{"id":"n-63a4f3580f22ae58","revision":0,"first_index":0,"fingerprint":"f6-50c545bfd2603dc2","name":"derivative of basis_map_A013610 of Triangle read by rows: coefficients of 1; 1(X+2); 1(X+2)(X+3); 1(X+2)(X+3)(X+4); ....","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":[[],["6"],["51","108"],["399","1764","1944"],["3294","22554","51354","38880"],["29538","276462","968436","1504656","874800"],["289224","3452292","16456041","39153456","46495620","22044960"],["3086856","44953380","272386881","878974416","1593064260","1537478496","617258880"],["35767440","616224024","4544480700","18595671876","45596029140","66990066984","54603161424","19046845440"],["447873840","8924510664","77719368780","386331618636","1198894015890","2378347785864","2945287120104","2081623691520","642831033600"],["6033113280","136653801408","1374395703228","8055631649952","30322907459415","76016376459612","126909835197012","136058884256448","84994251950880","23570471232000"],["87052501440","2210883574464","25246449417996","170694205204320","756701221220235","2298162496221684","4842515668864860","6990198714868032","6615310087747584","3706186204051200","933390660787200"],["1340161925760","37742465154816","482786980958088","3702534242596272","18915159600793890","67587892100210574","172361612942194662","313700623248986208","399309309843301080","338550043992445440","172062857404817280","39711529931673600"],["21933173001600","678710579086080","9619580019124728","82574084572597008","478110441748212630","1967142425383170666","5897230907043238056","12978866769960136896","20811994096164114528","23712685119240732000","18221954893523238528","8479280233704913920","1806874611891148800"],["380353441305600","12833354835521280","199726244394306048","1898360154309035808","12294559325179983780","57292242660534212676","197630671845308268861","510948029253014433216","990047035767857762808","1420002666716057438400","1465329191698788050448","1030200814875873968640","442274905183372934400","87563923499340288000"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","6","159","4107","116082","3653892","127891593","4947203169","210028247028","9720407018898","487445440823235","26340703644459015","1526423432915057670","94447033807002715440","6215695340932307509965"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["6"]],"recurrence":{"denominator":{"coeffs":[["0"],["1"]]},"terms":[{"coeff":{"coeffs":[["-1","0","0"],["-3","-6","0"],["-3","-12","-9"],["-1","-6","-9"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1","0"],["4","6"],["2","6"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":191097,"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-685852d2ffd6d75b"]},{"id":191098,"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-685852d2ffd6d75b"]},{"id":191099,"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-685852d2ffd6d75b"]},{"id":191100,"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-685852d2ffd6d75b"]}],"region_exact":{},"incoming_edges":[{"id":"e-685852d2ffd6d75b","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-63a4f3580f22ae58","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":"af42d64c51050330903869cef932924437a2c1811765aa6f750a460c7aea76df","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-70594f2875298da0","role":"sequence","name":"basis_map_A013610 of Triangle read by rows: coefficients of 1; 1(X+2); 1(X+2)(X+3); 1(X+2)(X+3)(X+4); ....","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:         0;\n  1:         6;\n  2:        51,        108;\n  3:       399,       1764,        1944;\n  4:      3294,      22554,       51354,        38880;\n  5:     29538,     276462,      968436,      1504656,        874800;\n  6:    289224,    3452292,    16456041,     39153456,      46495620,      22044960;\n  7:   3086856,   44953380,   272386881,    878974416,    1593064260,    1537478496,     617258880;\n  8:  35767440,  616224024,  4544480700,  18595671876,   45596029140,   66990066984,   54603161424,   19046845440;\n  9: 447873840, 8924510664, 77719368780, 386331618636, 1198894015890, 2378347785864, 2945287120104, 2081623691520, 642831033600;\n","recurrence_sentence":"The row polynomials satisfy n*P(n,x) = -(n^3+3*n^2+3*n+1+(6*n^3+12*n^2+6*n)*x+(9*n^3+9*n^2)*x^2)*P(n-2,x) + (2*n^2+4*n+1+(6*n^2+6*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 6.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 6;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + 3*n^2 + 3*n + 1 + (6*n^3 + 12*n^2 + 6*n)*x + (9*n^3 + 9*n^2)*x^2)*P[n - 2, x] + (2*n^2 + 4*n + 1 + (6*n^2 + 6*n)*x)*P[n - 1, x])/(n)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 6; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 + 3*n^2 + 3*n + 1 + (6*n^3 + 12*n^2 + 6*n)*x + (9*n^3 + 9*n^2)*x^2)*v[i - 2] + (2*n^2 + 4*n + 1 + (6*n^2 + 6*n)*x)*v[i - 1])/(n)); 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-2feb6dc01f424b0a","steps":["basis A013610","D"],"edges":["e-45b4054949a52789","e-685852d2ffd6d75b"],"text":"A145324 \u2192 basis A013610 \u2192 D"},"headline":"A145324 \u2192 basis A013610 \u2192 D","chain":null}