{"id":"n-a3fa6cd5be25841d","revision":1,"first_index":0,"fingerprint":"f6-8904488c668d87b5","name":"one_plus_derivative of Square array read by antidiagonals downwards: T(n,k), n>=0, k>=0, in which column 0 is equal to A057427: 0, 1, 1, 1, ..., and for k > 0 column k lists two zeros followed by the partial sums of column k-1.","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","1"],["0","2","1"],["0","0","3","1"],["0","0","3","5","1"],["0","0","0","8","7","1"],["0","0","0","0","15","9","1"],["0","0","0","0","5","25","11","1"],["0","0","0","0","0","18","38","13","1"],["0","0","0","0","0","0","42","54","15","1"],["0","0","0","0","0","0","7","81","73","17","1"],["0","0","0","0","0","0","0","32","139","95","19","1"],["0","0","0","0","0","0","0","0","90","220","120","21","1"],["0","0","0","0","0","0","0","0","9","201","328","148","23","1"],["0","0","0","0","0","0","0","0","0","50","390","467","179","25","1"],["0","0","0","0","0","0","0","0","0","0","165","687","641","213","27","1"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["2","3","4","9","16","25","42","70","112","179","286","452","710","1112","1734"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1","1"],["0","2","1"],["0","0","3","1"],["0","0","3","5","1"],["0","0","0","8","7","1"]],"recurrence":{"denominator":{"coeffs":[["2"]]},"terms":[{"coeff":{"coeffs":[["0","0","0","12","4"]]},"deriv_order":0,"offset":5,"sign":"none"},{"coeff":{"coeffs":[["0","0","0","0","4"]]},"deriv_order":1,"offset":5,"sign":"none"},{"coeff":{"coeffs":[["0","0","0","2","1"]]},"deriv_order":0,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["0","0","0","0","1"]]},"deriv_order":1,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["0","0","6","4"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["0","0","0","4"]]},"deriv_order":1,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["0","4","-4","1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","-4","1"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","4","-1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":160762,"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-8460e4eda37fa7e6"]},{"id":160763,"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-8460e4eda37fa7e6"]},{"id":160764,"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-8460e4eda37fa7e6"]}],"region_exact":{},"incoming_edges":[{"id":"e-8460e4eda37fa7e6","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-a3fa6cd5be25841d","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":"2dc08f0d838534a021b90c1b213f66bd529f43db2cf451b8915efd5fa2229a5e","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-67a75673619cc276","role":"sequence","name":"Square array read by antidiagonals downwards: T(n,k), n>=0, k>=0, in which column 0 is equal to A057427: 0, 1, 1, 1, ..., and for k > 0 column k lists two zeros followed by the partial sums of column k-1.","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, 1;\n  1: 0, 2, 1;\n  2: 0, 0, 3, 1;\n  3: 0, 0, 3, 5,  1;\n  4: 0, 0, 0, 8,  7,  1;\n  5: 0, 0, 0, 0, 15,  9,  1;\n  6: 0, 0, 0, 0,  5, 25, 11,  1;\n  7: 0, 0, 0, 0,  0, 18, 38, 13,  1;\n  8: 0, 0, 0, 0,  0,  0, 42, 54, 15,  1;\n  9: 0, 0, 0, 0,  0,  0,  7, 81, 73, 17, 1;\n","recurrence_sentence":"The row polynomials satisfy 2*P(n,x) = (12*x^3+4*x^4)*P(n-5,x) + 4*x^4*d/dx P(n-5,x) + (2*x^3+x^4)*P(n-4,x) + x^4*d/dx P(n-4,x) + (6*x^2+4*x^3)*P(n-3,x) + 4*x^3*d/dx P(n-3,x) + (4*x-4*x^2+x^3)*P(n-2,x) - (4*x^2-x^3)*d/dx P(n-2,x) + (4*x-x^2)*P(n-1,x) - x^2*d/dx P(n-1,x) for n >= 5, with P(0,x) = 1 + x, P(1,x) = 2*x + x^2, P(2,x) = 3*x^2 + x^3, P(3,x) = 3*x^2 + 5*x^3 + x^4, P(4,x) = 8*x^3 + 7*x^4 + x^5.","mathematica":"P[0, x_] := 1 + x;\nP[1, x_] := 2*x + x^2;\nP[2, x_] := 3*x^2 + x^3;\nP[3, x_] := 3*x^2 + 5*x^3 + x^4;\nP[4, x_] := 8*x^3 + 7*x^4 + x^5;\nP[n_, x_] := P[n, x] = Expand[Together[((12*x^3 + 4*x^4)*P[n - 5, x] + 4*x^4*D[P[n - 5, x], x] + (2*x^3 + x^4)*P[n - 4, x] + x^4*D[P[n - 4, x], x] + (6*x^2 + 4*x^3)*P[n - 3, x] + 4*x^3*D[P[n - 3, x], x] + (4*x - 4*x^2 + x^3)*P[n - 2, x] - (4*x^2 - x^3)*D[P[n - 2, x], x] + (4*x - x^2)*P[n - 1, x] - x^2*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 + x; v[2] = 2*x + x^2; v[3] = 3*x^2 + x^3; v[4] = 3*x^2 + 5*x^3 + x^4; v[5] = 8*x^3 + 7*x^4 + x^5; for(n = 5, N, my(i = n + 1); v[i] = ((12*x^3 + 4*x^4)*v[i - 5] + 4*x^4*deriv(v[i - 5], x) + (2*x^3 + x^4)*v[i - 4] + x^4*deriv(v[i - 4], x) + (6*x^2 + 4*x^3)*v[i - 3] + 4*x^3*deriv(v[i - 3], x) + (4*x - 4*x^2 + x^3)*v[i - 2] - (4*x^2 - x^3)*deriv(v[i - 2], x) + (4*x - x^2)*v[i - 1] - x^2*deriv(v[i - 1], x))/(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-0d55e0bb3c384646","steps":["reverse","D+1"],"edges":["e-0e9e334768ea0a35","e-8460e4eda37fa7e6"],"text":"A102547 \u2192 reverse \u2192 D+1"},"headline":"A102547 \u2192 reverse \u2192 D+1","chain":null}
