{"id":"n-da081fb7592ee6c9","revision":0,"first_index":0,"fingerprint":"f6-e589bedfdbabf881","name":"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":-1,"palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":false,"rows":[[],["-1"],["-10","2"],["-73","30","-3"],["-470","294","-60","4"],["-2828","2380","-738","100","-5"],["-16310","17262","-7200","1480","-150","6"],["-91371","116550","-61155","16900","-2595","210","-7"],["-501150","748230","-473400","164560","-33950","4158","-280","8"],["-2704755","4626900","-3428190","1437100","-372855","61320","-6244","360","-9"],["-14413900","27795780","-23607900","11591860","-3626700","749868","-102480","8928","-450","10"],["-76030744","163179800","-156304980","87999400","-32250470","8040060","-1381002","161400","-12285","550","-11"],["-397678540","940164708","-1002686700","636811780","-267624900","78150996","-16182600","2376360","-242550","16390","-660","12"],["-2065389557","5333000820","-6267375582","4432975100","-2101888755","703864800","-170708160","30214800","-3873870","350900","-21318","780","-13"],["-10662468530","29855618506","-38333692440","29884564496","-15783037750","5963439510","-1658533800","343792680","-53118450","6042410","-491920","27144","-910","14"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","-1","-8","-46","-232","-1091","-4912","-21468","-91824","-386373","-1604984","-6598282","-26895704","-108850055","-437859040"],"nonn":false,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["-1"]],"recurrence":{"denominator":{"coeffs":[["-2"],["2"]]},"terms":[{"coeff":{"coeffs":[["10","-2"],["5","-1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-21","10","-1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":179999,"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-f7f8feb970c53eb9"]},{"id":180000,"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-f7f8feb970c53eb9"]}],"region_exact":{},"incoming_edges":[{"id":"e-f7f8feb970c53eb9","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-da081fb7592ee6c9","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":"44a84ad8402e7216050efed3b8d99d0b97a27cc7c44995f48050924075acb05a","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:        0;\n  1:       -1;\n  2:      -10,       2;\n  3:      -73,      30,       -3;\n  4:     -470,     294,      -60,       4;\n  5:    -2828,    2380,     -738,     100,      -5;\n  6:   -16310,   17262,    -7200,    1480,    -150,     6;\n  7:   -91371,  116550,   -61155,   16900,   -2595,   210,    -7;\n  8:  -501150,  748230,  -473400,  164560,  -33950,  4158,  -280,   8;\n  9: -2704755, 4626900, -3428190, 1437100, -372855, 61320, -6244, 360, -9;\n","recurrence_sentence":"The row polynomials satisfy (2*n-2)*P(n,x) = (5*n+10-(n+2)*x)*P(n-1,x) - (21-10*x+x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = -1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := -1;\nP[n_, x_] := P[n, x] = Expand[Together[((5*n + 10 - (n + 2)*x)*P[n - 1, x] - (21 - 10*x + x^2)*D[P[n - 1, x], x])/(2*n - 2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = -1; for(n = 2, N, my(i = n + 1); v[i] = ((5*n + 10 - (n + 2)*x)*v[i - 1] - (21 - 10*x + x^2)*deriv(v[i - 1], x))/(2*n - 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"],"edges":["e-f7f8feb970c53eb9"],"text":"A179900 \u2192 D"},"headline":"A179900 \u2192 D","chain":null}