{"id":"n-84ea1b0f8a235f9e","revision":1,"first_index":1,"fingerprint":"f6-1e8466a1ae3eeaa5","name":"one_plus_derivative of Riordan array (1, 3+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":false,"rows":[["1"],["3","3"],["1","19","9"],["0","12","87","27"],["0","2","82","351","81"],["0","0","27","441","1323","243"],["0","0","3","217","2079","4779","729"],["0","0","0","48","1362","9018","16767","2187"],["0","0","0","4","451","7380","36936","57591","6561"],["0","0","0","0","75","3255","36261","145071","194643","19683"],["0","0","0","0","5","811","19980","166131","551853","649539","59049"],["0","0","0","0","0","108","6633","109809","722196","2047032","2145447","177147"],["0","0","0","0","0","6","1324","45549","556794","3013686","7440174","7026831","531441"],["0","0","0","0","0","0","147","12117","277074","2655018","12170655","26591733","22851963","1594323"],["0","0","0","0","0","0","7","2017","92106","1541106","12061305","47849373","93710763","73870299","4782969"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","6","29","126","516","2034","7807","29382","108923","398988","1447368","5208372","18615805","66153030","233909945"],"nonn":true,"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":[["-6","-24","96","-27"],["15","39","-123","27"],["-12","-15","27","0"],["3","0","0","0"]]},"terms":[{"coeff":{"coeffs":[["16","68","-20","-171","81"],["-28","-117","-26","63","0"],["14","37","-18","0","0"],["-2","0","0","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","8","30","-29","-90","81"],["0","-12","-47","-9","81","0"],["0","4","9","0","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":164247,"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-a3e4828d005b5462"]},{"id":164248,"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-a3e4828d005b5462"]},{"id":164249,"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-a3e4828d005b5462"]}],"region_exact":{},"incoming_edges":[{"id":"e-a3e4828d005b5462","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-84ea1b0f8a235f9e","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":"08755a4f890d0bb3044dfdb463c89059d70d939ae2c7133d0eb3a9eef0cf4b61","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-5d39164c4108e043","role":"sequence","name":"Riordan array (1, 3+x).","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: 1;\n   2: 3,  3;\n   3: 1, 19,  9;\n   4: 0, 12, 87,  27;\n   5: 0,  2, 82, 351,   81;\n   6: 0,  0, 27, 441, 1323,  243;\n   7: 0,  0,  3, 217, 2079, 4779,   729;\n   8: 0,  0,  0,  48, 1362, 9018, 16767,   2187;\n   9: 0,  0,  0,   4,  451, 7380, 36936,  57591,   6561;\n  10: 0,  0,  0,   0,   75, 3255, 36261, 145071, 194643, 19683;\n","recurrence_sentence":"The row polynomials satisfy (3*n^3-12*n^2+15*n-6-(15*n^2-39*n+24)*x+(27*n^2-123*n+96)*x^2+(27*n-27)*x^3)*P(n,x) = -(2*n^3-14*n^2+28*n-16-(37*n^2-117*n+68)*x+(18*n^2+26*n+20)*x^2-(63*n-171)*x^3-81*x^4)*P(n-1,x) + ((4*n^2-12*n+8)*x+(9*n^2-47*n+30)*x^2-(9*n+29)*x^3+(81*n-90)*x^4+81*x^5)*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^3 - 14*n^2 + 28*n - 16 - (37*n^2 - 117*n + 68)*x + (18*n^2 + 26*n + 20)*x^2 - (63*n - 171)*x^3 - 81*x^4)*P[n - 1, x] + ((4*n^2 - 12*n + 8)*x + (9*n^2 - 47*n + 30)*x^2 - (9*n + 29)*x^3 + (81*n - 90)*x^4 + 81*x^5)*D[P[n - 1, x], x])/(3*n^3 - 12*n^2 + 15*n - 6 - (15*n^2 - 39*n + 24)*x + (27*n^2 - 123*n + 96)*x^2 + (27*n - 27)*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^3 - 14*n^2 + 28*n - 16 - (37*n^2 - 117*n + 68)*x + (18*n^2 + 26*n + 20)*x^2 - (63*n - 171)*x^3 - 81*x^4)*v[i - 1] + ((4*n^2 - 12*n + 8)*x + (9*n^2 - 47*n + 30)*x^2 - (9*n + 29)*x^3 + (81*n - 90)*x^4 + 81*x^5)*deriv(v[i - 1], x))/(3*n^3 - 12*n^2 + 15*n - 6 - (15*n^2 - 39*n + 24)*x + (27*n^2 - 123*n + 96)*x^2 + (27*n - 27)*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-5d39164c4108e043","steps":["D+1"],"edges":["e-a3e4828d005b5462"],"text":"A099097 \u2192 D+1"},"headline":"A099097 \u2192 D+1","chain":null}