{"id":"n-3ee2e71fbf42ce63","revision":0,"first_index":0,"fingerprint":"f6-f99aa997effff5f8","name":"derivative of basis_map_A013611 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":[[],["8"],["40","128"],["160","1024","1536"],["576","5504","16896","16384"],["1952","24832","115200","229376","163840"],["6368","101120","628224","1900544","2785280","1572864"],["20224","385024","2998272","12189696","27197440","31457280","14680064"],["62976","1397760","13086720","66863104","200867840","353894400","337641472","134217728"],["193152","4896768","53563392","329678848","1246658560","2960130048","4301258752","3489660928","1207959552"],["585344","16687104","208834560","1504018432","6860472320","20524302336","40208695296","49660559360","35030827008","10737418240"],["1756672","55623680","783802368","6466306048","34551234560","124797321216","308207943680","513248591872","550829555712","343597383680","94489280512"],["5229568","182093824","2853003264","26528776192","162526658560","688195633152","2053168431104","4311341858816","6237903126528","5916317450240","3307124817920","824633720832"],["15462912","587214848","10126221312","104782757888","724106608640","3517933682688","12310686990336","31238907756544","56998779420672","72864120176640","61890478735360","31336081391616","7146825580544"],["45456896","1869697024","35190153216","401068130304","3086381875200","16923937996800","67984917790720","202427177369600","446425611632640","719664720117760","823663058223104","633318697598976","293019848802304","61572651155456"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","8","168","2720","39360","535200","6994400","88928000","1108032000","13594000000","164752400000","1977028800000","23530780800000","278143432000000","3268525176000000"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["8"]],"recurrence":{"denominator":{"coeffs":[["4","16"],["-6","-16"],["2","0"]]},"terms":[{"coeff":{"coeffs":[["-56","-240","-256"],["36","80","0"],["-4","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-9","-72","-176","-128"],["3","16","16","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":186295,"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-7f1b4c57a31942c3"]},{"id":186296,"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-7f1b4c57a31942c3"]},{"id":186297,"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-7f1b4c57a31942c3"]}],"region_exact":{},"incoming_edges":[{"id":"e-7f1b4c57a31942c3","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-3ee2e71fbf42ce63","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":"5b5b85526f848164a43b41617916895209a7125184c305d170ee1ad16bf98177","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-8588fa6ee4014167","role":"sequence","name":"basis_map_A013611 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:      8;\n  2:     40,     128;\n  3:    160,    1024,     1536;\n  4:    576,    5504,    16896,     16384;\n  5:   1952,   24832,   115200,    229376,     163840;\n  6:   6368,  101120,   628224,   1900544,    2785280,    1572864;\n  7:  20224,  385024,  2998272,  12189696,   27197440,   31457280,   14680064;\n  8:  62976, 1397760, 13086720,  66863104,  200867840,  353894400,  337641472,  134217728;\n  9: 193152, 4896768, 53563392, 329678848, 1246658560, 2960130048, 4301258752, 3489660928, 1207959552;\n","recurrence_sentence":"The row polynomials satisfy (2*n^2-6*n+4-(16*n-16)*x)*P(n,x) = -(4*n^2-36*n+56-(80*n-240)*x+256*x^2)*P(n-1,x) + (3*n-9+(16*n-72)*x+(16*n-176)*x^2-128*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 8.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 8;\nP[n_, x_] := P[n, x] = Expand[Together[(-(4*n^2 - 36*n + 56 - (80*n - 240)*x + 256*x^2)*P[n - 1, x] + (3*n - 9 + (16*n - 72)*x + (16*n - 176)*x^2 - 128*x^3)*D[P[n - 1, x], x])/(2*n^2 - 6*n + 4 - (16*n - 16)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 8; for(n = 2, N, my(i = n + 1); v[i] = (-(4*n^2 - 36*n + 56 - (80*n - 240)*x + 256*x^2)*v[i - 1] + (3*n - 9 + (16*n - 72)*x + (16*n - 176)*x^2 - 128*x^3)*deriv(v[i - 1], x))/(2*n^2 - 6*n + 4 - (16*n - 16)*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 A013611","D"],"edges":["e-5d40644b540aa393","e-7f1b4c57a31942c3"],"text":"A099040 \u2192 basis A013611 \u2192 D"},"headline":"A099040 \u2192 basis A013611 \u2192 D","chain":null}