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begins:\n  0: 1;\n  1: 0, 1;\n  2: 0, 1,   1;\n  3: 0, 1,   4,    1;\n  4: 0, 1,  10,   10,     1;\n  5: 0, 1,  20,   50,    20,     1;\n  6: 0, 1,  35,  175,   175,    35,     1;\n  7: 0, 1,  56,  490,   980,   490,    56,    1;\n  8: 0, 1,  84, 1176,  4116,  4116,  1176,   84,   1;\n  9: 0, 1, 120, 2520, 14112, 24696, 14112, 2520, 120, 1;\n","recurrence_sentence":"The row polynomials satisfy (n^2+5*n+6)*P(n,x) = -(n^2-2*n+1-(8*n^2+5*n-1)*x)*P(n-1,x) + ((9*n+3)*x-(9*n+3)*x^2)*d/dx P(n-1,x) + (2*n^2-6*n+4+(4*n^2-12*n+8)*x+(2*n^2-6*n+4)*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^2 - 2*n + 1 - (8*n^2 + 5*n - 1)*x)*P[n - 1, x] + ((9*n + 3)*x - (9*n + 3)*x^2)*D[P[n - 1, x], x] + (2*n^2 - 6*n + 4 + (4*n^2 - 12*n + 8)*x + (2*n^2 - 6*n + 4)*x^2)*P[n - 2, x])/(n^2 + 5*n + 6)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^2 - 2*n + 1 - (8*n^2 + 5*n - 1)*x)*v[i - 1] + ((9*n + 3)*x - (9*n + 3)*x^2)*deriv(v[i - 1], x) + (2*n^2 - 6*n + 4 + (4*n^2 - 12*n + 8)*x + (2*n^2 - 6*n + 4)*x^2)*v[i - 2])/(n^2 + 5*n + 6)); 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}},{"oeis_id":"A056939","target":"triangle","first_row":1,"oeis_offset":0,"reversed":false,"trim_zeros":"low","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0: 1;\n  1: 1,   1;\n  2: 1,   4,    1;\n  3: 1,  10,   10,     1;\n  4: 1,  20,   50,    20,      1;\n  5: 1,  35,  175,   175,     35,      1;\n  6: 1,  56,  490,   980,    490,     56,     1;\n  7: 1,  84, 1176,  4116,   4116,   1176,    84,    1;\n  8: 1, 120, 2520, 14112,  24696,  14112,  2520,  120,   1;\n  9: 1, 165, 4950, 41580, 116424, 116424, 41580, 4950, 165, 1;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in ascending powers of x, starting at the lowest nonzero coefficient, where (n^2+7*n+12)*P(n,x) = -(n^2-(8*n^2+21*n+12)*x)*P(n-1,x) + ((9*n+12)*x-(9*n+12)*x^2)*d/dx P(n-1,x) + (2*n^2-2*n+(4*n^2-4*n)*x+(2*n^2-2*n)*x^2)*P(n-2,x) for n >= 2, with P(0,x) = x, P(1,x) = x + x^2.","mathematica":"P[0, x_] := x;\nP[1, x_] := x + x^2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^2 - (8*n^2 + 21*n + 12)*x)*P[n - 1, x] + ((9*n + 12)*x - (9*n + 12)*x^2)*D[P[n - 1, x], x] + (2*n^2 - 2*n + (4*n^2 - 4*n)*x + (2*n^2 - 2*n)*x^2)*P[n - 2, x])/(n^2 + 7*n + 12)]];\nTable[PadRight[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x], n + 1], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = x; v[2] = x + x^2; for(n = 2, N, my(i = n + 1); v[i] = (-(n^2 - (8*n^2 + 21*n + 12)*x)*v[i - 1] + ((9*n + 12)*x - (9*n + 12)*x^2)*deriv(v[i - 1], x) + (2*n^2 - 2*n + (4*n^2 - 4*n)*x + (2*n^2 - 2*n)*x^2)*v[i - 2])/(n^2 + 7*n + 12)); v};\nrow(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); c = concat(c, vector(n + 1 - #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":false,"trim_zeros":"low","regular_triangle":true}},{"oeis_id":"A056939","target":"reversed_triangle","first_row":1,"oeis_offset":0,"reversed":true,"trim_zeros":"low","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0: 1;\n  1: 1,   1;\n  2: 1,   4,    1;\n  3: 1,  10,   10,     1;\n  4: 1,  20,   50,    20,      1;\n  5: 1,  35,  175,   175,     35,      1;\n  6: 1,  56,  490,   980,    490,     56,     1;\n  7: 1,  84, 1176,  4116,   4116,   1176,    84,    1;\n  8: 1, 120, 2520, 14112,  24696,  14112,  2520,  120,   1;\n  9: 1, 165, 4950, 41580, 116424, 116424, 41580, 4950, 165, 1;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, down to the lowest nonzero coefficient, where (n^2+7*n+12)*P(n,x) = -(n^2-(8*n^2+21*n+12)*x)*P(n-1,x) + ((9*n+12)*x-(9*n+12)*x^2)*d/dx P(n-1,x) + (2*n^2-2*n+(4*n^2-4*n)*x+(2*n^2-2*n)*x^2)*P(n-2,x) for n >= 2, with P(0,x) = x, P(1,x) = x + x^2.","mathematica":"P[0, x_] := x;\nP[1, x_] := x + x^2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^2 - (8*n^2 + 21*n + 12)*x)*P[n - 1, x] + ((9*n + 12)*x - (9*n + 12)*x^2)*D[P[n - 1, x], x] + (2*n^2 - 2*n + (4*n^2 - 4*n)*x + (2*n^2 - 2*n)*x^2)*P[n - 2, x])/(n^2 + 7*n + 12)]];\nTable[Reverse[PadRight[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x], n + 1]], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = x; v[2] = x + x^2; for(n = 2, N, my(i = n + 1); v[i] = (-(n^2 - (8*n^2 + 21*n + 12)*x)*v[i - 1] + ((9*n + 12)*x - (9*n + 12)*x^2)*deriv(v[i - 1], x) + (2*n^2 - 2*n + (4*n^2 - 4*n)*x + (2*n^2 - 2*n)*x^2)*v[i - 2])/(n^2 + 7*n + 12)); v};\nrow(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); 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":"low","regular_triangle":true}},{"oeis_id":"A359363","target":"triangle","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0: 1;\n  1: 0, 1;\n  2: 0, 1,   1;\n  3: 0, 1,   4,    1;\n  4: 0, 1,  10,   10,     1;\n  5: 0, 1,  20,   50,    20,     1;\n  6: 0, 1,  35,  175,   175,    35,     1;\n  7: 0, 1,  56,  490,   980,   490,    56,    1;\n  8: 0, 1,  84, 1176,  4116,  4116,  1176,   84,   1;\n  9: 0, 1, 120, 2520, 14112, 24696, 14112, 2520, 120, 1;\n","recurrence_sentence":"The row polynomials satisfy (n^2+5*n+6)*P(n,x) = -(n^2-2*n+1-(8*n^2+5*n-1)*x)*P(n-1,x) + ((9*n+3)*x-(9*n+3)*x^2)*d/dx P(n-1,x) + (2*n^2-6*n+4+(4*n^2-12*n+8)*x+(2*n^2-6*n+4)*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^2 - 2*n + 1 - (8*n^2 + 5*n - 1)*x)*P[n - 1, x] + ((9*n + 3)*x - (9*n + 3)*x^2)*D[P[n - 1, x], x] + (2*n^2 - 6*n + 4 + (4*n^2 - 12*n + 8)*x + (2*n^2 - 6*n + 4)*x^2)*P[n - 2, x])/(n^2 + 5*n + 6)]];\nTable[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] = x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^2 - 2*n + 1 - (8*n^2 + 5*n - 1)*x)*v[i - 1] + ((9*n + 3)*x - (9*n + 3)*x^2)*deriv(v[i - 1], x) + (2*n^2 - 6*n + 4 + (4*n^2 - 12*n + 8)*x + (2*n^2 - 6*n + 4)*x^2)*v[i - 2])/(n^2 + 5*n + 6)); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); 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":true}}],"derivation":{"root":"n-afb2f273dc5c03d1","steps":[],"edges":[],"text":"A359363"},"headline":"A359363: Triangle read by rows. The coefficients of the Baxter polynomials p(0, x) = 1 and p(n, x) = x*hypergeom([-1 - n, -n, 1 - n], [2, 3], -x) for n >= 1.","chain":null}