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begins:\n  0:    1;\n  1:    3,    3;\n  2:    7,   14,     7;\n  3:   15,   45,    45,    15;\n  4:   31,  124,   186,   124,     31;\n  5:   63,  315,   630,   630,    315,     63;\n  6:  127,  762,  1905,  2540,   1905,    762,   127;\n  7:  255, 1785,  5355,  8925,   8925,   5355,  1785,   255;\n  8:  511, 4088, 14308, 28616,  35770,  28616, 14308,  4088,  511;\n  9: 1023, 9207, 36828, 85932, 128898, 128898, 85932, 36828, 9207, 1023;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3+3*x)*P(n-1,x) - (2+4*x+2*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 3*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + 3*x;\nP[n_, x_] := P[n, x] = Expand[(3 + 3*x)*P[n - 1, x] - (2 + 4*x + 2*x^2)*P[n - 2, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (3 + 3*x)*v[i - 1] - (2 + 4*x + 2*x^2)*v[i - 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}},{"oeis_id":"A134346","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:    3,    3;\n  2:    7,   14,     7;\n  3:   15,   45,    45,    15;\n  4:   31,  124,   186,   124,     31;\n  5:   63,  315,   630,   630,    315,     63;\n  6:  127,  762,  1905,  2540,   1905,    762,   127;\n  7:  255, 1785,  5355,  8925,   8925,   5355,  1785,   255;\n  8:  511, 4088, 14308, 28616,  35770,  28616, 14308,  4088,  511;\n  9: 1023, 9207, 36828, 85932, 128898, 128898, 85932, 36828, 9207, 1023;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3+3*x)*P(n-1,x) - (2+4*x+2*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 3*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + 3*x;\nP[n_, x_] := P[n, x] = Expand[(3 + 3*x)*P[n - 1, x] - (2 + 4*x + 2*x^2)*P[n - 2, x]];\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] = 3 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (3 + 3*x)*v[i - 1] - (2 + 4*x + 2*x^2)*v[i - 2]); 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}},{"oeis_id":"A134346","target":"reversed_triangle","first_row":0,"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,    3;\n  2:    7,   14,     7;\n  3:   15,   45,    45,    15;\n  4:   31,  124,   186,   124,     31;\n  5:   63,  315,   630,   630,    315,     63;\n  6:  127,  762,  1905,  2540,   1905,    762,   127;\n  7:  255, 1785,  5355,  8925,   8925,   5355,  1785,   255;\n  8:  511, 4088, 14308, 28616,  35770,  28616, 14308,  4088,  511;\n  9: 1023, 9207, 36828, 85932, 128898, 128898, 85932, 36828, 9207, 1023;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (3+3*x)*P(n-1,x) - (2+4*x+2*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 3*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + 3*x;\nP[n_, x_] := P[n, x] = Expand[(3 + 3*x)*P[n - 1, x] - (2 + 4*x + 2*x^2)*P[n - 2, x]];\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] = 3 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (3 + 3*x)*v[i - 1] - (2 + 4*x + 2*x^2)*v[i - 2]); 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":0,"offset":0,"reversed":true,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-8d96419786c3ee2a","steps":[],"edges":[],"text":"A134346"},"headline":"A134346: Triangle read by rows: T(n,k) = (2^(n+1)-1)*binomial(n,k).","chain":null}