{"id":"n-a6a369db31b832e1","revision":0,"first_index":0,"fingerprint":"f6-fc7fdb4b275e3246","name":"one_plus_derivative of Triangle T(n,k) read by rows: coefficient of [x^k] of the polynomial p_n(x)=(5-x)*p_{n-1}(x)-p_{n-2}(x), p_0=1, p_1=5-x.","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":[["1"],["4","-1"],["14","-8","1"],["42","-43","12","-1"],["81","-176","87","-16","1"],["-188","-448","452","-146","20","-1"],["-3661","952","1431","-920","220","-24","1"],["-30766","25179","-2880","-3485","1630","-309","28","-1"],["-210774","247080","-99285","6760","7190","-2632","413","-32","1"],["-1313480","1922145","-1114740","294370","-13580","-13251","3976","-532","36","-1"],["-7747901","13381880","-9710010","3722560","-728735","24528","22498","-5712","666","-40","1"],["-44092024","87149056","-74715080","35897740","-10250620","1589966","-40992","-35886","7890","-815","44","-1"],["-244650939","542486168","-532604346","302582880","-108421955","24626016","-3157434","64560","54495","-10560","979","-48","1"],["-1332190272","3267611263","-3600875172","2343849906","-993644980","283487049","-53397360","5827920","-97020","-79530","13772","-1158","52","-1"],["-7149499706","19193149976","-23405883187","17106667016","-8311896626","2806831960","-664627215","106859280","-10144365","140360","112321","-17576","1352","-56","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","7","10","-23","-311","-2001","-10604","-51279","-235057","-1040265","-4490722","-19030183","-79495531","-328306465"],"nonn":false,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0","0","0"],["-74","26","-2"],["0","0","0"],["2","0","0"]]},"terms":[{"coeff":{"coeffs":[["-159","91","-17","1"],["-149","89","-17","1"],["15","-3","0","0"],["5","-1","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["546","-491","157","-21","1"],["-21","10","-1","0","0"],["-21","10","-1","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":180001,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, +inf)","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)","region_exact":null,"edges":["e-84a997eaa02be6ea"]},{"id":180002,"position":1,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-84a997eaa02be6ea"]}],"region_exact":{},"incoming_edges":[{"id":"e-84a997eaa02be6ea","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-a6a369db31b832e1","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":"6adf639337e67120737d4f84704f07942f78ebd6fb7d2503b09d3599489a2844","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-490435c99149594f","role":"sequence","name":"Triangle T(n,k) read by rows: coefficient of [x^k] of the polynomial p_n(x)=(5-x)*p_{n-1}(x)-p_{n-2}(x), p_0=1, p_1=5-x.","uses":{"claim":"real_rooted","from_index":0,"region":{}},"uses_text":"Every row P_n, n >= 0, 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":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:        1;\n  1:        4,      -1;\n  2:       14,      -8,        1;\n  3:       42,     -43,       12,     -1;\n  4:       81,    -176,       87,    -16,      1;\n  5:     -188,    -448,      452,   -146,     20,     -1;\n  6:    -3661,     952,     1431,   -920,    220,    -24,    1;\n  7:   -30766,   25179,    -2880,  -3485,   1630,   -309,   28,   -1;\n  8:  -210774,  247080,   -99285,   6760,   7190,  -2632,  413,  -32,  1;\n  9: -1313480, 1922145, -1114740, 294370, -13580, -13251, 3976, -532, 36, -1;\n","recurrence_sentence":"The row polynomials satisfy (2*n^3-74*n+26*n*x-2*n*x^2)*P(n,x) = (5*n^3+15*n^2-149*n-159-(n^3+3*n^2-89*n-91)*x-(17*n+17)*x^2+(n+1)*x^3)*P(n-1,x) - (21*n^2+21*n-546-(10*n^2+10*n-491)*x+(n^2+n-157)*x^2+21*x^3-x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((5*n^3 + 15*n^2 - 149*n - 159 - (n^3 + 3*n^2 - 89*n - 91)*x - (17*n + 17)*x^2 + (n + 1)*x^3)*P[n - 1, x] - (21*n^2 + 21*n - 546 - (10*n^2 + 10*n - 491)*x + (n^2 + n - 157)*x^2 + 21*x^3 - x^4)*D[P[n - 1, x], x])/(2*n^3 - 74*n + 26*n*x - 2*n*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((5*n^3 + 15*n^2 - 149*n - 159 - (n^3 + 3*n^2 - 89*n - 91)*x - (17*n + 17)*x^2 + (n + 1)*x^3)*v[i - 1] - (21*n^2 + 21*n - 546 - (10*n^2 + 10*n - 491)*x + (n^2 + n - 157)*x^2 + 21*x^3 - x^4)*deriv(v[i - 1], x))/(2*n^3 - 74*n + 26*n*x - 2*n*x^2)); 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-490435c99149594f","steps":["D+1"],"edges":["e-84a997eaa02be6ea"],"text":"A179900 \u2192 D+1"},"headline":"A179900 \u2192 D+1","chain":null}