{"id":"n-5240258528480355","revision":1,"first_index":0,"fingerprint":"f6-441b88ccebee6207","name":"one_plus_derivative of Riordan array (1, 3+3x).","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","3"],["3","21","9"],["0","36","99","27"],["0","18","252","405","81"],["0","0","243","1377","1539","243"],["0","0","81","1971","6561","5589","729"],["0","0","0","1296","12474","28674","19683","2187"],["0","0","0","324","12231","68040","118098","67797","6561"],["0","0","0","0","6075","88695","336069","465831","229635","19683"],["0","0","0","0","1215","65853","546750","1546209","1778031","767637","59049"],["0","0","0","0","0","26244","540189","3015873","6744708","6613488","2539107","177147"],["0","0","0","0","0","4374","322218","3720087","15339618","28225422","24091992","8325909","531441"],["0","0","0","0","0","0","107163","2954637","22687938","73338858","114259815","86270589","27103491","1594323"],["0","0","0","0","0","0","15309","1471851","22504230","126482958","333922095","450130527","304515693","87687765","4782969"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","6","33","162","756","3402","14931","64314","273051","1145988","4764744","19656756","80561061","328316814","1331513397"],"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"],["0","-3","-5","3"],["-1","1","3","0"],["1","0","0","0"]]},"terms":[{"coeff":{"coeffs":[["0","-12","-16","0","9"],["-6","-1","10","3","0"],["8","7","-6","0","0"],["-2","0","0","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-8","-2","15","9"],["0","-4","1","15","9","0"],["0","4","3","0","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":164540,"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-b19f0f91a9476669"]},{"id":164541,"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-b19f0f91a9476669"]},{"id":164542,"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-b19f0f91a9476669"]}],"region_exact":{},"incoming_edges":[{"id":"e-b19f0f91a9476669","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-5240258528480355","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":"fa1281201ed4d149a87e9820cebe153d9e2461ee089ca29ce6557d649475e75d","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-016d7724d51f6416","role":"sequence","name":"Riordan array (1, 3+3x).","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-bf5eb7454d643bb2","kind":"oeis_match","node_a_id":"n-5240258528480355","node_b_id":null,"fingerprint":null,"oeis_id":"A099432","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":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A099432","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A099432","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-5240258528480355","oeis_id":"A099432","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":23,"oeis_terms_available":23,"term_count":200,"raw_size":12851}],"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,  3;\n  2: 3, 21,   9;\n  3: 0, 36,  99,   27;\n  4: 0, 18, 252,  405,    81;\n  5: 0,  0, 243, 1377,  1539,   243;\n  6: 0,  0,  81, 1971,  6561,  5589,    729;\n  7: 0,  0,   0, 1296, 12474, 28674,  19683,   2187;\n  8: 0,  0,   0,  324, 12231, 68040, 118098,  67797,   6561;\n  9: 0,  0,   0,    0,  6075, 88695, 336069, 465831, 229635, 19683;\n","recurrence_sentence":"The row polynomials satisfy (n^3-n^2+(n^2-3*n)*x+(3*n^2-5*n)*x^2+3*n*x^3)*P(n,x) = -(2*n^3-8*n^2+6*n-(7*n^2-n-12)*x+(6*n^2-10*n+16)*x^2-3*n*x^3-9*x^4)*P(n-1,x) + ((4*n^2-4*n)*x+(3*n^2+n-8)*x^2+(15*n-2)*x^3+(9*n+15)*x^4+9*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 - 8*n^2 + 6*n - (7*n^2 - n - 12)*x + (6*n^2 - 10*n + 16)*x^2 - 3*n*x^3 - 9*x^4)*P[n - 1, x] + ((4*n^2 - 4*n)*x + (3*n^2 + n - 8)*x^2 + (15*n - 2)*x^3 + (9*n + 15)*x^4 + 9*x^5)*D[P[n - 1, x], x])/(n^3 - n^2 + (n^2 - 3*n)*x + (3*n^2 - 5*n)*x^2 + 3*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 - 8*n^2 + 6*n - (7*n^2 - n - 12)*x + (6*n^2 - 10*n + 16)*x^2 - 3*n*x^3 - 9*x^4)*v[i - 1] + ((4*n^2 - 4*n)*x + (3*n^2 + n - 8)*x^2 + (15*n - 2)*x^3 + (9*n + 15)*x^4 + 9*x^5)*deriv(v[i - 1], x))/(n^3 - n^2 + (n^2 - 3*n)*x + (3*n^2 - 5*n)*x^2 + 3*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-016d7724d51f6416","steps":["D+1"],"edges":["e-b19f0f91a9476669"],"text":"A099093 \u2192 D+1"},"headline":"A099093 \u2192 D+1","chain":null}
