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DELTA (0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["2"],["4","1"],["8","3"],["15","9","1"],["28","23","4"],["51","54","15","1"],["92","120","46","5"],["164","255","127","22","1"],["290","525","324","78","6"],["509","1053","779","245","30","1"],["888","2069","1790","701","120","7"],["1541","3996","3966","1870","420","39","1"],["2662","7608","8532","4722","1326","173","8"],["4580","14309","17910","11408","3871","665","49","1"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","2","5","11","25","55","121","263","569","1223","2617","5575","11833","25031","52793"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["-20","-20","14","8"],["13","25","3","4"],["5","0","4","0"],["-1","1","0","0"]]},"terms":[{"coeff":{"coeffs":[["-44","-51","17","14","-8"],["30","60","17","2","8"],["11","2","6","8","0"],["-3","1","2","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-20","-56","-67","-63","-48","-16"],["5","9","-16","-36","-16","0"],["5","4","-5","-4","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":171778,"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-ce7b34fc9023be1d"]},{"id":171779,"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-ce7b34fc9023be1d"]},{"id":171780,"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-ce7b34fc9023be1d"]}],"region_exact":{},"incoming_edges":[{"id":"e-ce7b34fc9023be1d","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-708cc2529cf30076","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":"212be6490ac2dee1ca1acc8c5eeb949e9e06faeee3a4f1eee946ed76802fe64d","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-72cbbb2f152c8f9c","role":"sequence","name":"Triangle T(n,k), read by rows, given by (2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.","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:   2;\n  2:   4,   1;\n  3:   8,   3;\n  4:  15,   9,   1;\n  5:  28,  23,   4;\n  6:  51,  54,  15,  1;\n  7:  92, 120,  46,  5;\n  8: 164, 255, 127, 22, 1;\n  9: 290, 525, 324, 78, 6;\n","recurrence_sentence":"The row polynomials satisfy -(n^3-5*n^2-13*n+20-(n^3+25*n-20)*x-(4*n^2+3*n+14)*x^2-(4*n+8)*x^3)*P(n,x) = -(3*n^3-11*n^2-30*n+44-(n^3+2*n^2+60*n-51)*x-(2*n^3+6*n^2+17*n+17)*x^2-(8*n^2+2*n+14)*x^3-(8*n-8)*x^4)*P(n-1,x) + (5*n^2+5*n-20+(4*n^2+9*n-56)*x-(5*n^2+16*n+67)*x^2-(4*n^2+36*n+63)*x^3-(16*n+48)*x^4-16*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(3*n^3 - 11*n^2 - 30*n + 44 - (n^3 + 2*n^2 + 60*n - 51)*x - (2*n^3 + 6*n^2 + 17*n + 17)*x^2 - (8*n^2 + 2*n + 14)*x^3 - (8*n - 8)*x^4)*P[n - 1, x] + (5*n^2 + 5*n - 20 + (4*n^2 + 9*n - 56)*x - (5*n^2 + 16*n + 67)*x^2 - (4*n^2 + 36*n + 63)*x^3 - (16*n + 48)*x^4 - 16*x^5)*D[P[n - 1, x], x])/(-n^3 + 5*n^2 + 13*n - 20 + (n^3 + 25*n - 20)*x + (4*n^2 + 3*n + 14)*x^2 + (4*n + 8)*x^3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(3*n^3 - 11*n^2 - 30*n + 44 - (n^3 + 2*n^2 + 60*n - 51)*x - (2*n^3 + 6*n^2 + 17*n + 17)*x^2 - (8*n^2 + 2*n + 14)*x^3 - (8*n - 8)*x^4)*v[i - 1] + (5*n^2 + 5*n - 20 + (4*n^2 + 9*n - 56)*x - (5*n^2 + 16*n + 67)*x^2 - (4*n^2 + 36*n + 63)*x^3 - (16*n + 48)*x^4 - 16*x^5)*deriv(v[i - 1], x))/(-n^3 + 5*n^2 + 13*n - 20 + (n^3 + 25*n - 20)*x + (4*n^2 + 3*n + 14)*x^2 + (4*n + 8)*x^3)); 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-72cbbb2f152c8f9c","steps":["D+1"],"edges":["e-ce7b34fc9023be1d"],"text":"A209599 \u2192 D+1"},"headline":"A209599 \u2192 D+1","chain":null}