{"id":"n-cbfa3e440ba864d3","revision":1,"first_index":0,"fingerprint":"f6-4b74653ef77df9d6","name":"one_plus_derivative of Riordan array (1+2x,x(1+x)).","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"],["3","5","1"],["2","10","7","1"],["0","10","20","9","1"],["0","4","29","33","11","1"],["0","0","21","63","49","13","1"],["0","0","6","66","116","68","15","1"],["0","0","0","36","159","192","90","17","1"],["0","0","0","8","127","325","295","115","19","1"],["0","0","0","0","55","341","594","429","143","21","1"],["0","0","0","0","10","218","771","1001","598","174","23","1"],["0","0","0","0","0","78","650","1547","1586","806","208","25","1"],["0","0","0","0","0","12","345","1617","2842","2394","1057","245","27","1"],["0","0","0","0","0","0","105","1135","3542","4878","3475","1355","285","29","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","9","20","40","78","147","272","495","890","1584","2796","4901","8540","14805"],"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":[["0","0","0","0"],["2","-9","1","1"],["-3","3","1","0"],["1","0","0","0"]]},"terms":[{"coeff":{"coeffs":[["16","-36","-2","6","1"],["-28","21","4","-1","0"],["14","-3","-2","0","0"],["-2","0","0","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","8","-22","6","7","1"],["0","-12","9","7","1","0"],["0","4","1","0","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":160926,"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-3009f31f72baba04"]},{"id":160927,"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-3009f31f72baba04"]},{"id":160928,"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-3009f31f72baba04"]}],"region_exact":{},"incoming_edges":[{"id":"e-3009f31f72baba04","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-cbfa3e440ba864d3","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":"04264c40aefffabf99cd73d2172f7505eebf471857adcada60472e733ecae5c7","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-c27b11147fce8bfa","role":"sequence","name":"Riordan array (1+2x,x(1+x)).","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":[{"id":"l-9aaf6bbccc4a901e","kind":"oeis_match","node_a_id":"n-cbfa3e440ba864d3","node_b_id":null,"fingerprint":null,"oeis_id":"A023607","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A023607","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A023607","relation":"match_row_sums"}],"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: 3,  5,  1;\n  3: 2, 10,  7,  1;\n  4: 0, 10, 20,  9,   1;\n  5: 0,  4, 29, 33,  11,   1;\n  6: 0,  0, 21, 63,  49,  13,   1;\n  7: 0,  0,  6, 66, 116,  68,  15,   1;\n  8: 0,  0,  0, 36, 159, 192,  90,  17,  1;\n  9: 0,  0,  0,  8, 127, 325, 295, 115, 19, 1;\n","recurrence_sentence":"The row polynomials satisfy (n^3-3*n^2+2*n+(3*n^2-9*n)*x+(n^2+n)*x^2+n*x^3)*P(n,x) = -(2*n^3-14*n^2+28*n-16+(3*n^2-21*n+36)*x+(2*n^2-4*n+2)*x^2+(n-6)*x^3-x^4)*P(n-1,x) + ((4*n^2-12*n+8)*x+(n^2+9*n-22)*x^2+(7*n+6)*x^3+(n+7)*x^4+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[(-(2*n^3 - 14*n^2 + 28*n - 16 + (3*n^2 - 21*n + 36)*x + (2*n^2 - 4*n + 2)*x^2 + (n - 6)*x^3 - x^4)*P[n - 1, x] + ((4*n^2 - 12*n + 8)*x + (n^2 + 9*n - 22)*x^2 + (7*n + 6)*x^3 + (n + 7)*x^4 + x^5)*D[P[n - 1, x], x])/(n^3 - 3*n^2 + 2*n + (3*n^2 - 9*n)*x + (n^2 + n)*x^2 + n*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] = (-(2*n^3 - 14*n^2 + 28*n - 16 + (3*n^2 - 21*n + 36)*x + (2*n^2 - 4*n + 2)*x^2 + (n - 6)*x^3 - x^4)*v[i - 1] + ((4*n^2 - 12*n + 8)*x + (n^2 + 9*n - 22)*x^2 + (7*n + 6)*x^3 + (n + 7)*x^4 + x^5)*deriv(v[i - 1], x))/(n^3 - 3*n^2 + 2*n + (3*n^2 - 9*n)*x + (n^2 + n)*x^2 + n*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-c27b11147fce8bfa","steps":["D+1"],"edges":["e-3009f31f72baba04"],"text":"A113214 \u2192 D+1"},"headline":"A113214 \u2192 D+1","chain":null}
