{"id":"n-9e89907ee1f3ed20","revision":0,"first_index":1,"fingerprint":"f6-bbde493405ec4710","name":"derivative of Triangle read by rows, Lah numbers with level 2.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[[],["1"],["4","2"],["36","40","3"],["576","1072","168","4"],["14400","39744","9840","480","5"],["518400","1976256","689376","52000","1100","6"],["25401600","127517184","58803840","6119168","198100","2184","7"],["1625702400","10379695104","6071659776","818824448","36175360","606816","3920","8"],["131681894400","1041220915200","750240375552","126006266880","7137166400","162346368","1589952","6528","9"],["13168189440000","126251094528000","109596445452288","22295379603456","1556392448000","45092532480","596431808","3703680","10260","10"],["1593350922240000","18206493990912000","18711175923228672","4516022996213760","378057525304320","13411359252480","224976747072","1881628672","7870500","15400","11"],["229442532802560000","3080084186308608000","3694700221939187712","1041053408712400896","102427755223173120","4329551854331904","88539051845888","936384232960","5265032256","15536400","22264","12"],["38775788043632640000","604155385582092288000","835927676215780147200","271435939592960212992","30902938020387532800","1525688107071344640","37013741546419456","475741181007872","3375311486976","13369653440","28867696","31200","13"],["7600054456551997440000","136012513332058030080000","214903718189612856115200","79559556777319875084288","10351846843096760893440","587856932277220454400","16585466076817684480","252103279734665216","2168859588330240","10822892766400","31334411680","50992032","42588","14"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","1","6","79","1820","64469","3237138","218042083","18932667832","2056450561289","272913194149982","43418739583412887","8152127137556931412","1782760908433361632285","449032386279987158959978"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["0"],["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-36","-18","-1"],["60","12","0"],["-37","-2","0"],["10","0","0"],["-1","0","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","27","4"],["0","-36","-2"],["0","15","0"],["0","-2","0"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","-9","-1"],["0","0","6","0"],["0","0","-1","0"]]},"deriv_order":2,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["10","2"],["-8","0"],["2","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-1"],["0","2"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","1"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":190309,"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-2838d0ff29cf8023"]},{"id":190310,"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-2838d0ff29cf8023"]},{"id":190311,"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-2838d0ff29cf8023"]}],"region_exact":{},"incoming_edges":[{"id":"e-2838d0ff29cf8023","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"0\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-9e89907ee1f3ed20","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":"3fe297515d52d06389c9b3e3048c4fcbc012f7a44da9fdfc79f17a0abf3909d7","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-313c8ccf28f71545","role":"sequence","name":"Triangle read by rows, Lah numbers with level 2.","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:            0;\n   2:            1;\n   3:            4,             2;\n   4:           36,            40,            3;\n   5:          576,          1072,          168,            4;\n   6:        14400,         39744,         9840,          480,          5;\n   7:       518400,       1976256,       689376,        52000,       1100,         6;\n   8:     25401600,     127517184,     58803840,      6119168,     198100,      2184,       7;\n   9:   1625702400,   10379695104,   6071659776,    818824448,   36175360,    606816,    3920,    8;\n  10: 131681894400, 1041220915200, 750240375552, 126006266880, 7137166400, 162346368, 1589952, 6528, 9;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^4-10*n^3+37*n^2-60*n+36+(2*n^2-12*n+18)*x+x^2)*P(n-2,x) - ((2*n^3-15*n^2+36*n-27)*x+(2*n-4)*x^2)*d/dx P(n-2,x) - ((n^2-6*n+9)*x^2+x^3)*d^2/dx^2 P(n-2,x) + (2*n^2-8*n+10+2*x)*P(n-1,x) + (2*n-1)*x*d/dx P(n-1,x) + x^2*d^2/dx^2 P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 1.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[-(n^4 - 10*n^3 + 37*n^2 - 60*n + 36 + (2*n^2 - 12*n + 18)*x + x^2)*P[n - 2, x] - ((2*n^3 - 15*n^2 + 36*n - 27)*x + (2*n - 4)*x^2)*D[P[n - 2, x], x] - ((n^2 - 6*n + 9)*x^2 + x^3)*D[P[n - 2, x], {x, 2}] + (2*n^2 - 8*n + 10 + 2*x)*P[n - 1, x] + (2*n - 1)*x*D[P[n - 1, x], x] + x^2*D[P[n - 1, x], {x, 2}]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = -(n^4 - 10*n^3 + 37*n^2 - 60*n + 36 + (2*n^2 - 12*n + 18)*x + x^2)*v[i - 2] - ((2*n^3 - 15*n^2 + 36*n - 27)*x + (2*n - 4)*x^2)*deriv(v[i - 2], x) - ((n^2 - 6*n + 9)*x^2 + x^3)*deriv(deriv(v[i - 2], x), x) + (2*n^2 - 8*n + 10 + 2*x)*v[i - 1] + (2*n - 1)*x*deriv(v[i - 1], x) + x^2*deriv(deriv(v[i - 1], x), 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-313c8ccf28f71545","steps":["D"],"edges":["e-2838d0ff29cf8023"],"text":"A390433 \u2192 D"},"headline":"A390433 \u2192 D","chain":null}