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T(n, k) is the coefficient of x^k in a(n+3) where a(1) = a(2) = a(3) = 1 and a(m+2) = (m*x + 2)*a(m+1) - a(m) for all m in Z.","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","2"],["1","8","6"],["1","20","48","24"],["1","40","210","304","120"],["1","70","672","1984","2140","720"],["1","112","1764","9040","19040","16848","5040"],["1","168","4032","32560","115040","192912","147672","40320"],["1","240","8316","99088","536640","1473360","2090816","1430784","362880"],["1","330","15840","265408","2072980","8549520","19504240","24326784","15212448","3628800"],["1","440","28314","642928","6926920","40501440","136662064","270128384","303738336","176284800","39916800"],["1","572","48048","1435720","20621120","163888704","773588368","2233967360","3935102976","4061298240","2212879680","479001600"],["1","728","78078","2997280","55851120","584220096","3704956528","14807018240","37725026976","60414351360","57994979328","29926609920","6227020800"],["1","910","122304","5912192","139786920","1875828960","15496996144","82425137536","288067592736","661684460800","977702303424","881848470528","433920464640","87178291200"],["1","1120","185640","11110112","327186080","5515858080","57927575440","397995195136","1836606074688","5744006012800","12087350323840","16665950965248","14236315711488","6716542049280","1307674368000"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","15","93","675","5587","51845","532705","6002125","73576351","974830427","13881832389","211443110455","3430345368295","59056222616953"],"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","2"],["1","8","6"],["1","20","48","24"]],"recurrence":{"denominator":{"coeffs":[["2"],["-3"],["1"]]},"terms":[{"coeff":{"coeffs":[["0"],["1"],["-1"]]},"deriv_order":0,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["0","0"],["-6","4"],["4","-6"],["0","2"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["-2","0","0"],["12","-8","2"],["-6","12","-5"],["0","-4","4"],["0","0","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["4","0"],["-10","4"],["4","-6"],["0","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":180423,"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-10fe478f95f8b63d"]},{"id":180424,"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-10fe478f95f8b63d"]},{"id":180425,"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-10fe478f95f8b63d"]},{"id":180426,"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-10fe478f95f8b63d"]}],"region_exact":{},"incoming_edges":[{"id":"e-10fe478f95f8b63d","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-67fe0ee394103768","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":"c922b47828ee560772fbce53039afed528353d603a0d91ed1f7617c512f290b2","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-f3e0631eb8740f76","role":"sequence","name":"Triangular array read by rows. T(n, k) is the coefficient of x^k in a(n+3) where a(1) = a(2) = a(3) = 1 and a(m+2) = (m*x + 2)*a(m+1) - a(m) for all m in Z.","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,   2;\n  2: 1,   8,     6;\n  3: 1,  20,    48,     24;\n  4: 1,  40,   210,    304,     120;\n  5: 1,  70,   672,   1984,    2140,     720;\n  6: 1, 112,  1764,   9040,   19040,   16848,     5040;\n  7: 1, 168,  4032,  32560,  115040,  192912,   147672,    40320;\n  8: 1, 240,  8316,  99088,  536640, 1473360,  2090816,  1430784,   362880;\n  9: 1, 330, 15840, 265408, 2072980, 8549520, 19504240, 24326784, 15212448, 3628800;\n","recurrence_sentence":"The row polynomials satisfy (n^2-3*n+2)*P(n,x) = -(n^2-n)*P(n-4,x) + (4*n^2-6*n+(2*n^3-6*n^2+4*n)*x)*P(n-3,x) - (6*n^2-12*n+2+(4*n^3-12*n^2+8*n)*x+(n^4-4*n^3+5*n^2-2*n)*x^2)*P(n-2,x) + (4*n^2-10*n+4+(2*n^3-6*n^2+4*n)*x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1 + 2*x, P(2,x) = 1 + 8*x + 6*x^2, P(3,x) = 1 + 20*x + 48*x^2 + 24*x^3.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 2*x;\nP[2, x_] := 1 + 8*x + 6*x^2;\nP[3, x_] := 1 + 20*x + 48*x^2 + 24*x^3;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^2 - n)*P[n - 4, x] + (4*n^2 - 6*n + (2*n^3 - 6*n^2 + 4*n)*x)*P[n - 3, x] - (6*n^2 - 12*n + 2 + (4*n^3 - 12*n^2 + 8*n)*x + (n^4 - 4*n^3 + 5*n^2 - 2*n)*x^2)*P[n - 2, x] + (4*n^2 - 10*n + 4 + (2*n^3 - 6*n^2 + 4*n)*x)*P[n - 1, x])/(n^2 - 3*n + 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 + 2*x; v[3] = 1 + 8*x + 6*x^2; v[4] = 1 + 20*x + 48*x^2 + 24*x^3; for(n = 4, N, my(i = n + 1); v[i] = (-(n^2 - n)*v[i - 4] + (4*n^2 - 6*n + (2*n^3 - 6*n^2 + 4*n)*x)*v[i - 3] - (6*n^2 - 12*n + 2 + (4*n^3 - 12*n^2 + 8*n)*x + (n^4 - 4*n^3 + 5*n^2 - 2*n)*x^2)*v[i - 2] + (4*n^2 - 10*n + 4 + (2*n^3 - 6*n^2 + 4*n)*x)*v[i - 1])/(n^2 - 3*n + 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-f3e0631eb8740f76","steps":["\u03b8+1"],"edges":["e-10fe478f95f8b63d"],"text":"A358735 \u2192 \u03b8+1"},"headline":"A358735 \u2192 \u03b8+1","chain":null}