{"id":"n-1fd60f77276be098","revision":1,"first_index":0,"fingerprint":"f6-34cd73f5fc28d5d0","name":"theta_plus_one_squared of Triangular array read by rows. T(n,k) = A008277(n,k)*2^k; n >= 1, 1 <= k <= n.","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":[["2"],["2","16"],["2","48","72"],["2","112","432","256"],["2","240","1800","2560","800"],["2","496","6480","16640","12000","2304"],["2","1008","21672","89600","112000","48384","6272"],["2","2032","69552","435456","840000","612864","175616","16384"],["2","4080","217800","1989120","5560800","6096384","2897664","589824","41472"],["2","8176","671760","8730880","34020000","52593408","36879360","12288000","1866240","102400"],["2","16368","2052072","37312000","197384000","413538048","401326464","194641920","47900160","5632000","247808"],["2","32752","6229872","156544256","1103520000","3049694208","3935027712","2605498368","923788800","174592000","16355328","589824"],["2","65520","18837000","648327680","6006800800","21476302848","35847139328","31123243008","14909266944","4026880000","602421248","46006272","1384448"],["2","131056","56805840","2660286720","32060028000","146157403392","309393244160","342627450880","212964111360","77081804800","16371683328","1985937408","125984768","3211264"],["2","262128","171007272","10843123200","168613536000","969277300992","2563625640576","3549230530560","2783952737280","1296655360000","366626484224","62798561280","6299238400","337182720","7372800"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["2","18","122","802","5402","37922","278938","2151906","17397146","147160226","1300050842","11971873122","114706675098","1141488082978","11778438337434"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one_squared","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["2"],["2","16"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","-8","36"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","-2","8","16"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","-1","8"]]},"deriv_order":2,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1","8"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2","-8"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":165904,"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-ca8ea9a3beb129bc"]},{"id":165905,"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-ca8ea9a3beb129bc"]},{"id":165906,"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-ca8ea9a3beb129bc"]},{"id":165907,"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-ca8ea9a3beb129bc"]}],"region_exact":{},"incoming_edges":[{"id":"e-ca8ea9a3beb129bc","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"2\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-1fd60f77276be098","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":"ed0d22487e16682e57668484e140b9a8d43b97a85a9ef2dec7120d63a30b72b9","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-d5fb64ec7bd8ef70","role":"sequence","name":"Triangular array read by rows. T(n,k) = A008277(n,k)*2^k; n >= 1, 1 <= k <= n.","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: 2;\n  1: 2,   16;\n  2: 2,   48,     72;\n  3: 2,  112,    432,     256;\n  4: 2,  240,   1800,    2560,      800;\n  5: 2,  496,   6480,   16640,    12000,     2304;\n  6: 2, 1008,  21672,   89600,   112000,    48384,     6272;\n  7: 2, 2032,  69552,  435456,   840000,   612864,   175616,    16384;\n  8: 2, 4080, 217800, 1989120,  5560800,  6096384,  2897664,   589824,   41472;\n  9: 2, 8176, 671760, 8730880, 34020000, 52593408, 36879360, 12288000, 1866240, 102400;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(8*x-36*x^2)*P(n-2,x) - (2*x-8*x^2-16*x^3)*d/dx P(n-2,x) - (x^2-8*x^3)*d^2/dx^2 P(n-2,x) + (1+8*x)*P(n-1,x) + (2*x-8*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 2, P(1,x) = 2 + 16*x.","mathematica":"P[0, x_] := 2;\nP[1, x_] := 2 + 16*x;\nP[n_, x_] := P[n, x] = Expand[-(8*x - 36*x^2)*P[n - 2, x] - (2*x - 8*x^2 - 16*x^3)*D[P[n - 2, x], x] - (x^2 - 8*x^3)*D[P[n - 2, x], {x, 2}] + (1 + 8*x)*P[n - 1, x] + (2*x - 8*x^2)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 2; v[2] = 2 + 16*x; for(n = 2, N, my(i = n + 1); v[i] = -(8*x - 36*x^2)*v[i - 2] - (2*x - 8*x^2 - 16*x^3)*deriv(v[i - 2], x) - (x^2 - 8*x^3)*deriv(deriv(v[i - 2], x), x) + (1 + 8*x)*v[i - 1] + (2*x - 8*x^2)*deriv(v[i - 1], 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-d5fb64ec7bd8ef70","steps":["(\u03b8+1)\u00b2"],"edges":["e-ca8ea9a3beb129bc"],"text":"A227450 \u2192 (\u03b8+1)\u00b2"},"headline":"A227450 \u2192 (\u03b8+1)\u00b2","chain":null}
