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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:         3,             7,               2;\n  2:        26,           196,             229,               72,                 6;\n  3:       252,          5238,           14964,            12525,              3942,                490,                 20;\n  4:      2568,        140046,          849486,          1446813,            967041,             294220,              42860,               2870,                 70;\n  5:     26928,       3784122,        45879342,        143327859,         172396431,           96848383,           28087334,            4394810,             369362,             15498,               252;\n  6:    287648,     103446622,      2433280978,      13231699853,       26503333389,        24560881196,        11953669939,         3286092558,          529675608,          50437800,           2760366,            79464,             924;\n  7:   3112896,    2857227954,    128377246062,    1179569964895,     3766915267031,      5424335669871,      4082246816362,      1756737648109,       456973363004,       74162294136,        7596757056,        486477222,        18722748,         392964,         3432;\n  8:  34013312,   79596165718,   6774762227050,  103320259073289,   512118407101865,   1105158713246224,   1226564760483055,    776462189904247,    299867423103421,    73821188886636,    11900589122472,    1272391447866,     90096809418,     4145701560,    118346514,     1891890,     12870;\n  9: 374416128, 2232931262922, 358420712443062, 8972825636866539, 67835869600154691, 214257356209845907, 340186246320531770, 304835492952661454, 166444937238088711, 58348944732306135, 13608912548141990, 2161144838184002, 236645734622902, 17912626417686, 928891953132, 32196184020, 708975410, 8921770, 48620;\n","recurrence_sentence":"The row polynomials satisfy (n^2-2*n)*P(n,x) = -(4*n^2-12*n+8+(5*n^2-15*n+10)*x+(n^2-3*n+2)*x^2)*P(n-2,x) - ((5*n^2-15*n+10)*x+(2*n^2-6*n+4)*x^2)*d/dx P(n-2,x) - (n^2-3*n+2)*x^2*d^2/dx^2 P(n-2,x) + (12*n^2-30*n+12+(18*n^2-45*n+18)*x+(4*n^2-10*n+4)*x^2)*P(n-1,x) + ((18*n^2-45*n+18)*x+(8*n^2-20*n+8)*x^2)*d/dx P(n-1,x) + (4*n^2-10*n+4)*x^2*d^2/dx^2 P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 3 + 7*x + 2*x^2, P(2,x) = 26 + 196*x + 229*x^2 + 72*x^3 + 6*x^4.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + 7*x + 2*x^2;\nP[2, x_] := 26 + 196*x + 229*x^2 + 72*x^3 + 6*x^4;\nP[n_, x_] := P[n, x] = Expand[Together[(-(4*n^2 - 12*n + 8 + (5*n^2 - 15*n + 10)*x + (n^2 - 3*n + 2)*x^2)*P[n - 2, x] - ((5*n^2 - 15*n + 10)*x + (2*n^2 - 6*n + 4)*x^2)*D[P[n - 2, x], x] - (n^2 - 3*n + 2)*x^2*D[P[n - 2, x], {x, 2}] + (12*n^2 - 30*n + 12 + (18*n^2 - 45*n + 18)*x + (4*n^2 - 10*n + 4)*x^2)*P[n - 1, x] + ((18*n^2 - 45*n + 18)*x + (8*n^2 - 20*n + 8)*x^2)*D[P[n - 1, x], x] + (4*n^2 - 10*n + 4)*x^2*D[P[n - 1, x], {x, 2}])/(n^2 - 2*n)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 7*x + 2*x^2; v[3] = 26 + 196*x + 229*x^2 + 72*x^3 + 6*x^4; for(n = 3, N, my(i = n + 1); v[i] = (-(4*n^2 - 12*n + 8 + (5*n^2 - 15*n + 10)*x + (n^2 - 3*n + 2)*x^2)*v[i - 2] - ((5*n^2 - 15*n + 10)*x + (2*n^2 - 6*n + 4)*x^2)*deriv(v[i - 2], x) - (n^2 - 3*n + 2)*x^2*deriv(deriv(v[i - 2], x), x) + (12*n^2 - 30*n + 12 + (18*n^2 - 45*n + 18)*x + (4*n^2 - 10*n + 4)*x^2)*v[i - 1] + ((18*n^2 - 45*n + 18)*x + (8*n^2 - 20*n + 8)*x^2)*deriv(v[i - 1], x) + (4*n^2 - 10*n + 4)*x^2*deriv(deriv(v[i - 1], x), x))/(n^2 - 2*n)); 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-97a17a1dc7998ff8","steps":["basis A008277"],"edges":["e-14e4fe32a6cd6e38"],"text":"A316649 \u2192 basis A008277"},"headline":"A316649 \u2192 basis A008277","chain":null}