{"id":"n-5b50e616320aab8d","revision":0,"first_index":1,"fingerprint":"f6-205f5c838c09afd7","name":"derivative of Triangle T(n,k) of arctangent numbers: expansion of arctan(x)^n/n!.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":false,"rows":[["0","2"],["-8","0","3"],["0","-40","0","4"],["184","0","-120","0","5"],["0","1568","0","-280","0","6"],["-8448","0","7392","0","-560","0","7"],["0","-104704","0","25536","0","-1008","0","8"],["648576","0","-689280","0","72240","0","-1680","0","9"],["0","10720512","0","-3217280","0","177408","0","-2640","0","10"],["-74972160","0","91901568","0","-11968000","0","391776","0","-3960","0","11"],["0","-1565051904","0","547216384","0","-37779456","0","796224","0","-5720","0","12"],["12174658560","0","-16684222464","0","2551028480","0","-105193088","0","1513512","0","-8008","0","13"],["0","309188763648","0","-121839011840","0","9937095168","0","-265133440","0","2722720","0","-10920","0","14"],["-2643856588800","0","3967469862912","0","-688014745600","0","33683826176","0","-616112640","0","4676672","0","-14560","0","15"],["0","-79493016453120","0","34531322658816","0","-3210520535040","0","102127826944","0","-1338022400","0","7722624","0","-19040","0","16"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["2","-5","-36","69","1294","-1609","-80168","29865","7678010","5349235","-1054824460","-2062222995","197024425350","668670904175","-48071416821200"],"nonn":false,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["0","2"],["-8","0","3"],["0","-40","0","4"],["184","0","-120","0","5"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0"],["2"],["-5"],["4"],["-1"]]},"deriv_order":0,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["0","0"],["0","-2"],["0","2"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["0","0","-1"],["0","0","0"],["-2","0","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":183979,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, +inf)","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)","region_exact":null,"edges":["e-f36d6681046ffd69"]},{"id":183980,"position":1,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2","region_exact":null,"edges":["e-f36d6681046ffd69"]}],"region_exact":{},"incoming_edges":[{"id":"e-f36d6681046ffd69","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-5b50e616320aab8d","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":"1d88087e1dfaed427a2451eea3f8e14309311912954dc4d49bbb87cd8b396248","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-52650ae6398d7ffe","role":"sequence","name":"Triangle T(n,k) of arctangent numbers: expansion of arctan(x)^n/n!.","uses":{"claim":"real_rooted","from_index":1,"region":{}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)"}]}],"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,        2;\n   2:        -8,        0,        3;\n   3:         0,      -40,        0,        4;\n   4:       184,        0,     -120,        0,         5;\n   5:         0,     1568,        0,     -280,         0,      6;\n   6:     -8448,        0,     7392,        0,      -560,      0,      7;\n   7:         0,  -104704,        0,    25536,         0,  -1008,      0,     8;\n   8:    648576,        0,  -689280,        0,     72240,      0,  -1680,     0,     9;\n   9:         0, 10720512,        0, -3217280,         0, 177408,      0, -2640,     0, 10;\n  10: -74972160,        0, 91901568,        0, -11968000,      0, 391776,     0, -3960,  0, 11;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^4-4*n^3+5*n^2-2*n)*P(n-4,x) + (2*n^2-2*n)*x*P(n-3,x) - (2*n^2+x^2)*P(n-2,x) + 2*x*P(n-1,x) for n >= 5, with P(1,x) = 2*x, P(2,x) = -8 + 3*x^2, P(3,x) = -40*x + 4*x^3, P(4,x) = 184 - 120*x^2 + 5*x^4.","mathematica":"P[1, x_] := 2*x;\nP[2, x_] := -8 + 3*x^2;\nP[3, x_] := -40*x + 4*x^3;\nP[4, x_] := 184 - 120*x^2 + 5*x^4;\nP[n_, x_] := P[n, x] = Expand[-(n^4 - 4*n^3 + 5*n^2 - 2*n)*P[n - 4, x] + (2*n^2 - 2*n)*x*P[n - 3, x] - (2*n^2 + x^2)*P[n - 2, x] + 2*x*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 2*x; v[2] = -8 + 3*x^2; v[3] = -40*x + 4*x^3; v[4] = 184 - 120*x^2 + 5*x^4; for(n = 5, N, my(i = n); v[i] = -(n^4 - 4*n^3 + 5*n^2 - 2*n)*v[i - 4] + (2*n^2 - 2*n)*x*v[i - 3] - (2*n^2 + x^2)*v[i - 2] + 2*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-52650ae6398d7ffe","steps":["D"],"edges":["e-f36d6681046ffd69"],"text":"A049218 \u2192 D"},"headline":"A049218 \u2192 D","chain":null}