{"id":"n-a90658b17f2703fe","revision":0,"first_index":0,"fingerprint":"f6-65ecce61f1255cba","name":"theta_plus_one of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*9^j.","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":true,"rows":[["1"],["20","18"],["400","720","243"],["8000","21600","14580","2916"],["160000","576000","583200","233280","32805"],["3200000","14400000","19440000","11664000","3280500","354294"],["64000000","345600000","583200000","466560000","196830000","42515280","3720087"],["1280000000","8064000000","16329600000","16329600000","9185400000","2976069600","520812180","38263752"],["25600000000","184320000000","435456000000","522547200000","367416000000","158723712000","41664974400","6122200320","387420489"],["512000000000","4147200000000","11197440000000","15676416000000","13226976000000","7142567040000","2499898464000","550998028800","69735688020","3874204890"],["10240000000000","92160000000000","279936000000000","447897600000000","440899200000000","285702681600000","124994923200000","36733201920000","6973568802000","774840978000","38354628411"],["204800000000000","2027520000000000","6842880000000000","12317184000000000","13856832000000000","10475764992000000","5499776620800000","2020326105600000","511395045480000","85232507580000","8438018250420","376572715308"],["4096000000000000","44236800000000000","164229120000000000","328458240000000000","415704960000000000","359169085440000000","219991064832000000","96975653068800000","30683702728800000","6818600606400000","1012562190050400","90377451673920","3671583974253"],["81920000000000000","958464000000000000","3881779200000000000","8539914240000000000","12009254400000000000","11672995276800000000","8171096693760000000","4202278299648000000","1595552541897600000","443209039416000000","87755389804368000","11749068717609600","954611833305780","35586121596606"],["1638400000000000000","20643840000000000000","90574848000000000000","217379635200000000000","336259123200000000000","363159853056000000000","285988384281600000000","168091131985920000000","74459118621888000000","24819706207296000000","6142877286305760000","1096579746976896000","133645656662809200","9964114047049680","343151886824415"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","38","1363","47096","1585285","52338794","1702425367","54723745532","1742237507209","55027105425710","1726350371128411","53850525862425728","1671469837901698573","51656958347998479986","1590397450508583339295"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["20","0"],["0","9"]]},"terms":[{"coeff":{"coeffs":[["400","360","81"],["0","180","81"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":192408,"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-77f20a7af4af80fa"]},{"id":192409,"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-77f20a7af4af80fa"]},{"id":192410,"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-77f20a7af4af80fa"]},{"id":192411,"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-77f20a7af4af80fa"]}],"region_exact":{},"incoming_edges":[{"id":"e-77f20a7af4af80fa","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-a90658b17f2703fe","premise_count":1,"theorem":"Laguerre's theorem on multiplier sequences: if f is a real polynomial whose roots are all real and <= 0, then (f(k))_(k >= 0) is a multiplier sequence, so x^k -> f(k) x^k maps every real-rooted polynomial to a real-rooted polynomial or zero (Laguerre; see T. Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"certificate_hash":"48f607b03a66af1c50e421e97939932e1c475e1f04e0ed8ec66a6c7fb968410d","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-d2d581dc25d5445a","role":"sequence","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*9^j.","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:           20,            18;\n  2:          400,           720,            243;\n  3:         8000,         21600,          14580,           2916;\n  4:       160000,        576000,         583200,         233280,          32805;\n  5:      3200000,      14400000,       19440000,       11664000,        3280500,        354294;\n  6:     64000000,     345600000,      583200000,      466560000,      196830000,      42515280,       3720087;\n  7:   1280000000,    8064000000,    16329600000,    16329600000,     9185400000,    2976069600,     520812180,     38263752;\n  8:  25600000000,  184320000000,   435456000000,   522547200000,   367416000000,  158723712000,   41664974400,   6122200320,   387420489;\n  9: 512000000000, 4147200000000, 11197440000000, 15676416000000, 13226976000000, 7142567040000, 2499898464000, 550998028800, 69735688020, 3874204890;\n","recurrence_sentence":"The row polynomials satisfy (20+9*n*x)*P(n,x) = (400+(180*n+360)*x+(81*n+81)*x^2)*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[((400 + (180*n + 360)*x + (81*n + 81)*x^2)*P[n - 1, x])/(20 + 9*n*x)]];\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] = ((400 + (180*n + 360)*x + (81*n + 81)*x^2)*v[i - 1])/(20 + 9*n*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-4e096b5929fd3f2e","steps":["x \u21a6 x+1","\u03b8+1"],"edges":["e-e91f5a6c0f5b0271","e-77f20a7af4af80fa"],"text":"A038323 \u2192 x \u21a6 x+1 \u2192 \u03b8+1"},"headline":"A038323 \u2192 x \u21a6 x+1 \u2192 \u03b8+1","chain":null}