*x^2+(256*n^3-2432*n^2+7424*n-7296)*x^3+(256*n^3-2560*n^2+7936*n-7680)*x^4)*d/dx P(n-3,x) - (24*n^2-104*n+120+(192*n^2-832*n+896)*x+(384*n^2-1536*n+1536)*x^2)*P(n-2,x) + (n^3-6*n^2+13*n-8+(12*n^3-84*n^2+192*n-136)*x+(48*n^3-384*n^2+944*n-736)*x^2+(64*n^3-576*n^2+1536*n-1280)*x^3)*d/dx P(n-2,x) + (16*n^2-48*n+40+(64*n^2-256*n+256)*x)*P(n-1,x) - (n^2-5*n+8+(8*n^2-48*n+72)*x+(16*n^2-112*n+160)*x^2)*d/dx P(n-1,x) for n >= 4, with P(1,x) = 0, P(2,x) = 4, P(3,x) = 20 + 32*x.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 4;\nP[3, x_] := 20 + 32*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(8*n^3 - 72*n^2 + 232*n - 264 + (96*n^3 - 960*n^2 + 3168*n - 3456)*x + (384*n^3 - 4224*n^2 + 14208*n - 14976)*x^2 + (512*n^3 - 6144*n^2 + 20992*n - 21504)*x^3)*P[n - 3, x] - (n^3 - 8*n^2 + 23*n - 24 + (16*n^3 - 136*n^2 + 400*n - 408)*x + (96*n^3 - 864*n^2 + 2592*n - 2592)*x^2 + (256*n^3 - 2432*n^2 + 7424*n - 7296)*x^3 + (256*n^3 - 2560*n^2 + 7936*n - 7680)*x^4)*D[P[n - 3, x], x] - (24*n^2 - 104*n + 120 + (192*n^2 - 832*n + 896)*x + (384*n^2 - 1536*n + 1536)*x^2)*P[n - 2, x] + (n^3 - 6*n^2 + 13*n - 8 + (12*n^3 - 84*n^2 + 192*n - 136)*x + (48*n^3 - 384*n^2 + 944*n - 736)*x^2 + (64*n^3 - 576*n^2 + 1536*n - 1280)*x^3)*D[P[n - 2, x], x] + (16*n^2 - 48*n + 40 + (64*n^2 - 256*n + 256)*x)*P[n - 1, x] - (n^2 - 5*n + 8 + (8*n^2 - 48*n + 72)*x + (16*n^2 - 112*n + 160)*x^2)*D[P[n - 1, x], x])/(8*n - 16)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 4; v[3] = 20 + 32*x; for(n = 4, N, my(i = n); v[i] = (-(8*n^3 - 72*n^2 + 232*n - 264 + (96*n^3 - 960*n^2 + 3168*n - 3456)*x + (384*n^3 - 4224*n^2 + 14208*n - 14976)*x^2 + (512*n^3 - 6144*n^2 + 20992*n - 21504)*x^3)*v[i - 3] - (n^3 - 8*n^2 + 23*n - 24 + (16*n^3 - 136*n^2 + 400*n - 408)*x + (96*n^3 - 864*n^2 + 2592*n - 2592)*x^2 + (256*n^3 - 2432*n^2 + 7424*n - 7296)*x^3 + (256*n^3 - 2560*n^2 + 7936*n - 7680)*x^4)*deriv(v[i - 3], x) - (24*n^2 - 104*n + 120 + (192*n^2 - 832*n + 896)*x + (384*n^2 - 1536*n + 1536)*x^2)*v[i - 2] + (n^3 - 6*n^2 + 13*n - 8 + (12*n^3 - 84*n^2 + 192*n - 136)*x + (48*n^3 - 384*n^2 + 944*n - 736)*x^2 + (64*n^3 - 576*n^2 + 1536*n - 1280)*x^3)*deriv(v[i - 2], x) + (16*n^2 - 48*n + 40 + (64*n^2 - 256*n + 256)*x)*v[i - 1] - (n^2 - 5*n + 8 + (8*n^2 - 48*n + 72)*x + (16*n^2 - 112*n + 160)*x^2)*deriv(v[i - 1], x))/(8*n - 16)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-c9dce8c25621fdea","steps":["basis A013611","D"],"edges":["e-5274dd88a300ad29","e-97ced682499ca77e"],"text":"A046716 \u2192 basis A013611 \u2192 D"},"headline":"A046716 \u2192 basis A013611 \u2192 D","chain":null}