{"id":"n-dd5e7f7fbd0b1911","revision":0,"first_index":1,"fingerprint":"f6-4b1e3aef476c99e5","name":"one_plus_derivative of Triangle of coefficients of n!*(1 - x)^n*L_n(x/(1 - x)), where L_n(x) is the Laguerre polynomial.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":false,"rows":[["1"],["-1","-2"],["-6","6","7"],["-30","90","-39","-34"],["-168","816","-1128","292","209"],["-1080","7200","-16200","14100","-2505","-1546"],["-7920","66960","-207000","294600","-184230","24306","13327"],["-65520","670320","-2628360","5145000","-5281710","2554566","-263431","-130922"],["-604800","7257600","-34433280","85518720","-120869280","96290880","-37756320","3154824","1441729"],["-6168960","84913920","-472469760","1419949440","-2547145440","2796208128","-1811236896","594890568","-41368977","-17572114"],["-68947200","1070496000","-6831216000","24077088000","-52270142400","72850518720","-65076328800","35400358800","-9975900150","589410910","234662231"],["-838252800","14489798400","-104282640000","421521408000","-1073233022400","1810500310080","-2053895991840","1541873718000","-721295175150","177630009810","-9064804551","-3405357682"],["-11017036800","209802700800","-1680816614400","7662029760000","-22372907356800","44266597562880","-60790675841280","57971133570240","-37448245593000","15340831659120","-3349709211816","149641946796","53334454417"],["-155675520000","3238050816000","-28575798451200","144997217164800","-477474203606400","1083472040851200","-1745235184492800","2014939231081920","-1653829432964280","936066615300960","-340633723939080","66725881283100","-2638693215769","-896324308634"],["-2353813862400","53091579340800","-511692980198400","2859927431961600","-10482764522630400","26826132533840640","-49558839280030080","67116696264858240","-66702333386248560","47980137245760720","-24135790951680408","7892725180046616","-1400509104345078","49490245341642","16083557845279"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","-3","7","-13","21","-31","43","-57","73","-91","111","-133","157","-183","211"],"nonn":false,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["-5","15","-8","1"],["2","-7","2","0"],["0","1","0","0"]]},"terms":[{"coeff":{"coeffs":[["3","-5","-16","29","-14","2"],["-5","11","9","-23","10","-1"],["2","-7","-2","8","-2","0"],["0","1","0","-1","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-2","11","-21","18","-7","1"],["0","2","-9","14","-9","2","0"],["0","0","1","-2","1","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":185718,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, +inf)","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)","region_exact":null,"edges":["e-b38f9fe792a24ed8"]}],"region_exact":{},"incoming_edges":[{"id":"e-b38f9fe792a24ed8","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-dd5e7f7fbd0b1911","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":"a5f57fe236681e0380d0ff5af39f761b3ec9ec2dc70e6320e45b251fe6a49b06","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{}}],"premises":[{"position":0,"node_id":"n-db839974123087ee","role":"sequence","name":"Triangle of coefficients of n!*(1 - x)^n*L_n(x/(1 - x)), where L_n(x) is the Laguerre polynomial.","uses":{"claim":"real_rooted","from_index":1,"region":{}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)"}]}],"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:        1;\n   2:       -1,       -2;\n   3:       -6,        6,          7;\n   4:      -30,       90,        -39,        -34;\n   5:     -168,      816,      -1128,        292,         209;\n   6:    -1080,     7200,     -16200,      14100,       -2505,      -1546;\n   7:    -7920,    66960,    -207000,     294600,     -184230,      24306,       13327;\n   8:   -65520,   670320,   -2628360,    5145000,    -5281710,    2554566,     -263431,   -130922;\n   9:  -604800,  7257600,  -34433280,   85518720,  -120869280,   96290880,   -37756320,   3154824,   1441729;\n  10: -6168960, 84913920, -472469760, 1419949440, -2547145440, 2796208128, -1811236896, 594890568, -41368977, -17572114;\n","recurrence_sentence":"The row polynomials satisfy (2*n-5+(n^2-7*n+15)*x+(2*n-8)*x^2+x^3)*P(n,x) = (2*n^2-5*n+3+(n^3-7*n^2+11*n-5)*x-(2*n^2-9*n+16)*x^2-(n^3-8*n^2+23*n-29)*x^3-(2*n^2-10*n+14)*x^4-(n-2)*x^5)*P(n-1,x) + ((2*n-2)*x+(n^2-9*n+11)*x^2-(2*n^2-14*n+21)*x^3+(n^2-9*n+18)*x^4+(2*n-7)*x^5+x^6)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n^2 - 5*n + 3 + (n^3 - 7*n^2 + 11*n - 5)*x - (2*n^2 - 9*n + 16)*x^2 - (n^3 - 8*n^2 + 23*n - 29)*x^3 - (2*n^2 - 10*n + 14)*x^4 - (n - 2)*x^5)*P[n - 1, x] + ((2*n - 2)*x + (n^2 - 9*n + 11)*x^2 - (2*n^2 - 14*n + 21)*x^3 + (n^2 - 9*n + 18)*x^4 + (2*n - 7)*x^5 + x^6)*D[P[n - 1, x], x])/(2*n - 5 + (n^2 - 7*n + 15)*x + (2*n - 8)*x^2 + x^3)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((2*n^2 - 5*n + 3 + (n^3 - 7*n^2 + 11*n - 5)*x - (2*n^2 - 9*n + 16)*x^2 - (n^3 - 8*n^2 + 23*n - 29)*x^3 - (2*n^2 - 10*n + 14)*x^4 - (n - 2)*x^5)*v[i - 1] + ((2*n - 2)*x + (n^2 - 9*n + 11)*x^2 - (2*n^2 - 14*n + 21)*x^3 + (n^2 - 9*n + 18)*x^4 + (2*n - 7)*x^5 + x^6)*deriv(v[i - 1], x))/(2*n - 5 + (n^2 - 7*n + 15)*x + (2*n - 8)*x^2 + x^3)); 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-db839974123087ee","steps":["D+1"],"edges":["e-b38f9fe792a24ed8"],"text":"A123202 \u2192 D+1"},"headline":"A123202 \u2192 D+1","chain":null}