{"id":"n-c6a8b7ddc1412b03","revision":0,"first_index":1,"fingerprint":"f6-5dc0e20091c9a09f","name":"one_plus_derivative of Triangle, T(n, k) = (1/2)*(n+2)! * [x^k]( p(x, n) ), where p(x,0) = 1, p(x,1) = -x, P(x, n) = (1/(n+1))*( (2*n-x)*P(x, n-1) - n*P(x, n-2) ), read by rows.","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":[["-24","-8","4"],["-125","35","35","-5"],["-552","756","138","-84","6"],["-1617","8435","-1036","-840","161","-7"],["7040","80368","-34856","-4448","2656","-272","8"],["228735","702567","-607383","46233","31905","-6471","423","-9"],["3585240","5470300","-9012290","2151660","215750","-118300","13530","-620","10"],["48536587","30676679","-125269738","51557154","-2749736","-1687268","335874","-25498","869","-11"],["624578592","-88851960","-1677493164","1046342880","-166939656","-14055216","6924960","-812064","44520","-1176","12"],["7831769595","-7744368853","-21640627489","19878278823","-5126396106","204786582","115172694","-21843354","1755663","-73281","1547","-13"],["95396903000","-200008430972","-261828402542","366618622412","-132541452330","15615565896","1149313620","-506530248","58656612","-3490060","115066","-1988","14"],["1095241536615","-4230884895525","-2718845522040","6663814015260","-3195236168925","587452419315","-18356881320","-9660663360","1724479245","-140553375","6494400","-173820","2505","-15"],["10774931831808","-83329679830304","-15483433998032","119917838732736","-74655452695424","18498384356896","-1710552608208","-112943754624","44640085632","-5012138208","308767888","-11453504","254208","-3104","16"],["55070381210263","-1591060647282865","333755019652229","2129894705412005","-1721706966426289","540908722873799","-76398426914795","1926348557493","962476174605","-160988997323","13007652119","-632860105","19317389","-361675","3791","-17"],["-1382516849157480","-29881299667357332","18251142658791174","36892037742279372","-39557670715245834","15280861251656268","-2840459317641666","214790271202188","12947392191294","-4613786532828","497659929426","-30932360316","1225182114","-31371228","502506","-4572","18"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["-28","-60","264","5096","50496","396000","2305280","1374912","-270262272","-6501544192","-116049131520","-1824885911040","-26060982452224","-326796981989376","-3014313097936896"],"nonn":false,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["-24","-8","4"],["-125","35","35","-5"],["-552","756","138","-84","6"],["-1617","8435","-1036","-840","161","-7"]],"recurrence":{"denominator":{"coeffs":[["2"],["3"],["1"]]},"terms":[{"coeff":{"coeffs":[["0"],["0"],["6"],["5"],["-5"],["-5"],["-1"]]},"deriv_order":0,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["0","0"],["-12","-12"],["2","-22"],["32","-12"],["22","-2"],["4","0"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["-6","0","-6"],["-17","24","-5"],["-47","20","-1"],["-32","4","0"],["-6","0","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["6","-6"],["20","-8"],["18","-2"],["4","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":188362,"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-6761fdc76704a2aa"]},{"id":188363,"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-6761fdc76704a2aa"]}],"region_exact":{},"incoming_edges":[{"id":"e-6761fdc76704a2aa","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-c6a8b7ddc1412b03","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":"77ff56bcb5de05285ce3345268fde0d63a0a428884310edb33f19bef1390c0aa","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-cfb1b41332b76420","role":"sequence","name":"Triangle, T(n, k) = (1/2)*(n+2)! * [x^k]( p(x, n) ), where p(x,0) = 1, p(x,1) = -x, P(x, n) = (1/(n+1))*( (2*n-x)*P(x, n-1) - n*P(x, n-2) ), read by rows.","uses":{"claim":"real_rooted","from_index":1,"region":{}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:        -24,          -8,            4;\n   2:       -125,          35,           35,          -5;\n   3:       -552,         756,          138,         -84,           6;\n   4:      -1617,        8435,        -1036,        -840,         161,        -7;\n   5:       7040,       80368,       -34856,       -4448,        2656,      -272,         8;\n   6:     228735,      702567,      -607383,       46233,       31905,     -6471,       423,        -9;\n   7:    3585240,     5470300,     -9012290,     2151660,      215750,   -118300,     13530,      -620,      10;\n   8:   48536587,    30676679,   -125269738,    51557154,    -2749736,  -1687268,    335874,    -25498,     869,    -11;\n   9:  624578592,   -88851960,  -1677493164,  1046342880,  -166939656, -14055216,   6924960,   -812064,   44520,  -1176,   12;\n  10: 7831769595, -7744368853, -21640627489, 19878278823, -5126396106, 204786582, 115172694, -21843354, 1755663, -73281, 1547, -13;\n","recurrence_sentence":"The row polynomials satisfy (n^2+3*n+2)*P(n,x) = -(n^6+5*n^5+5*n^4-5*n^3-6*n^2)*P(n-4,x) + (4*n^5+22*n^4+32*n^3+2*n^2-12*n-(2*n^4+12*n^3+22*n^2+12*n)*x)*P(n-3,x) - (6*n^4+32*n^3+47*n^2+17*n+6-(4*n^3+20*n^2+24*n)*x+(n^2+5*n+6)*x^2)*P(n-2,x) + (4*n^3+18*n^2+20*n+6-(2*n^2+8*n+6)*x)*P(n-1,x) for n >= 5, with P(1,x) = -24 - 8*x + 4*x^2, P(2,x) = -125 + 35*x + 35*x^2 - 5*x^3, P(3,x) = -552 + 756*x + 138*x^2 - 84*x^3 + 6*x^4, P(4,x) = -1617 + 8435*x - 1036*x^2 - 840*x^3 + 161*x^4 - 7*x^5.","mathematica":"P[1, x_] := -24 - 8*x + 4*x^2;\nP[2, x_] := -125 + 35*x + 35*x^2 - 5*x^3;\nP[3, x_] := -552 + 756*x + 138*x^2 - 84*x^3 + 6*x^4;\nP[4, x_] := -1617 + 8435*x - 1036*x^2 - 840*x^3 + 161*x^4 - 7*x^5;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^6 + 5*n^5 + 5*n^4 - 5*n^3 - 6*n^2)*P[n - 4, x] + (4*n^5 + 22*n^4 + 32*n^3 + 2*n^2 - 12*n - (2*n^4 + 12*n^3 + 22*n^2 + 12*n)*x)*P[n - 3, x] - (6*n^4 + 32*n^3 + 47*n^2 + 17*n + 6 - (4*n^3 + 20*n^2 + 24*n)*x + (n^2 + 5*n + 6)*x^2)*P[n - 2, x] + (4*n^3 + 18*n^2 + 20*n + 6 - (2*n^2 + 8*n + 6)*x)*P[n - 1, x])/(n^2 + 3*n + 2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = -24 - 8*x + 4*x^2; v[2] = -125 + 35*x + 35*x^2 - 5*x^3; v[3] = -552 + 756*x + 138*x^2 - 84*x^3 + 6*x^4; v[4] = -1617 + 8435*x - 1036*x^2 - 840*x^3 + 161*x^4 - 7*x^5; for(n = 5, N, my(i = n); v[i] = (-(n^6 + 5*n^5 + 5*n^4 - 5*n^3 - 6*n^2)*v[i - 4] + (4*n^5 + 22*n^4 + 32*n^3 + 2*n^2 - 12*n - (2*n^4 + 12*n^3 + 22*n^2 + 12*n)*x)*v[i - 3] - (6*n^4 + 32*n^3 + 47*n^2 + 17*n + 6 - (4*n^3 + 20*n^2 + 24*n)*x + (n^2 + 5*n + 6)*x^2)*v[i - 2] + (4*n^3 + 18*n^2 + 20*n + 6 - (2*n^2 + 8*n + 6)*x)*v[i - 1])/(n^2 + 3*n + 2)); 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-cfb1b41332b76420","steps":["D+1"],"edges":["e-6761fdc76704a2aa"],"text":"A136532 \u2192 D+1"},"headline":"A136532 \u2192 D+1","chain":null}