{"id":"n-eb5c2993685be52f","revision":0,"first_index":1,"fingerprint":"f6-d0e0bf31a1431b92","name":"derivative of Triangle of numbers used for basis change between certain falling factorials.","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":[[],["4"],["12","60"],["24","336","1008"],["40","1080","8640","20160"],["60","2640","39600","237600","475200"],["84","5460","131040","1441440","7207200","12972960"],["112","10080","352800","6115200","55036800","242161920","403603200"],["144","17136","822528","20563200","287884800","2245501440","8982005760","14114580480"],["180","27360","1723680","58605120","1172102400","14065228800","98456601600","365695948800","548543923200"],["220","41580","3326400","147470400","3981700800","67688913600","722015078400","4641525504000","16245339264000","23465490048000"],["264","60720","6011280","336631680","11782108800","268632080640","4029481209600","39143531750400","234861190502400","782870635008000","1096018889011200"],["312","85800","10296000","710424000","31258656000","919004486400","18380089728000","249444074880000","2244996673920000","12721647818880000","40709273020416000","55512645027840000"],["364","117936","16864848","1405404000","75891816000","2792818828800","71682349939200","1290282298905600","16128528736320000","136196464884480000","735460910376192000","2273242813890048000","3030990418520064000"],["420","158340","26601120","2633510880","171178207200","7703019324000","246496618368000","5669422222464000","93545466670656000","1091363777824320000","8730910222594560000","45241989335262720000","135725968005788160000","177487804315261440000"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","4","72","1368","29920","755100","21758184","707280112","25651375488","1027994161140","45146191347400","2157204484374984","111457337689612512","6193383971188980748","368377504919100489960"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["0"],["4"]],"recurrence":{"denominator":{"coeffs":[["-2"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0","0"],["1","-7","6"],["0","4","-5"],["0","0","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","0","0"],["0","0","-1","2"],["0","0","0","-1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":189377,"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-53e96264afefea15"]},{"id":189378,"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-53e96264afefea15"]},{"id":189379,"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-53e96264afefea15"]}],"region_exact":{},"incoming_edges":[{"id":"e-53e96264afefea15","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-eb5c2993685be52f","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":"6e738214530538b286ccc65aa81e07e27dd4b34e1e16f125aa27b0b08c36325e","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-ee693a689121a12c","role":"sequence","name":"Triangle of numbers used for basis change between certain falling factorials.","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:   4;\n   3:  12,    60;\n   4:  24,   336,    1008;\n   5:  40,  1080,    8640,    20160;\n   6:  60,  2640,   39600,   237600,     475200;\n   7:  84,  5460,  131040,  1441440,    7207200,    12972960;\n   8: 112, 10080,  352800,  6115200,   55036800,   242161920,   403603200;\n   9: 144, 17136,  822528, 20563200,  287884800,  2245501440,  8982005760,  14114580480;\n  10: 180, 27360, 1723680, 58605120, 1172102400, 14065228800, 98456601600, 365695948800, 548543923200;\n","recurrence_sentence":"The row polynomials satisfy (n-2)*P(n,x) = (n+(4*n^2-7*n)*x+(n^3-5*n^2+6*n)*x^2)*P(n-1,x) - (n*x^2+(n^2-2*n)*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 4.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 4;\nP[n_, x_] := P[n, x] = Expand[Together[((n + (4*n^2 - 7*n)*x + (n^3 - 5*n^2 + 6*n)*x^2)*P[n - 1, x] - (n*x^2 + (n^2 - 2*n)*x^3)*D[P[n - 1, x], x])/(n - 2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 4; for(n = 3, N, my(i = n); v[i] = ((n + (4*n^2 - 7*n)*x + (n^3 - 5*n^2 + 6*n)*x^2)*v[i - 1] - (n*x^2 + (n^2 - 2*n)*x^3)*deriv(v[i - 1], x))/(n - 2)); 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-ee693a689121a12c","steps":["D"],"edges":["e-53e96264afefea15"],"text":"A089503 \u2192 D"},"headline":"A089503 \u2192 D","chain":null}