{"id":"n-5ce84951924c70ee","revision":0,"first_index":0,"fingerprint":"f6-61423eae1de6db11","name":"theta_plus_one_squared of Triangle read by rows, 0 <= k <= n, T(n,k) = Sum_{j=0..n} A(n,j)*binomial(n-j,k) where A(n,j) are the Eulerian numbers A173018.","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","4"],["2","12","9"],["6","48","63","16"],["24","240","450","240","25"],["120","1440","3510","2880","775","36"],["720","10080","30240","33600","15050","2268","49"],["5040","80640","287280","403200","255150","69552","6223","64"],["40320","725760","2993760","5080320","4170600","1678320","296450","16320","81"],["362880","7257600","34020000","67737600","68481000","36741600","10026870","1194240","41391","100"],["3628800","79833600","419126400","958003200","1151766000","775383840","290154480","55968000","4617162","102300","121"],["39916800","958003200","5568393600","14370048000","20037402000","16262104320","7783073760","2118758400","297189486","17305200","247687","144"],["479001600","12454041600","79394515200","228324096000","362378016000","344665601280","201640158720","71587676160","14596525944","1519518000","63313250","589680","169"],["6227020800","174356582400","1209598790400","3835844812800","6828966144000","7453743897600","5164506547200","2273093222400","616601545560","96180084000","7544406870","227223360","1384279","196"],["87178291200","2615348736000","19615115520000","67999067136000","134218244160000","165420807552000","132624415123200","69875476070400","23837794540560","5048319276000","612067135680","36595540800","802784138","3211068","225"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","23","133","979","8761","92007","1107149","15001931","225863281","3738583903","67452442597","1317103053603","27666891661865","621991235077271"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one_squared","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["1","4"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","0","-7"],["0","4","-2"],["-1","-4","9"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","-1","-8","-13"],["0","2","0","-14"],["0","0","4","4"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","-1","-4","-3"],["0","0","0","-4","-4"]]},"deriv_order":2,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1","0"],["2","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-2","-4"],["0","0","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":179371,"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-73c0ffe1fc5a513a"]},{"id":179372,"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-73c0ffe1fc5a513a"]},{"id":179373,"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-73c0ffe1fc5a513a"]}],"region_exact":{},"incoming_edges":[{"id":"e-73c0ffe1fc5a513a","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-5ce84951924c70ee","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":"09b54e744577dccc0bc6700a5c6d90dc101fd9fda56d8de7cfd64b57bcc9b135","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-45ec3349543b0d37","role":"sequence","name":"Triangle read by rows, 0 <= k <= n, T(n,k) = Sum_{j=0..n} A(n,j)*binomial(n-j,k) where A(n,j) are the Eulerian numbers A173018.","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,       4;\n  2:      2,      12,        9;\n  3:      6,      48,       63,       16;\n  4:     24,     240,      450,      240,       25;\n  5:    120,    1440,     3510,     2880,      775,       36;\n  6:    720,   10080,    30240,    33600,    15050,     2268,       49;\n  7:   5040,   80640,   287280,   403200,   255150,    69552,     6223,      64;\n  8:  40320,  725760,  2993760,  5080320,  4170600,  1678320,   296450,   16320,    81;\n  9: 362880, 7257600, 34020000, 67737600, 68481000, 36741600, 10026870, 1194240, 41391, 100;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^2-1+(4*n^2-4*n)*x-(9*n^2-2*n-7)*x^2)*P(n-2,x) + ((2*n-1)*x+(4*n^2-8)*x^2+(4*n^2-14*n-13)*x^3)*d/dx P(n-2,x) - (x^2+(4*n+4)*x^3+(4*n+3)*x^4)*d^2/dx^2 P(n-2,x) + (2*n+1+4*n*x)*P(n-1,x) - (2*x+(4*n+4)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 4*x;\nP[n_, x_] := P[n, x] = Expand[-(n^2 - 1 + (4*n^2 - 4*n)*x - (9*n^2 - 2*n - 7)*x^2)*P[n - 2, x] + ((2*n - 1)*x + (4*n^2 - 8)*x^2 + (4*n^2 - 14*n - 13)*x^3)*D[P[n - 2, x], x] - (x^2 + (4*n + 4)*x^3 + (4*n + 3)*x^4)*D[P[n - 2, x], {x, 2}] + (2*n + 1 + 4*n*x)*P[n - 1, x] - (2*x + (4*n + 4)*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] = 1; v[2] = 1 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 - 1 + (4*n^2 - 4*n)*x - (9*n^2 - 2*n - 7)*x^2)*v[i - 2] + ((2*n - 1)*x + (4*n^2 - 8)*x^2 + (4*n^2 - 14*n - 13)*x^3)*deriv(v[i - 2], x) - (x^2 + (4*n + 4)*x^3 + (4*n + 3)*x^4)*deriv(deriv(v[i - 2], x), x) + (2*n + 1 + 4*n*x)*v[i - 1] - (2*x + (4*n + 4)*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-45ec3349543b0d37","steps":["(\u03b8+1)\u00b2"],"edges":["e-73c0ffe1fc5a513a"],"text":"A130850 \u2192 (\u03b8+1)\u00b2"},"headline":"A130850 \u2192 (\u03b8+1)\u00b2","chain":null}