{"id":"n-8a7a8aaa440e4f07","revision":0,"first_index":1,"fingerprint":"f6-a6bffaf3050004e1","name":"derivative of Triangle read by rows: The n-th derivative of the logistic function written in terms of y, where y = 1/(1 + exp(-x)).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-2,"palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":false,"rows":[[],["-1"],["-3","4"],["-7","24","-18"],["-15","100","-180","96"],["-31","360","-1170","1440","-600"],["-63","1204","-6300","13440","-12600","4320"],["-127","3864","-30618","100800","-159600","120960","-35280"],["-255","12100","-139860","667296","-1587600","1995840","-1270080","322560"],["-511","37320","-613890","4082400","-13696200","25401600","-26460000","14515200","-3265920"],["-1023","114004","-2623500","23686080","-107692200","276423840","-419126400","372556800","-179625600","36288000"],["-2047","346104","-11007018","132422400","-794191200","2710350720","-5610472560","7185024000","-5568393600","2395008000","-439084800"],["-4095","1046500","-45585540","720816096","-5592787200","24690631680","-67018311360","115960965120","-128432304000","88216128000","-34248614400","5748019200"],["-8191","3155880","-187051410","3847203360","-38061823800","213102489600","-737786649600","1656387532800","-2458427811840","2397403008000","-1478398521600","523069747200","-80951270400"],["-16383","9500404","-762407100","20233624320","-252415963800","1765762558560","-7642630195200","21652965734400","-41355201888000","53687297664000","-46749358656000","26153487360000","-8499883392000","1220496076800"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","-1","1","-1","1","-1","1","-1","1","-1","1","-1","1","-1","1"],"nonn":false,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["0"],["-1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-2","4","-2"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","-1","2","-1"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["3","-4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1","-1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":179257,"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-42bdce22a2faaa51"]},{"id":179258,"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-42bdce22a2faaa51"]}],"region_exact":{},"incoming_edges":[{"id":"e-42bdce22a2faaa51","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-8a7a8aaa440e4f07","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":"767826d2236da4e1467ecbfd36354745b9288affa6e8b6181a222ab703e39a98","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-2ae6a2215aff79d4","role":"sequence","name":"Triangle read by rows: The n-th derivative of the logistic function written in terms of y, where y = 1/(1 + exp(-x)).","uses":{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":false,"value":"0"},"upper":{"closed":true,"value":"1"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (0, 1]"}]}],"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:   -1;\n   3:   -3,     4;\n   4:   -7,    24,     -18;\n   5:  -15,   100,    -180,      96;\n   6:  -31,   360,   -1170,    1440,      -600;\n   7:  -63,  1204,   -6300,   13440,    -12600,     4320;\n   8: -127,  3864,  -30618,  100800,   -159600,   120960,    -35280;\n   9: -255, 12100, -139860,  667296,  -1587600,  1995840,  -1270080,   322560;\n  10: -511, 37320, -613890, 4082400, -13696200, 25401600, -26460000, 14515200, -3265920;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(2-4*x+2*x^2)*P(n-2,x) - (x-2*x^2+x^3)*d/dx P(n-2,x) + (3-4*x)*P(n-1,x) + (x-x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = -1.","mathematica":"P[1, x_] := 0;\nP[2, x_] := -1;\nP[n_, x_] := P[n, x] = Expand[-(2 - 4*x + 2*x^2)*P[n - 2, x] - (x - 2*x^2 + x^3)*D[P[n - 2, x], x] + (3 - 4*x)*P[n - 1, x] + (x - x^2)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 0; v[2] = -1; for(n = 3, N, my(i = n); v[i] = -(2 - 4*x + 2*x^2)*v[i - 2] - (x - 2*x^2 + x^3)*deriv(v[i - 2], x) + (3 - 4*x)*v[i - 1] + (x - x^2)*deriv(v[i - 1], x)); 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-2ae6a2215aff79d4","steps":["D"],"edges":["e-42bdce22a2faaa51"],"text":"A163626 \u2192 D"},"headline":"A163626 \u2192 D","chain":null}