{"id":"n-3ef784a499ce7f14","revision":0,"first_index":0,"fingerprint":"f6-26250b2c4c73936c","name":"derivative of basis_map_A013610 of Riordan array (1, 2+2x).","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"],["30","72"],["120","576","648"],["432","3096","7128","5184"],["1464","13968","48600","72576","38880"],["4776","56880","265032","601344","660960","279936"],["15168","216576","1264896","3856896","6454080","5598720","1959552"],["47232","786240","5520960","21155904","47666880","62985600","45069696","13436928"],["144864","2754432","22597056","104312448","295837920","526839552","574148736","349360128","90699264"],["439008","9386496","88102080","475880832","1628022240","3652884864","5367212928","4971663360","2630278656","604661760"],["1317504","31288320","330666624","2045979648","8199169920","22211241984","41140794240","51382812672","41358864384","19349176320","3990767616"],["3922176","102427776","1203610752","8393870592","38568337920","122484037248","274064902272","431621001216","468370999296","333168629760","139676866560","26121388032"],["11597184","330308352","4271999616","33153919488","171833892480","626116614912","1643278347648","3127418118144","4279735471104","4103234703360","2613952788480","992612745216","169789022208"],["34092672","1051704576","14845845888","126900463104","732412886400","3012097363200","9074891064960","20265574809600","33519726996480","40526849802240","34787521308672","20061226008576","6961349910528","1097098297344"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","6","102","1344","15840","175488","1868928","19365888","196669440","1966694400","19428532224","190042079232","1843779993600","17765739528192","170181580554240"],"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":[["2","6"],["-3","-6"],["1","0"]]},"terms":[{"coeff":{"coeffs":[["-28","-90","-72"],["18","30","0"],["-2","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-6","-36","-66","-36"],["2","8","6","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":184624,"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-31a3567f9640d035"]},{"id":184625,"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-31a3567f9640d035"]},{"id":184626,"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-31a3567f9640d035"]}],"region_exact":{},"incoming_edges":[{"id":"e-31a3567f9640d035","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-3ef784a499ce7f14","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":"7a30007c51fccf6184e9c394d0688d0bdba1f1a426538a3af7f9d85845023351","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-0892e4c474e4320d","role":"sequence","name":"basis_map_A013610 of Riordan array (1, 2+2x).","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:     30,      72;\n  3:    120,     576,      648;\n  4:    432,    3096,     7128,      5184;\n  5:   1464,   13968,    48600,     72576,     38880;\n  6:   4776,   56880,   265032,    601344,    660960,    279936;\n  7:  15168,  216576,  1264896,   3856896,   6454080,   5598720,   1959552;\n  8:  47232,  786240,  5520960,  21155904,  47666880,  62985600,  45069696,  13436928;\n  9: 144864, 2754432, 22597056, 104312448, 295837920, 526839552, 574148736, 349360128, 90699264;\n","recurrence_sentence":"The row polynomials satisfy (n^2-3*n+2-(6*n-6)*x)*P(n,x) = -(2*n^2-18*n+28-(30*n-90)*x+72*x^2)*P(n-1,x) + (2*n-6+(8*n-36)*x+(6*n-66)*x^2-36*x^3)*d/dx 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[(-(2*n^2 - 18*n + 28 - (30*n - 90)*x + 72*x^2)*P[n - 1, x] + (2*n - 6 + (8*n - 36)*x + (6*n - 66)*x^2 - 36*x^3)*D[P[n - 1, x], x])/(n^2 - 3*n + 2 - (6*n - 6)*x)]];\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] = (-(2*n^2 - 18*n + 28 - (30*n - 90)*x + 72*x^2)*v[i - 1] + (2*n - 6 + (8*n - 36)*x + (6*n - 66)*x^2 - 36*x^3)*deriv(v[i - 1], x))/(n^2 - 3*n + 2 - (6*n - 6)*x)); 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-f5a001953ee28e8f","steps":["basis A013610","D"],"edges":["e-937d6a83cea8d959","e-31a3567f9640d035"],"text":"A099040 \u2192 basis A013610 \u2192 D"},"headline":"A099040 \u2192 basis A013610 \u2192 D","chain":null}