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Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"adfa2219d713af9c0a2fb497d5fe1f5becebfd9c8338a847cc875488a7be75b7","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-eb51f16fedb3c293","role":"sequence","name":"Triangle read by rows, T(n, k) = Sum_{i=0..n} L'(n, n-i) * binomial(i, k), for k = 0..n-1.","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)"},{"position":1,"node_id":"n-eb51f16fedb3c293","role":"sequence","name":"Triangle read by rows, T(n, k) = Sum_{i=0..n} L'(n, n-i) * binomial(i, k), for k = 0..n-1.","uses":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"uses_text":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1"}]}],"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:         4,          2;\n  1:        16,         20,           6;\n  2:        86,        174,         114,          24;\n  3:       574,       1616,        1668,         744,         120;\n  4:      4552,      16490,       23460,       16320,        5520,         720;\n  5:     41684,     185172,      337530,      322200,      169200,       46080,        5040;\n  6:    431986,    2277590,     5079606,     6198360,     4456200,     1879920,      428400,       40320;\n  7:   4990906,   30500864,    80632104,   120227184,   110344080,    63645120,    22438080,     4394880,     362880;\n  8:  63537644,  442083186,  1353571848,  2390379264,  2678598720,  1970740800,   949173120,   287400960,   49351680,   3628800;\n  9: 883014232, 6897209780, 24030641070, 49125180240, 65163752640, 58504965120, 35924212800, 14852678400, 3940876800, 602380800, 39916800;\n","recurrence_sentence":"The row polynomials satisfy (n^2+3*n+2+(2*n^2+4*n)*x+(n^2+n)*x^2)*P(n,x) = (2*n^3+9*n^2+13*n+4+(6*n^3+22*n^2+22*n+2)*x+(6*n^3+17*n^2+11*n)*x^2+(2*n^3+4*n^2+2*n)*x^3)*P(n-1,x) - (n^2+3*n+4+(4*n^2+10*n+10)*x+(6*n^2+12*n+8)*x^2+(4*n^2+6*n+2)*x^3+(n^2+n)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 4 + 2*x.","mathematica":"P[0, x_] := 4 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n^3 + 9*n^2 + 13*n + 4 + (6*n^3 + 22*n^2 + 22*n + 2)*x + (6*n^3 + 17*n^2 + 11*n)*x^2 + (2*n^3 + 4*n^2 + 2*n)*x^3)*P[n - 1, x] - (n^2 + 3*n + 4 + (4*n^2 + 10*n + 10)*x + (6*n^2 + 12*n + 8)*x^2 + (4*n^2 + 6*n + 2)*x^3 + (n^2 + n)*x^4)*D[P[n - 1, x], x])/(n^2 + 3*n + 2 + (2*n^2 + 4*n)*x + (n^2 + n)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 4 + 2*x; for(n = 1, N, my(i = n + 1); v[i] = ((2*n^3 + 9*n^2 + 13*n + 4 + (6*n^3 + 22*n^2 + 22*n + 2)*x + (6*n^3 + 17*n^2 + 11*n)*x^2 + (2*n^3 + 4*n^2 + 2*n)*x^3)*v[i - 1] - (n^2 + 3*n + 4 + (4*n^2 + 10*n + 10)*x + (6*n^2 + 12*n + 8)*x^2 + (4*n^2 + 6*n + 2)*x^3 + (n^2 + n)*x^4)*deriv(v[i - 1], x))/(n^2 + 3*n + 2 + (2*n^2 + 4*n)*x + (n^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-d55979c30b97975a","steps":["x \u21a6 x+1","P\u2099 + P\u2099\u208a\u2081"],"edges":["e-cee0a9f908e74de4","e-b96b8d87f5e7b835"],"text":"A089231 \u2192 x \u21a6 x+1 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A089231 \u2192 x \u21a6 x+1 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}