{"id":"n-49b042466eca8b16","revision":0,"first_index":0,"fingerprint":"f6-d8dc46745c680172","name":"one_plus_derivative of basis_map_A013610 of Triangle of coefficients of Chebyshev's S(n,x+4) polynomials (exponents of x in increasing order).","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","3"],["54","48","9"],["334","489","216","27"],["1961","4056","2943","864","81"],["11124","29904","30636","14742","3240","243"],["61579","204288","272079","184680","66420","11664","729"],["334718","1323063","2175660","1919295","972810","279207","40824","2187"],["1793826","8237520","16148835","17589960","11582190","4681152","1117557","139968","6561"],["9505540","49756365","113382180","147258810","119705040","62823033","21106008","4312764","472392","19683"],["49907699","293403720","762494310","1151411760","1116023265","722911392","315232722","90594288","16179426","1574640","59049"],["260030952","1696710432","4954543560","8534696220","9618852060","7428576582","3997118664","1490409882","374173830","59344245","5196312","177147"],["1346063221","9654517992","31303282086","60605934480","77928289245","69978646224","44886692214","20647204560","6723023895","1498269960","213698331","17006112","531441"],["6929367956","54193176051","193217644404","415477363338","600526713240","615269232693","458858319120","251573409360","101029296060","29192447790","5848449156","757834866","55269864","1594323"],["35500374414","300687972144","1169359979157","2765652956136","4440440604294","5114403037440","4351771321065","2773797119280","1327483423035","472968318240","122821979049","22347448344","2652953472","178564176","4782969"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","11","111","1066","9905","89889","801439","7047764","61297569","528341815","4519792271","38419829886","324803159761","2732930118221","22900070833215"],"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","-2","-6"],["0","0","0"],["6","0","0"]]},"terms":[{"coeff":{"coeffs":[["19","-11","-27","-9"],["49","7","-27","-9"],["45","27","0","0"],["15","9","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-14","-69","-97","-51","-9"],["21","30","9","0","0"],["21","30","9","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":183895,"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-747bcb741c0f054b"]},{"id":183896,"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-747bcb741c0f054b"]},{"id":183897,"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-747bcb741c0f054b"]},{"id":183898,"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-747bcb741c0f054b"]}],"region_exact":{},"incoming_edges":[{"id":"e-747bcb741c0f054b","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-49b042466eca8b16","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":"bd527644253bcd47496adffcded302f3397823819e358d9909d3699212ea9ee1","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-0778fc04c5956f01","role":"sequence","name":"basis_map_A013610 of Triangle of coefficients of Chebyshev's S(n,x+4) polynomials (exponents of x in increasing order).","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,        3;\n  2:      54,       48,         9;\n  3:     334,      489,       216,        27;\n  4:    1961,     4056,      2943,       864,        81;\n  5:   11124,    29904,     30636,     14742,      3240,      243;\n  6:   61579,   204288,    272079,    184680,     66420,    11664,      729;\n  7:  334718,  1323063,   2175660,   1919295,    972810,   279207,    40824,    2187;\n  8: 1793826,  8237520,  16148835,  17589960,  11582190,  4681152,  1117557,  139968,   6561;\n  9: 9505540, 49756365, 113382180, 147258810, 119705040, 62823033, 21106008, 4312764, 472392, 19683;\n","recurrence_sentence":"The row polynomials satisfy (6*n^3+10*n-2*n*x-6*n*x^2)*P(n,x) = (15*n^3+45*n^2+49*n+19+(9*n^3+27*n^2+7*n-11)*x-(27*n+27)*x^2-(9*n+9)*x^3)*P(n-1,x) + (21*n^2+21*n-14+(30*n^2+30*n-69)*x+(9*n^2+9*n-97)*x^2-51*x^3-9*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[((15*n^3 + 45*n^2 + 49*n + 19 + (9*n^3 + 27*n^2 + 7*n - 11)*x - (27*n + 27)*x^2 - (9*n + 9)*x^3)*P[n - 1, x] + (21*n^2 + 21*n - 14 + (30*n^2 + 30*n - 69)*x + (9*n^2 + 9*n - 97)*x^2 - 51*x^3 - 9*x^4)*D[P[n - 1, x], x])/(6*n^3 + 10*n - 2*n*x - 6*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] = ((15*n^3 + 45*n^2 + 49*n + 19 + (9*n^3 + 27*n^2 + 7*n - 11)*x - (27*n + 27)*x^2 - (9*n + 9)*x^3)*v[i - 1] + (21*n^2 + 21*n - 14 + (30*n^2 + 30*n - 69)*x + (9*n^2 + 9*n - 97)*x^2 - 51*x^3 - 9*x^4)*deriv(v[i - 1], x))/(6*n^3 + 10*n - 2*n*x - 6*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-30cb3cad216ed189","steps":["basis A013610","D+1"],"edges":["e-99fce6e1b50f281f","e-747bcb741c0f054b"],"text":"A207823 \u2192 basis A013610 \u2192 D+1"},"headline":"A207823 \u2192 basis A013610 \u2192 D+1","chain":null}