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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"],["3","1"],["15","12","2"],["93","128","52","6"],["675","1368","958","268","24"],["5587","15243","15704","7520","1628","120"],["51845","179796","249012","174564","64288","11448","720"],["532705","2257064","3967648","3726444","1999448","602824","91656","5040"],["6002125","30187204","64744876","76991156","55154608","24128920","6189440","823968","40320"],["73576351","429754749","1092786520","1582489552","1431699396","833669512","309268720","69276240","8222112","362880"],["974830427","6500480464","19176091014","32873420348","36148189624","26520686784","13060548240","4219445152","840949344","90204480","3628800"],["13881832389","104240147488","350734063564","696677182578","905512243888","806009390848","499085997744","213561747936","61250727296","11015819424","1079340480","39916800"],["211443110455","1767953274268","6693168603050","15148623474664","22786379069440","23939941166336","17955577649616","9643207060832","3656186572192","944499146496","154980039168","13989922560","479001600"],["3430345368295","31639999294503","133283793884384","339129728611344","580658839802424","705193309420640","623477474200304","404999746345776","192420614528224","65603902433920","15438258480192","2331694174464","195281038080","6227020800"],["59056222616953","596139565915352","2768466171070936","7832338304348408","15063665769552816","20804602615999696","21239803755225392","16244647548279136","9323118378949952","3977699911987520","1233647916246656","266865053063808","37368352797696","2920745157120","87178291200"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","29","279","3293","45802","731673","13182829","264262617","5831106032","140408474677","3663088410435","102916428090677","3097809214603350","99450427489502641"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["3","1"],["15","12","2"],["93","128","52","6"]],"recurrence":{"denominator":{"coeffs":[["2"],["-3"],["1"]]},"terms":[{"coeff":{"coeffs":[["0"],["1"],["-1"]]},"deriv_order":0,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["0","0"],["-2","4"],["-2","-6"],["2","2"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["-2","0","0"],["6","-4","2"],["1","2","-5"],["0","4","4"],["-1","-2","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["4","0"],["-6","4"],["-2","-6"],["2","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":185574,"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-f2aec896582b898d"]},{"id":185575,"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-f2aec896582b898d"]},{"id":185576,"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-f2aec896582b898d"]},{"id":185577,"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-f2aec896582b898d"]}],"region_exact":{},"incoming_edges":[{"id":"e-f2aec896582b898d","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-33c9b9d3f3d7386f","premise_count":1,"theorem":"Laguerre's theorem on differential operators: if phi is a real polynomial with only real roots, then phi(D) maps every real-rooted polynomial to a real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"2ba06bad182cc42af8e4254e181bb46697fb394068324a430b9cc7b85946c0b7","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-4a57e8264fe984f4","role":"sequence","name":"basis_map_A007318 of 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:        3,         1;\n  2:       15,        12,          2;\n  3:       93,       128,         52,          6;\n  4:      675,      1368,        958,        268,         24;\n  5:     5587,     15243,      15704,       7520,       1628,       120;\n  6:    51845,    179796,     249012,     174564,      64288,     11448,       720;\n  7:   532705,   2257064,    3967648,    3726444,    1999448,    602824,     91656,     5040;\n  8:  6002125,  30187204,   64744876,   76991156,   55154608,  24128920,   6189440,   823968,   40320;\n  9: 73576351, 429754749, 1092786520, 1582489552, 1431699396, 833669512, 309268720, 69276240, 8222112, 362880;\n","recurrence_sentence":"The row polynomials satisfy (n^2-3*n+2)*P(n,x) = -(n^2-n)*P(n-4,x) + (2*n^3-2*n^2-2*n+(2*n^3-6*n^2+4*n)*x)*P(n-3,x) - (n^4-n^2-6*n+2+(2*n^4-4*n^3-2*n^2+4*n)*x+(n^4-4*n^3+5*n^2-2*n)*x^2)*P(n-2,x) + (2*n^3-2*n^2-6*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) = 3 + x, P(2,x) = 15 + 12*x + 2*x^2, P(3,x) = 93 + 128*x + 52*x^2 + 6*x^3.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + x;\nP[2, x_] := 15 + 12*x + 2*x^2;\nP[3, x_] := 93 + 128*x + 52*x^2 + 6*x^3;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^2 - n)*P[n - 4, x] + (2*n^3 - 2*n^2 - 2*n + (2*n^3 - 6*n^2 + 4*n)*x)*P[n - 3, x] - (n^4 - n^2 - 6*n + 2 + (2*n^4 - 4*n^3 - 2*n^2 + 4*n)*x + (n^4 - 4*n^3 + 5*n^2 - 2*n)*x^2)*P[n - 2, x] + (2*n^3 - 2*n^2 - 6*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] = 3 + x; v[3] = 15 + 12*x + 2*x^2; v[4] = 93 + 128*x + 52*x^2 + 6*x^3; for(n = 4, N, my(i = n + 1); v[i] = (-(n^2 - n)*v[i - 4] + (2*n^3 - 2*n^2 - 2*n + (2*n^3 - 6*n^2 + 4*n)*x)*v[i - 3] - (n^4 - n^2 - 6*n + 2 + (2*n^4 - 4*n^3 - 2*n^2 + 4*n)*x + (n^4 - 4*n^3 + 5*n^2 - 2*n)*x^2)*v[i - 2] + (2*n^3 - 2*n^2 - 6*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":["basis A007318","D+1"],"edges":["e-3f96abc359582f71","e-f2aec896582b898d"],"text":"A358735 \u2192 basis A007318 \u2192 D+1"},"headline":"A358735 \u2192 basis A007318 \u2192 D+1","chain":null}