{"id":"n-ee02330825df0350","revision":0,"first_index":0,"fingerprint":"f6-a07a15ae93e03f31","name":"derivative of Triangle of central factorial numbers T(2*n,2*n-2*k), k >= 0, n >= 1 (in Riordan's notation).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[[],["1"],["5","2"],["14","42","3"],["30","294","255","4"],["55","1254","4224","1364","5"],["91","4004","34320","52052","6825","6"],["140","10556","184470","784212","592410","32766","7"],["204","24276","754494","6933212","16276650","6430794","152915","8"],["285","50388","2538780","43148924","232939525","318942474","67676252","699048","9"],["385","96558","7376028","209015884","2174641105","7307128224","6021269100","696797640","3145725","10"],["506","173558","19107825","838436940","14976864000","101251623960","219145842300","110800004040","7058221830","13981010","11"],["650","296010","45141525","2902082040","82051819200","981891227160","4471714048620","6372109784040","2001458294550","70596199310","61516455","12"],["819","483210","98867340","8920952040","375887625750","7283470941720","60603162534600","190351202070120","181217045970675","35652076991310","698963889624","268435452","13"],["1015","760032","203240700","24871549560","1491006630750","43819748164620","604435659516360","3584128304007720","7890440729810535","5069458909510060","628175518936680","6862822988124","1163220305","14"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","1","7","59","583","6902","97298","1604561","30572553","665995685","16419470659","454104255980","13982830409572","476191098762673","17828838938336475"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["1"]],"recurrence":{"denominator":{"coeffs":[["1"],["-2"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0","0"],["0","0","0"],["-1","-8","-4"],["0","8","12"],["0","-2","-13"],["0","0","6"],["0","0","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","0","0"],["0","0","0","0"],["0","0","-6","-5"],["0","0","2","12"],["0","0","0","-9"],["0","0","0","2"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","0","0","0"],["0","0","0","0","0"],["0","0","0","-1","-1"],["0","0","0","0","2"],["0","0","0","0","-1"]]},"deriv_order":2,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1","2"],["-2","-8"],["2","12"],["0","-8"],["0","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","4"],["0","0","-10"],["0","0","8"],["0","0","-2"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","0","1"],["0","0","0","-2"],["0","0","0","1"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":182870,"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-63862432c6a86c22"]},{"id":182871,"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-63862432c6a86c22"]},{"id":182872,"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-63862432c6a86c22"]}],"region_exact":{},"incoming_edges":[{"id":"e-63862432c6a86c22","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"0\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-ee02330825df0350","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":"fa2d3185c6ebdc3e1ebc52527169e37cba62e62b245d5f4d04f713d4d899d302","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-c23f36ab5968876b","role":"sequence","name":"Triangle of central factorial numbers T(2*n,2*n-2*k), k >= 0, n >= 1 (in Riordan's notation).","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:   0;\n  1:   1;\n  2:   5,     2;\n  3:  14,    42,       3;\n  4:  30,   294,     255,        4;\n  5:  55,  1254,    4224,     1364,         5;\n  6:  91,  4004,   34320,    52052,      6825,         6;\n  7: 140, 10556,  184470,   784212,    592410,     32766,        7;\n  8: 204, 24276,  754494,  6933212,  16276650,   6430794,   152915,      8;\n  9: 285, 50388, 2538780, 43148924, 232939525, 318942474, 67676252, 699048, 9;\n","recurrence_sentence":"The row polynomials satisfy (n^2-2*n+1)*P(n,x) = -(n^2+(2*n^4-8*n^3+8*n^2)*x+(n^6-6*n^5+13*n^4-12*n^3+4*n^2)*x^2)*P(n-2,x) + ((2*n^3-6*n^2)*x^2+(2*n^5-9*n^4+12*n^3-5*n^2)*x^3)*d/dx P(n-2,x) - (n^2*x^3+(n^4-2*n^3+n^2)*x^4)*d^2/dx^2 P(n-2,x) + (2*n^2-2*n+1+(2*n^4-8*n^3+12*n^2-8*n+2)*x)*P(n-1,x) - (2*n^3-8*n^2+10*n-4)*x^2*d/dx P(n-1,x) + (n^2-2*n+1)*x^3*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^2 + (2*n^4 - 8*n^3 + 8*n^2)*x + (n^6 - 6*n^5 + 13*n^4 - 12*n^3 + 4*n^2)*x^2)*P[n - 2, x] + ((2*n^3 - 6*n^2)*x^2 + (2*n^5 - 9*n^4 + 12*n^3 - 5*n^2)*x^3)*D[P[n - 2, x], x] - (n^2*x^3 + (n^4 - 2*n^3 + n^2)*x^4)*D[P[n - 2, x], {x, 2}] + (2*n^2 - 2*n + 1 + (2*n^4 - 8*n^3 + 12*n^2 - 8*n + 2)*x)*P[n - 1, x] - (2*n^3 - 8*n^2 + 10*n - 4)*x^2*D[P[n - 1, x], x] + (n^2 - 2*n + 1)*x^3*D[P[n - 1, x], {x, 2}])/(n^2 - 2*n + 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(n^2 + (2*n^4 - 8*n^3 + 8*n^2)*x + (n^6 - 6*n^5 + 13*n^4 - 12*n^3 + 4*n^2)*x^2)*v[i - 2] + ((2*n^3 - 6*n^2)*x^2 + (2*n^5 - 9*n^4 + 12*n^3 - 5*n^2)*x^3)*deriv(v[i - 2], x) - (n^2*x^3 + (n^4 - 2*n^3 + n^2)*x^4)*deriv(deriv(v[i - 2], x), x) + (2*n^2 - 2*n + 1 + (2*n^4 - 8*n^3 + 12*n^2 - 8*n + 2)*x)*v[i - 1] - (2*n^3 - 8*n^2 + 10*n - 4)*x^2*deriv(v[i - 1], x) + (n^2 - 2*n + 1)*x^3*deriv(deriv(v[i - 1], x), x))/(n^2 - 2*n + 1)); 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-c23f36ab5968876b","steps":["D"],"edges":["e-63862432c6a86c22"],"text":"A008957 \u2192 D"},"headline":"A008957 \u2192 D","chain":null}