{"id":"n-901a4b04aee4ae2d","revision":0,"first_index":1,"fingerprint":"f6-daf35d08c219b1f8","name":"derivative of basis_map_A013610 of Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.","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":[[],["3"],["12","18"],["39","126","81"],["120","576","810","324"],["351","2232","4860","4212","1215"],["996","7866","23166","31428","19440","4374"],["2763","26046","96633","179172","173745","83106","15309"],["7536","82512","369036","868968","1156680","866052","336798","52488"],["20283","252864","1323216","3779784","6417630","6604740","4010958","1312200","177147"],["54012","755154","4524660","15197544","31424760","41343048","34506486","17583480","4960116","590490"],["142575","2208942","14908617","57566376","140432130","225523440","240841188","168538968","73870299","18305190","1948617"],["373608","6352560","47680326","208063404","585153720","1111074732","1450221570","1298710584","780863976","299968920","66252978","6377292"],["972951","18011160","148796028","724048956","2306851245","5059781964","7819377930","8563889592","6587565489","3467357280","1184759136","235959804","20726199"],["2520276","50454234","454909122","2442066732","8694277560","21641856534","38686975866","50207973768","47242270548","31796115030","14867947710","4572518364","829047960","66961566"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","3","30","246","1830","12870","87270","576774","3740070","23898822","150939750","944286342","5861093670","36138097734","221555895270"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["0"],["3"],["12","18"],["39","126","81"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-1","-6","-9"]]},"deriv_order":0,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["-4","-18","-18"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["-2","-6","-9"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["4","6"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":178988,"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-a675cee3cb3d0667"]},{"id":178989,"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-a675cee3cb3d0667"]},{"id":178990,"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-a675cee3cb3d0667"]}],"region_exact":{},"incoming_edges":[{"id":"e-a675cee3cb3d0667","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-901a4b04aee4ae2d","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":"66eee3eba3d6258916376beff96d028537c0e7032b28249fafa53b9e938bdd79","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-9c403911703f1b61","role":"sequence","name":"basis_map_A013610 of Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.","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:     3;\n   3:    12,     18;\n   4:    39,    126,      81;\n   5:   120,    576,     810,     324;\n   6:   351,   2232,    4860,    4212,    1215;\n   7:   996,   7866,   23166,   31428,   19440,    4374;\n   8:  2763,  26046,   96633,  179172,  173745,   83106,   15309;\n   9:  7536,  82512,  369036,  868968, 1156680,  866052,  336798,   52488;\n  10: 20283, 252864, 1323216, 3779784, 6417630, 6604740, 4010958, 1312200, 177147;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(1+6*x+9*x^2)*P(n-4,x) - (4+18*x+18*x^2)*P(n-3,x) - (2+6*x+9*x^2)*P(n-2,x) + (4+6*x)*P(n-1,x) for n >= 5, with P(1,x) = 0, P(2,x) = 3, P(3,x) = 12 + 18*x, P(4,x) = 39 + 126*x + 81*x^2.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 3;\nP[3, x_] := 12 + 18*x;\nP[4, x_] := 39 + 126*x + 81*x^2;\nP[n_, x_] := P[n, x] = Expand[-(1 + 6*x + 9*x^2)*P[n - 4, x] - (4 + 18*x + 18*x^2)*P[n - 3, x] - (2 + 6*x + 9*x^2)*P[n - 2, x] + (4 + 6*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 3; v[3] = 12 + 18*x; v[4] = 39 + 126*x + 81*x^2; for(n = 5, N, my(i = n); v[i] = -(1 + 6*x + 9*x^2)*v[i - 4] - (4 + 18*x + 18*x^2)*v[i - 3] - (2 + 6*x + 9*x^2)*v[i - 2] + (4 + 6*x)*v[i - 1]); 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-b0fc620949fb5990","steps":["basis A013610","D"],"edges":["e-7f7d545e75881e09","e-a675cee3cb3d0667"],"text":"A122542 \u2192 basis A013610 \u2192 D"},"headline":"A122542 \u2192 basis A013610 \u2192 D","chain":null}