{"id":"n-3c9286c555c35626","revision":0,"first_index":0,"fingerprint":"f6-403c190250d8ebb3","name":"theta_plus_one of Triangle read by rows: Joffe's central differences of zero, T(n,k), 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":[["1"],["1","12"],["1","60","270"],["1","252","3780","10080"],["1","1020","39690","302400","567000"],["1","4092","380160","6320160","31185000","44906400"],["1","16380","3513510","115315200","1135134000","4086482400","4767562800"],["1","65532","31990140","1976214240","34864830000","237015979200","667458792000","653837184000"],["1","262140","289385730","32813726400","982782801000","11293869787200","57868677266400","133382785536000","112541725296000"],["1","1048572","2610369720","535823356320","26415342135000","484415710178400","4034597695128000","16472774013696000","32074391709360000","23758808673600000"],["1","4194300","23516920350","8670627504000","690523617168000","19531060663514400","249124088291878800","1607573792208384000","5433401955565584000","9147141339336000000","6037113283961760000"],["1","16777212","211746654900","139608005090400","17750813226300000","757807654366224000","14280627933411840000","137050311952794240000","716809197384680400000","2061765657886334400000","3054779321684650560000","1817719926952849920000"],["1","67108860","1906097381370","2241633289896000","451623280943835000","28686939964707024000","780204681375727608000","10729706081561026560000","81651329933033820960000","357502951903177080000000","893522951592760288800000","1181517952519352448000000","639988890947982576000000"],["1","268435452","17156386381680","35937293607240480","11416673896152525000","1068498402580204596000","41275626180330381840000","793700688380005486080000","8460608383864431194400000","52987798176180403536000000","198957777221321290972800000","439170222951443304921600000","524150901686397729744000000","260524708530517214784000000"],["1","1073741820","154413517232790","575637202807430400","287438320169220402000","39370143065462645436000","2135944596576787098102000","56457501361051374904320000","821925010080427371037920000","7084908254211642494640000000","37538090560347316746206400000","123144890119282028245248000000","243206018382488546601216000000","264432579158474973005760000000","121423123082973201890400000000"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","13","331","14113","910111","82795813","10108024291","1595185055113","316102944060871","76851941578872013","22494584737792907851","7803180742082082406513","3165723128466028073173231","1485088073215315637639898613","797710167671972941765774985011"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["1","12"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","0","-6"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","-2","-9"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","0","-1","-2"]]},"deriv_order":2,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1","12"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","3","11"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","1","2"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":193815,"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-38c7794dc32b4713"]},{"id":193816,"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-38c7794dc32b4713"]},{"id":193817,"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-38c7794dc32b4713"]}],"region_exact":{},"incoming_edges":[{"id":"e-38c7794dc32b4713","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-3c9286c555c35626","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":"a85e676febdfdcbbf5cd561d8c2a03a17567beafe21c78e9a2786fe6693a9b84","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-4ef34abfaa61f0d6","role":"sequence","name":"Triangle read by rows: Joffe's central differences of zero, T(n,k), 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: 1;\n  1: 1,      12;\n  2: 1,      60,        270;\n  3: 1,     252,       3780,        10080;\n  4: 1,    1020,      39690,       302400,         567000;\n  5: 1,    4092,     380160,      6320160,       31185000,        44906400;\n  6: 1,   16380,    3513510,    115315200,     1135134000,      4086482400,       4767562800;\n  7: 1,   65532,   31990140,   1976214240,    34864830000,    237015979200,     667458792000,      653837184000;\n  8: 1,  262140,  289385730,  32813726400,   982782801000,  11293869787200,   57868677266400,   133382785536000,   112541725296000;\n  9: 1, 1048572, 2610369720, 535823356320, 26415342135000, 484415710178400, 4034597695128000, 16472774013696000, 32074391709360000, 23758808673600000;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -6*x^2*P(n-2,x) - (2*x^2+9*x^3)*d/dx P(n-2,x) - (x^3+2*x^4)*d^2/dx^2 P(n-2,x) + (1+12*x)*P(n-1,x) + (3*x+11*x^2)*d/dx P(n-1,x) + (x^2+2*x^3)*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 12*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 12*x;\nP[n_, x_] := P[n, x] = Expand[-6*x^2*P[n - 2, x] - (2*x^2 + 9*x^3)*D[P[n - 2, x], x] - (x^3 + 2*x^4)*D[P[n - 2, x], {x, 2}] + (1 + 12*x)*P[n - 1, x] + (3*x + 11*x^2)*D[P[n - 1, x], x] + (x^2 + 2*x^3)*D[P[n - 1, x], {x, 2}]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 12*x; for(n = 2, N, my(i = n + 1); v[i] = -6*x^2*v[i - 2] - (2*x^2 + 9*x^3)*deriv(v[i - 2], x) - (x^3 + 2*x^4)*deriv(deriv(v[i - 2], x), x) + (1 + 12*x)*v[i - 1] + (3*x + 11*x^2)*deriv(v[i - 1], x) + (x^2 + 2*x^3)*deriv(deriv(v[i - 1], x), 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-4ef34abfaa61f0d6","steps":["\u03b8+1"],"edges":["e-38c7794dc32b4713"],"text":"A241171 \u2192 \u03b8+1"},"headline":"A241171 \u2192 \u03b8+1","chain":null}