{"id":"n-04f443bf4d795a35","revision":1,"first_index":0,"fingerprint":"f6-843b6dd431a207b6","name":"one_plus_derivative of Coefficients of a generalized Jaco-Lucas polynomial (even indices) read by rows.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["2"],["5","2"],["15","12","2"],["48","60","21","2"],["151","264","152","32","2"],["463","1050","865","310","45","2"],["1390","3888","4206","2192","555","60","2"],["4105","13678","18396","12712","4739","910","77","2"],["11967","46320","74632","64544","32160","9184","1400","96","2"],["34524","152316","286299","298074","186939","71838","16422","2052","117","2"],["98747","489300","1051780","1282240","972255","472628","146250","27600","2895","140","2"],["280415","1542222","3733103","5220006","4650195","2743510","1078693","277002","44154","3960","165","2"],["791450","4784832","12883116","20334416","20832762","14483760","6935348","2271984","495099","67848","5280","192","2"],["2222093","14649050","43435080","76407760","88554050","70976620","40160250","16072940","4484480","843700","100815","6890","221","2"],["6210351","44342004","143584910","278611760","360531360","327549040","214142460","101530080","34707855","8388380","1381380","145600","8827","252","2"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["2","7","29","131","601","2735","12293","54619","240305","1048583","4543837","19573427","83886089","357913951","1521134261"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["2"],["5","2"]],"recurrence":{"denominator":{"coeffs":[["12","10","-8"],["-12","-26","8"],["-8","16","0"],["8","0","0"]]},"terms":[{"coeff":{"coeffs":[["66","58","-26","-24"],["-90","-138","-22","24"],["-8","56","48","0"],["32","24","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["15","12","-35","-48","-16"],["-30","-94","-96","-32","0"],["-20","-36","-16","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":164946,"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-58ea4b3a2d7c4054"]}],"region_exact":{},"incoming_edges":[{"id":"e-58ea4b3a2d7c4054","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-04f443bf4d795a35","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":"8f212b9d075a903cfa05d8161c155d0f1196d7f638096cbce1893b992d1098a8","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-9b40f045d129e237","role":"sequence","name":"Coefficients of a generalized Jaco-Lucas polynomial (even indices) read by rows.","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:     2;\n  1:     5,      2;\n  2:    15,     12,      2;\n  3:    48,     60,     21,      2;\n  4:   151,    264,    152,     32,      2;\n  5:   463,   1050,    865,    310,     45,     2;\n  6:  1390,   3888,   4206,   2192,    555,    60,     2;\n  7:  4105,  13678,  18396,  12712,   4739,   910,    77,    2;\n  8: 11967,  46320,  74632,  64544,  32160,  9184,  1400,   96,   2;\n  9: 34524, 152316, 286299, 298074, 186939, 71838, 16422, 2052, 117, 2;\n","recurrence_sentence":"The row polynomials satisfy (8*n^3-8*n^2-12*n+12+(16*n^2-26*n+10)*x+(8*n-8)*x^2)*P(n,x) = (32*n^3-8*n^2-90*n+66+(24*n^3+56*n^2-138*n+58)*x+(48*n^2-22*n-26)*x^2+(24*n-24)*x^3)*P(n-1,x) - (20*n^2+30*n-15+(36*n^2+94*n-12)*x+(16*n^2+96*n+35)*x^2+(32*n+48)*x^3+16*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 2, P(1,x) = 5 + 2*x.","mathematica":"P[0, x_] := 2;\nP[1, x_] := 5 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[((32*n^3 - 8*n^2 - 90*n + 66 + (24*n^3 + 56*n^2 - 138*n + 58)*x + (48*n^2 - 22*n - 26)*x^2 + (24*n - 24)*x^3)*P[n - 1, x] - (20*n^2 + 30*n - 15 + (36*n^2 + 94*n - 12)*x + (16*n^2 + 96*n + 35)*x^2 + (32*n + 48)*x^3 + 16*x^4)*D[P[n - 1, x], x])/(8*n^3 - 8*n^2 - 12*n + 12 + (16*n^2 - 26*n + 10)*x + (8*n - 8)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 2; v[2] = 5 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((32*n^3 - 8*n^2 - 90*n + 66 + (24*n^3 + 56*n^2 - 138*n + 58)*x + (48*n^2 - 22*n - 26)*x^2 + (24*n - 24)*x^3)*v[i - 1] - (20*n^2 + 30*n - 15 + (36*n^2 + 94*n - 12)*x + (16*n^2 + 96*n + 35)*x^2 + (32*n + 48)*x^3 + 16*x^4)*deriv(v[i - 1], x))/(8*n^3 - 8*n^2 - 12*n + 12 + (16*n^2 - 26*n + 10)*x + (8*n - 8)*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-9b40f045d129e237","steps":["D+1"],"edges":["e-58ea4b3a2d7c4054"],"text":"A122076 \u2192 D+1"},"headline":"A122076 \u2192 D+1","chain":null}