{"id":"n-489570c2b2a8d92c","revision":0,"first_index":1,"fingerprint":"f6-ec0e5cdcf61dbc5b","name":"one_plus_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":-1,"palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":false,"rows":[["1"],["0","-1"],["-2","1","2"],["-6","17","-6","-6"],["-14","85","-130","36","24"],["-30","329","-990","1050","-240","-120"],["-62","1141","-5698","11340","-9240","1800","720"],["-126","3737","-28686","90594","-134400","89040","-15120","-5040"],["-254","11845","-133810","620676","-1420776","1678320","-937440","141120","40320"],["-510","36809","-595230","3877770","-12675600","22662360","-22226400","10735200","-1451520","-362880"],["-1022","112981","-2566498","22811580","-101770680","254885400","-373055760","312681600","-133056000","16329600","3628800"],["-2046","344057","-10833966","128753394","-761085600","2551512480","-5158747440","6383527920","-4670265600","1776297600","-199584000","-39916800"],["-4094","1042405","-45062290","705620916","-5412583176","23572074240","-62903206080","106386920640","-113937183360","73945872000","-25427001600","2634508800","479001600"],["-8190","3147689","-185473470","3784852890","-37100022960","205490124840","-702269568000","1550989440000","-2251379370240","2124244362240","-1238658220800","388669881600","-37362124800","-6227020800"],["-16382","9484021","-757656898","19979488620","-247357557720","1715279365800","-7348336435440","20561161420800","-38648581171200","49092275232000","-41380628889600","21903545664000","-6320426112000","566658892800","87178291200"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","-1","1","-1","1","-1","1","-1","1","-1","1","-1","1","-1","1"],"nonn":false,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["0","-1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-2","5","-4","1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","-1","3","-3","1"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["3","-5","1"]]},"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":179259,"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-ce753e61dcfa144d"]},{"id":179260,"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-ce753e61dcfa144d"]}],"region_exact":{},"incoming_edges":[{"id":"e-ce753e61dcfa144d","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-489570c2b2a8d92c","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":"5e4c1d7efd40c6f7472f40271d9f880625d7a2597968c05d09f96ee35b8e1fa3","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:    1;\n   2:    0,    -1;\n   3:   -2,     1,       2;\n   4:   -6,    17,      -6,      -6;\n   5:  -14,    85,    -130,      36,        24;\n   6:  -30,   329,    -990,    1050,      -240,     -120;\n   7:  -62,  1141,   -5698,   11340,     -9240,     1800,       720;\n   8: -126,  3737,  -28686,   90594,   -134400,    89040,    -15120,    -5040;\n   9: -254, 11845, -133810,  620676,  -1420776,  1678320,   -937440,   141120,    40320;\n  10: -510, 36809, -595230, 3877770, -12675600, 22662360, -22226400, 10735200, -1451520, -362880;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(2-5*x+4*x^2-x^3)*P(n-2,x) - (x-3*x^2+3*x^3-x^4)*d/dx P(n-2,x) + (3-5*x+x^2)*P(n-1,x) + (x-x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = -x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := -x;\nP[n_, x_] := P[n, x] = Expand[-(2 - 5*x + 4*x^2 - x^3)*P[n - 2, x] - (x - 3*x^2 + 3*x^3 - x^4)*D[P[n - 2, x], x] + (3 - 5*x + x^2)*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] = 1; v[2] = -x; for(n = 3, N, my(i = n); v[i] = -(2 - 5*x + 4*x^2 - x^3)*v[i - 2] - (x - 3*x^2 + 3*x^3 - x^4)*deriv(v[i - 2], x) + (3 - 5*x + x^2)*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+1"],"edges":["e-ce753e61dcfa144d"],"text":"A163626 \u2192 D+1"},"headline":"A163626 \u2192 D+1","chain":null}