{"id":"n-b87d21ddf4200c60","revision":0,"first_index":0,"fingerprint":"f6-333a237dceb0e170","name":"one_plus_derivative of Triangular array A066667 or A008297 unsigned and transposed.","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","2"],["7","18","6"],["13","84","108","24"],["21","260","840","720","120"],["31","630","3900","8400","5400","720"],["43","1302","13230","54600","88200","45360","5040"],["57","2408","36456","246960","764400","987840","423360","40320"],["73","4104","86688","874944","4445280","11007360","11854080","4354560","362880"],["91","6570","184680","2600640","19686240","80015040","165110400","152409600","48988800","3628800"],["111","10010","361350","6771600","71517600","433097280","1466942400","2594592000","2095632000","598752000","39916800"],["133","14652","660660","15899400","223462800","1888064640","9528140160","27662342400","42810768000","30735936000","7903526400","479001600"],["157","20748","1142856","34354320","620076600","6972039360","49089680640","212341409280","539415676800","742053312000","479480601600","112086374400","6227020800"],["183","28574","1888068","69333264","1563121560","22570788240","211485193920","1276331696640","4830767061120","10908183686400","13505370278400","7933224499200","1699976678400","87178291200"],["211","38430","3000270","132164760","3639996360","65651105520","789977588400","6344555817600","33503707036800","112717898092800","229071857414400","257829796224000","138831428736000","27461161728000","1307674368000"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","31","229","1961","19081","207775","2501801","32989969","472630861","7307593151","121247816845","2148321709561","40476722545169","807927483311551"],"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":[["1","-2","1"],["-1","2","0"],["1","0","0"]]},"terms":[{"coeff":{"coeffs":[["1","-2","1","0"],["1","2","-2","2"],["1","0","4","0"],["0","2","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-1","0","-1"],["0","0","-1","-2","0"],["0","0","-1","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":181894,"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-b73e67fd94d5dbed"]},{"id":181895,"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-b73e67fd94d5dbed"]},{"id":181896,"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-b73e67fd94d5dbed"]},{"id":181897,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-b73e67fd94d5dbed"]}],"region_exact":{},"incoming_edges":[{"id":"e-b73e67fd94d5dbed","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-b87d21ddf4200c60","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":"4501c39ec0e7531b46011095ece28d706cf8dfd57ab34fe9db43005ea2cacf9b","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-d55979c30b97975a","role":"sequence","name":"Triangular array A066667 or A008297 unsigned and transposed.","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:  3,    2;\n  2:  7,   18,      6;\n  3: 13,   84,    108,      24;\n  4: 21,  260,    840,     720,      120;\n  5: 31,  630,   3900,    8400,     5400,      720;\n  6: 43, 1302,  13230,   54600,    88200,    45360,      5040;\n  7: 57, 2408,  36456,  246960,   764400,   987840,    423360,     40320;\n  8: 73, 4104,  86688,  874944,  4445280, 11007360,  11854080,   4354560,   362880;\n  9: 91, 6570, 184680, 2600640, 19686240, 80015040, 165110400, 152409600, 48988800, 3628800;\n","recurrence_sentence":"The row polynomials satisfy (n^2-n+1+(2*n-2)*x+x^2)*P(n,x) = (n^2+n+1+(2*n^3+2*n-2)*x+(4*n^2-2*n+1)*x^2+2*n*x^3)*P(n-1,x) - ((n^2+n+1)*x^2+2*n*x^3+x^4)*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[((n^2 + n + 1 + (2*n^3 + 2*n - 2)*x + (4*n^2 - 2*n + 1)*x^2 + 2*n*x^3)*P[n - 1, x] - ((n^2 + n + 1)*x^2 + 2*n*x^3 + x^4)*D[P[n - 1, x], x])/(n^2 - n + 1 + (2*n - 2)*x + x^2)]];\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] = ((n^2 + n + 1 + (2*n^3 + 2*n - 2)*x + (4*n^2 - 2*n + 1)*x^2 + 2*n*x^3)*v[i - 1] - ((n^2 + n + 1)*x^2 + 2*n*x^3 + x^4)*deriv(v[i - 1], x))/(n^2 - n + 1 + (2*n - 2)*x + 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-d55979c30b97975a","steps":["D+1"],"edges":["e-b73e67fd94d5dbed"],"text":"A089231 \u2192 D+1"},"headline":"A089231 \u2192 D+1","chain":null}
