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0]"}]}],"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:   1;\n   2:   3,     1;\n   3:   9,     8,      1;\n   4:  22,    46,     17,      1;\n   5:  45,   190,    165,     34,       1;\n   6:  81,   595,   1070,    518,      67,       1;\n   7: 133,  1526,   4900,   5012,    1533,     132,      1;\n   8: 204,  3388,  17346,  32466,   21294,    4422,    261,     1;\n   9: 297,  6756,  50862, 154896,  190155,   85530,  12645,   518,    1;\n  10: 415, 12405, 129480, 589008, 1200087, 1031145, 332350, 36166, 1031, 1;\n","recurrence_sentence":"The row polynomials satisfy 8*P(n,x) = -(18*n^2-129*n+194+(21*n^2-161*n+288)*x+(5*n^2-82*n+278)*x^2-(3*n^3-27*n^2+52*n+40)*x^3)*P(n-4,x) - (9*n-18+(9*n^2-99*n+144)*x-(3*n^2+21*n-184)*x^2-(3*n^3-33*n^2+92*n-46)*x^3+(3*n^2-12*n-8)*x^4)*d/dx P(n-4,x) + ((9*n-18)*x^2-18*x^3-(3*n^2-12*n)*x^4)*d^2/dx^2 P(n-4,x) + (18*n^2-72*n+104+(33*n^2-171*n+82)*x+(10*n^2-98*n+242)*x^2-(3*n^3-21*n^2+12*n+88)*x^3)*P(n-3,x) + (18*n-36+(9*n^2-81*n+90)*x-(6*n^2+15*n-152)*x^2-(3*n^3-27*n^2+68*n-2)*x^3+(3*n^2-12*n-24)*x^4)*d/dx P(n-3,x) - ((9*n-18)*x^2-18*x^3-(3*n^2-12*n-8)*x^4)*d^2/dx^2 P(n-3,x) - (27*n-6+(12*n^2-12*n-142)*x+(5*n^2-16*n+64)*x^2+(16*n-32)*x^3)*P(n-2,x) - (9*n-18-18*x-(3*n^2-6*n-4)*x^2-28*x^3-16*x^4)*d/dx P(n-2,x) + 8*x^4*d^2/dx^2 P(n-2,x) + (6*n+20+(16*n-44)*x+16*x^2)*P(n-1,x) for n >= 5, with P(1,x) = 1, P(2,x) = 3 + x, P(3,x) = 9 + 8*x + x^2, P(4,x) = 22 + 46*x + 17*x^2 + x^3.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 3 + x;\nP[3, x_] := 9 + 8*x + x^2;\nP[4, x_] := 22 + 46*x + 17*x^2 + x^3;\nP[n_, x_] := P[n, x] = Expand[Together[(-(18*n^2 - 129*n + 194 + (21*n^2 - 161*n + 288)*x + (5*n^2 - 82*n + 278)*x^2 - (3*n^3 - 27*n^2 + 52*n + 40)*x^3)*P[n - 4, x] - (9*n - 18 + (9*n^2 - 99*n + 144)*x - (3*n^2 + 21*n - 184)*x^2 - (3*n^3 - 33*n^2 + 92*n - 46)*x^3 + (3*n^2 - 12*n - 8)*x^4)*D[P[n - 4, x], x] + ((9*n - 18)*x^2 - 18*x^3 - (3*n^2 - 12*n)*x^4)*D[P[n - 4, x], {x, 2}] + (18*n^2 - 72*n + 104 + (33*n^2 - 171*n + 82)*x + (10*n^2 - 98*n + 242)*x^2 - (3*n^3 - 21*n^2 + 12*n + 88)*x^3)*P[n - 3, x] + (18*n - 36 + (9*n^2 - 81*n + 90)*x - (6*n^2 + 15*n - 152)*x^2 - (3*n^3 - 27*n^2 + 68*n - 2)*x^3 + (3*n^2 - 12*n - 24)*x^4)*D[P[n - 3, x], x] - ((9*n - 18)*x^2 - 18*x^3 - (3*n^2 - 12*n - 8)*x^4)*D[P[n - 3, x], {x, 2}] - (27*n - 6 + (12*n^2 - 12*n - 142)*x + (5*n^2 - 16*n + 64)*x^2 + (16*n - 32)*x^3)*P[n - 2, x] - (9*n - 18 - 18*x - (3*n^2 - 6*n - 4)*x^2 - 28*x^3 - 16*x^4)*D[P[n - 2, x], x] + 8*x^4*D[P[n - 2, x], {x, 2}] + (6*n + 20 + (16*n - 44)*x + 16*x^2)*P[n - 1, x])/(8)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 3 + x; v[3] = 9 + 8*x + x^2; v[4] = 22 + 46*x + 17*x^2 + x^3; for(n = 5, N, my(i = n); v[i] = (-(18*n^2 - 129*n + 194 + (21*n^2 - 161*n + 288)*x + (5*n^2 - 82*n + 278)*x^2 - (3*n^3 - 27*n^2 + 52*n + 40)*x^3)*v[i - 4] - (9*n - 18 + (9*n^2 - 99*n + 144)*x - (3*n^2 + 21*n - 184)*x^2 - (3*n^3 - 33*n^2 + 92*n - 46)*x^3 + (3*n^2 - 12*n - 8)*x^4)*deriv(v[i - 4], x) + ((9*n - 18)*x^2 - 18*x^3 - (3*n^2 - 12*n)*x^4)*deriv(deriv(v[i - 4], x), x) + (18*n^2 - 72*n + 104 + (33*n^2 - 171*n + 82)*x + (10*n^2 - 98*n + 242)*x^2 - (3*n^3 - 21*n^2 + 12*n + 88)*x^3)*v[i - 3] + (18*n - 36 + (9*n^2 - 81*n + 90)*x - (6*n^2 + 15*n - 152)*x^2 - (3*n^3 - 27*n^2 + 68*n - 2)*x^3 + (3*n^2 - 12*n - 24)*x^4)*deriv(v[i - 3], x) - ((9*n - 18)*x^2 - 18*x^3 - (3*n^2 - 12*n - 8)*x^4)*deriv(deriv(v[i - 3], x), x) - (27*n - 6 + (12*n^2 - 12*n - 142)*x + (5*n^2 - 16*n + 64)*x^2 + (16*n - 32)*x^3)*v[i - 2] - (9*n - 18 - 18*x - (3*n^2 - 6*n - 4)*x^2 - 28*x^3 - 16*x^4)*deriv(v[i - 2], x) + 8*x^4*deriv(deriv(v[i - 2], x), x) + (6*n + 20 + (16*n - 44)*x + 16*x^2)*v[i - 1])/(8)); 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-3cb22164e2f83fe1","steps":["\u03b8+1","reverse","D+1"],"edges":["e-907e3c6c6275331d","e-6878e8dc6c11eb74","e-3cb3ebd43147f8cb"],"text":"A008277 \u2192 \u03b8+1 \u2192 reverse \u2192 D+1"},"headline":"A008277 \u2192 \u03b8+1 \u2192 reverse \u2192 D+1","chain":null}