{"id":"n-6d54ebf30881a6d6","revision":0,"first_index":1,"fingerprint":"f6-3e125a4b64c05092","name":"one_plus_derivative of Triangle read by rows. A generalization of unsigned Lah numbers, called L[3,1].","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"],["3","1"],["20","12","1"],["200","168","27","1"],["2640","2816","660","48","1"],["43120","55440","16940","1820","75","1"],["837760","1256640","471240","66640","4080","108","1"],["18849600","32253760","14294280","2499000","202300","7980","147","1"],["481712000","924887040","472077760","98094080","9853200","515200","14168","192","1"],["13776963200","29307358080","16908091200","4064773440","486748080","31646160","1155336","23400","243","1"],["435853017600","1016990374400","653779526400","178303507200","24764376000","1922880960","87403680","2352480","36540","300","1"],["15109571276800","38355065548800","27168171430400","8281207334400","1307559052800","117936698880","6409603200","214980480","4440480","54560","363","1"],["569514609664000","1562097215078400","1208184564787200","406796149760000","71915747904000","7383862886400","466619112960","18727188480","482328000","7884800","78540","432","1"],["23187380536320000","68325481313689600","57272829924710400","21100516288051200","4126801654976000","474951597120000","34196514992640","1596938703360","49242473280","1004203200","13313300","109668","507","1"],["1014061442121728000","3194293542683443200","2883737226033664000","1153494890413465600","247177476517171200","31503011712972800","2544474022970880","135464455526400","4862826608640","118775937280","1965043080","21550256","149240","588","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","33","396","6165","117396","2636469","68107068","1987153641","64576759140","2311703475561","90346140421164","3826377986673213","174523805028663156","8526952511550231165"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0","-2"],["1","0"]]},"terms":[{"coeff":{"coeffs":[["-4","6","-2"],["-1","-5","0"],["3","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-6","-6"],["0","3","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":187949,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-b5b071b55fd67dd5"]},{"id":187950,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-b5b071b55fd67dd5"]},{"id":187951,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-b5b071b55fd67dd5"]}],"region_exact":{},"incoming_edges":[{"id":"e-b5b071b55fd67dd5","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-6d54ebf30881a6d6","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":"08fa0b59118a9b388b150ef788bb55571cff114b57094bdd380243218f8d9d8a","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-33e7d067844832dd","role":"sequence","name":"Triangle read by rows. A generalization of unsigned Lah numbers, called L[3,1].","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:           1;\n   2:           3,           1;\n   3:          20,          12,           1;\n   4:         200,         168,          27,          1;\n   5:        2640,        2816,         660,         48,         1;\n   6:       43120,       55440,       16940,       1820,        75,        1;\n   7:      837760,     1256640,      471240,      66640,      4080,      108,       1;\n   8:    18849600,    32253760,    14294280,    2499000,    202300,     7980,     147,     1;\n   9:   481712000,   924887040,   472077760,   98094080,   9853200,   515200,   14168,   192,   1;\n  10: 13776963200, 29307358080, 16908091200, 4064773440, 486748080, 31646160, 1155336, 23400, 243, 1;\n","recurrence_sentence":"The row polynomials satisfy (n-2*x)*P(n,x) = (3*n^2-n-4-(5*n-6)*x-2*x^2)*P(n-1,x) + ((3*n-6)*x-6*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((3*n^2 - n - 4 - (5*n - 6)*x - 2*x^2)*P[n - 1, x] + ((3*n - 6)*x - 6*x^2)*D[P[n - 1, x], x])/(n - 2*x)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((3*n^2 - n - 4 - (5*n - 6)*x - 2*x^2)*v[i - 1] + ((3*n - 6)*x - 6*x^2)*deriv(v[i - 1], x))/(n - 2*x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-33e7d067844832dd","steps":["D+1"],"edges":["e-b5b071b55fd67dd5"],"text":"A290596 \u2192 D+1"},"headline":"A290596 \u2192 D+1","chain":null}