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T(n, k) is the coefficient of x^k in a(n+3) where a(1) = a(2) = a(3) = 1 and a(m+2) = (m*x + 2)*a(m+1) - a(m) for all m in Z.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"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:       2,         4;\n  2:       7,        32,         18;\n  3:      33,       240,        306,         96;\n  4:     191,      1936,       3978,       2752,        600;\n  5:    1304,     17132,      49320,      54528,      25700,       4320;\n  6:   10241,    166416,     621864,     959552,     716200,     256608,      35280;\n  7:   90865,   1767360,    8168508,   16320192,   16915200,    9524448,    2762424,     322560;\n  8:  898409,  20414864,  112875696,  278416704,  372438200,  289850976,  131299616,   32090112,   3265920;\n  9: 9791634, 255138868, 1646865144, 4852274688, 7995139900, 8005635360, 4992207808, 1896109056, 401451552, 36288000;\n","recurrence_sentence":"The row polynomials satisfy (n^2-3*n+2)*P(n,x) = -(n^2-n)*P(n-4,x) + (2*n^3-2*n^2-2*n+(4*n^3-12*n^2+8*n)*x)*P(n-3,x) - (4*n^3-12*n^2+8*n)*x^2*d/dx P(n-3,x) - (n^4-n^2-6*n+2+(4*n^4-8*n^3-4*n^2+8*n)*x-(9*n^4-36*n^3+45*n^2-18*n)*x^2)*P(n-2,x) + ((4*n^4-8*n^3-4*n^2+8*n)*x^2+(4*n^4-16*n^3+20*n^2-8*n)*x^3)*d/dx P(n-2,x) + (2*n^3-2*n^2-6*n+4+(4*n^3-12*n^2+8*n)*x)*P(n-1,x) - (4*n^3-12*n^2+8*n)*x^2*d/dx P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2 + 4*x, P(2,x) = 7 + 32*x + 18*x^2, P(3,x) = 33 + 240*x + 306*x^2 + 96*x^3.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 4*x;\nP[2, x_] := 7 + 32*x + 18*x^2;\nP[3, x_] := 33 + 240*x + 306*x^2 + 96*x^3;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^2 - n)*P[n - 4, x] + (2*n^3 - 2*n^2 - 2*n + (4*n^3 - 12*n^2 + 8*n)*x)*P[n - 3, x] - (4*n^3 - 12*n^2 + 8*n)*x^2*D[P[n - 3, x], x] - (n^4 - n^2 - 6*n + 2 + (4*n^4 - 8*n^3 - 4*n^2 + 8*n)*x - (9*n^4 - 36*n^3 + 45*n^2 - 18*n)*x^2)*P[n - 2, x] + ((4*n^4 - 8*n^3 - 4*n^2 + 8*n)*x^2 + (4*n^4 - 16*n^3 + 20*n^2 - 8*n)*x^3)*D[P[n - 2, x], x] + (2*n^3 - 2*n^2 - 6*n + 4 + (4*n^3 - 12*n^2 + 8*n)*x)*P[n - 1, x] - (4*n^3 - 12*n^2 + 8*n)*x^2*D[P[n - 1, x], x])/(n^2 - 3*n + 2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 4*x; v[3] = 7 + 32*x + 18*x^2; v[4] = 33 + 240*x + 306*x^2 + 96*x^3; for(n = 4, N, my(i = n + 1); v[i] = (-(n^2 - n)*v[i - 4] + (2*n^3 - 2*n^2 - 2*n + (4*n^3 - 12*n^2 + 8*n)*x)*v[i - 3] - (4*n^3 - 12*n^2 + 8*n)*x^2*deriv(v[i - 3], x) - (n^4 - n^2 - 6*n + 2 + (4*n^4 - 8*n^3 - 4*n^2 + 8*n)*x - (9*n^4 - 36*n^3 + 45*n^2 - 18*n)*x^2)*v[i - 2] + ((4*n^4 - 8*n^3 - 4*n^2 + 8*n)*x^2 + (4*n^4 - 16*n^3 + 20*n^2 - 8*n)*x^3)*deriv(v[i - 2], x) + (2*n^3 - 2*n^2 - 6*n + 4 + (4*n^3 - 12*n^2 + 8*n)*x)*v[i - 1] - (4*n^3 - 12*n^2 + 8*n)*x^2*deriv(v[i - 1], x))/(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 - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-f3e0631eb8740f76","steps":["basis A007318","(\u03b8+1)\u00b2"],"edges":["e-3f96abc359582f71","e-4850f5709a8931b2"],"text":"A358735 \u2192 basis A007318 \u2192 (\u03b8+1)\u00b2"},"headline":"A358735 \u2192 basis A007318 \u2192 (\u03b8+1)\u00b2","chain":null}