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begins:\n   1: 1;\n   2: 1,  3;\n   3: 1,  6,   7;\n   4: 1,  9,  23,   17;\n   5: 1, 12,  48,   76,   41;\n   6: 1, 15,  82,  204,  233,    99;\n   7: 1, 18, 125,  428,  765,   682,   239;\n   8: 1, 21, 177,  775, 1907,  2649,  1935,   577;\n   9: 1, 24, 238, 1272, 4010,  7656,  8680,  5368,  1393;\n  10: 1, 27, 308, 1946, 7506, 18358, 28548, 27312, 14641, 3363;\n","recurrence_sentence":"The row polynomials satisfy (n-2+(3*n-6)*x+(3*n-6)*x^2+(n-2)*x^3)*P(n,x) = (3*n^2-8*n+4+(20*n^2-67*n+54)*x+(58*n^2-229*n+226)*x^2+(92*n^2-405*n+442)*x^3+(83*n^2-391*n+450)*x^4+(40*n^2-196*n+232)*x^5+(8*n^2-40*n+48)*x^6)*P(n-1,x) - (n-1+(14*n-27)*x+(70*n-166)*x^2+(176*n-462)*x^3+(249*n-693)*x^4+(202*n-583)*x^5+(88*n-260)*x^6+(16*n-48)*x^7)*d/dx P(n-1,x) + (x+13*x^2+58*x^3+130*x^4+165*x^5+121*x^6+48*x^7+8*x^8)*d^2/dx^2 P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 3*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 1 + 3*x;\nP[n_, x_] := P[n, x] = Expand[Together[((3*n^2 - 8*n + 4 + (20*n^2 - 67*n + 54)*x + (58*n^2 - 229*n + 226)*x^2 + (92*n^2 - 405*n + 442)*x^3 + (83*n^2 - 391*n + 450)*x^4 + (40*n^2 - 196*n + 232)*x^5 + (8*n^2 - 40*n + 48)*x^6)*P[n - 1, x] - (n - 1 + (14*n - 27)*x + (70*n - 166)*x^2 + (176*n - 462)*x^3 + (249*n - 693)*x^4 + (202*n - 583)*x^5 + (88*n - 260)*x^6 + (16*n - 48)*x^7)*D[P[n - 1, x], x] + (x + 13*x^2 + 58*x^3 + 130*x^4 + 165*x^5 + 121*x^6 + 48*x^7 + 8*x^8)*D[P[n - 1, x], {x, 2}])/(n - 2 + (3*n - 6)*x + (3*n - 6)*x^2 + (n - 2)*x^3)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 3*x; for(n = 3, N, my(i = n); v[i] = ((3*n^2 - 8*n + 4 + (20*n^2 - 67*n + 54)*x + (58*n^2 - 229*n + 226)*x^2 + (92*n^2 - 405*n + 442)*x^3 + (83*n^2 - 391*n + 450)*x^4 + (40*n^2 - 196*n + 232)*x^5 + (8*n^2 - 40*n + 48)*x^6)*v[i - 1] - (n - 1 + (14*n - 27)*x + (70*n - 166)*x^2 + (176*n - 462)*x^3 + (249*n - 693)*x^4 + (202*n - 583)*x^5 + (88*n - 260)*x^6 + (16*n - 48)*x^7)*deriv(v[i - 1], x) + (x + 13*x^2 + 58*x^3 + 130*x^4 + 165*x^5 + 121*x^6 + 48*x^7 + 8*x^8)*deriv(deriv(v[i - 1], x), x))/(n - 2 + (3*n - 6)*x + (3*n - 6)*x^2 + (n - 2)*x^3)); 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}},{"oeis_id":"A054458","target":"reversed_triangle","first_row":1,"oeis_offset":0,"reversed":true,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:    1;\n  1:    3,     1;\n  2:    7,     6,     1;\n  3:   17,    23,     9,     1;\n  4:   41,    76,    48,    12,     1;\n  5:   99,   233,   204,    82,    15,    1;\n  6:  239,   682,   765,   428,   125,   18,    1;\n  7:  577,  1935,  2649,  1907,   775,  177,   21,   1;\n  8: 1393,  5368,  8680,  7656,  4010, 1272,  238,  24,  1;\n  9: 3363, 14641, 27312, 28548, 18358, 7506, 1946, 308, 27, 1;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where (n-1+(3*n-3)*x+(3*n-3)*x^2+(n-1)*x^3)*P(n,x) = (3*n^2-2*n-1+(20*n^2-27*n+7)*x+(58*n^2-113*n+55)*x^2+(92*n^2-221*n+129)*x^3+(83*n^2-225*n+142)*x^4+(40*n^2-116*n+76)*x^5+(8*n^2-24*n+16)*x^6)*P(n-1,x) - (n+(14*n-13)*x+(70*n-96)*x^2+(176*n-286)*x^3+(249*n-444)*x^4+(202*n-381)*x^5+(88*n-172)*x^6+(16*n-32)*x^7)*d/dx P(n-1,x) + (x+13*x^2+58*x^3+130*x^4+165*x^5+121*x^6+48*x^7+8*x^8)*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 3*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 3*x;\nP[n_, x_] := P[n, x] = Expand[Together[((3*n^2 - 2*n - 1 + (20*n^2 - 27*n + 7)*x + (58*n^2 - 113*n + 55)*x^2 + (92*n^2 - 221*n + 129)*x^3 + (83*n^2 - 225*n + 142)*x^4 + (40*n^2 - 116*n + 76)*x^5 + (8*n^2 - 24*n + 16)*x^6)*P[n - 1, x] - (n + (14*n - 13)*x + (70*n - 96)*x^2 + (176*n - 286)*x^3 + (249*n - 444)*x^4 + (202*n - 381)*x^5 + (88*n - 172)*x^6 + (16*n - 32)*x^7)*D[P[n - 1, x], x] + (x + 13*x^2 + 58*x^3 + 130*x^4 + 165*x^5 + 121*x^6 + 48*x^7 + 8*x^8)*D[P[n - 1, x], {x, 2}])/(n - 1 + (3*n - 3)*x + (3*n - 3)*x^2 + (n - 1)*x^3)]];\nTable[Reverse[PadRight[CoefficientList[P[n, x], x], n + 1]], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = ((3*n^2 - 2*n - 1 + (20*n^2 - 27*n + 7)*x + (58*n^2 - 113*n + 55)*x^2 + (92*n^2 - 221*n + 129)*x^3 + (83*n^2 - 225*n + 142)*x^4 + (40*n^2 - 116*n + 76)*x^5 + (8*n^2 - 24*n + 16)*x^6)*v[i - 1] - (n + (14*n - 13)*x + (70*n - 96)*x^2 + (176*n - 286)*x^3 + (249*n - 444)*x^4 + (202*n - 381)*x^5 + (88*n - 172)*x^6 + (16*n - 32)*x^7)*deriv(v[i - 1], x) + (x + 13*x^2 + 58*x^3 + 130*x^4 + 165*x^5 + 121*x^6 + 48*x^7 + 8*x^8)*deriv(deriv(v[i - 1], x), x))/(n - 1 + (3*n - 3)*x + (3*n - 3)*x^2 + (n - 1)*x^3)); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c = Vecrev(c); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":1,"offset":0,"reversed":true,"trim_zeros":"none","regular_triangle":true}},{"oeis_id":"A209696","target":"triangle","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 1;\n   2: 1,  3;\n   3: 1,  6,   7;\n   4: 1,  9,  23,   17;\n   5: 1, 12,  48,   76,   41;\n   6: 1, 15,  82,  204,  233,    99;\n   7: 1, 18, 125,  428,  765,   682,   239;\n   8: 1, 21, 177,  775, 1907,  2649,  1935,   577;\n   9: 1, 24, 238, 1272, 4010,  7656,  8680,  5368,  1393;\n  10: 1, 27, 308, 1946, 7506, 18358, 28548, 27312, 14641, 3363;\n","recurrence_sentence":"The row polynomials satisfy (n-2+(3*n-6)*x+(3*n-6)*x^2+(n-2)*x^3)*P(n,x) = (3*n^2-8*n+4+(20*n^2-67*n+54)*x+(58*n^2-229*n+226)*x^2+(92*n^2-405*n+442)*x^3+(83*n^2-391*n+450)*x^4+(40*n^2-196*n+232)*x^5+(8*n^2-40*n+48)*x^6)*P(n-1,x) - (n-1+(14*n-27)*x+(70*n-166)*x^2+(176*n-462)*x^3+(249*n-693)*x^4+(202*n-583)*x^5+(88*n-260)*x^6+(16*n-48)*x^7)*d/dx P(n-1,x) + (x+13*x^2+58*x^3+130*x^4+165*x^5+121*x^6+48*x^7+8*x^8)*d^2/dx^2 P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 3*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 1 + 3*x;\nP[n_, x_] := P[n, x] = Expand[Together[((3*n^2 - 8*n + 4 + (20*n^2 - 67*n + 54)*x + (58*n^2 - 229*n + 226)*x^2 + (92*n^2 - 405*n + 442)*x^3 + (83*n^2 - 391*n + 450)*x^4 + (40*n^2 - 196*n + 232)*x^5 + (8*n^2 - 40*n + 48)*x^6)*P[n - 1, x] - (n - 1 + (14*n - 27)*x + (70*n - 166)*x^2 + (176*n - 462)*x^3 + (249*n - 693)*x^4 + (202*n - 583)*x^5 + (88*n - 260)*x^6 + (16*n - 48)*x^7)*D[P[n - 1, x], x] + (x + 13*x^2 + 58*x^3 + 130*x^4 + 165*x^5 + 121*x^6 + 48*x^7 + 8*x^8)*D[P[n - 1, x], {x, 2}])/(n - 2 + (3*n - 6)*x + (3*n - 6)*x^2 + (n - 2)*x^3)]];\nTable[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 3*x; for(n = 3, N, my(i = n); v[i] = ((3*n^2 - 8*n + 4 + (20*n^2 - 67*n + 54)*x + (58*n^2 - 229*n + 226)*x^2 + (92*n^2 - 405*n + 442)*x^3 + (83*n^2 - 391*n + 450)*x^4 + (40*n^2 - 196*n + 232)*x^5 + (8*n^2 - 40*n + 48)*x^6)*v[i - 1] - (n - 1 + (14*n - 27)*x + (70*n - 166)*x^2 + (176*n - 462)*x^3 + (249*n - 693)*x^4 + (202*n - 583)*x^5 + (88*n - 260)*x^6 + (16*n - 48)*x^7)*deriv(v[i - 1], x) + (x + 13*x^2 + 58*x^3 + 130*x^4 + 165*x^5 + 121*x^6 + 48*x^7 + 8*x^8)*deriv(deriv(v[i - 1], x), x))/(n - 2 + (3*n - 6)*x + (3*n - 6)*x^2 + (n - 2)*x^3)); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); 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":true}}],"derivation":{"root":"n-e073e5e0b021f36a","steps":["x \u21a6 x/(x+1)"],"edges":["e-4b3874b4d357a754"],"text":"A035607 \u2192 x \u21a6 x/(x+1)"},"headline":"A035607 \u2192 x \u21a6 x/(x+1)","chain":null}