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v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (5 + (2*n + 3)*x)*v[i - 1] + (2*x - 2*x^2)*deriv(v[i - 1], x)); 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":"A257615","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:       5,         5;\n  2:      25,        70,         25;\n  3:     125,       715,        715,         125;\n  4:     625,      6380,      12870,        6380,         625;\n  5:    3125,     52785,     186010,      186010,       52785,        3125;\n  6:   15625,    416370,    2360295,     4092220,     2360295,      416370,       15625;\n  7:   78125,   3180215,   27488205,    75698255,    75698255,    27488205,     3180215,      78125;\n  8:  390625,  23745880,  301457500,  1245003880,  1968154630,  1245003880,   301457500,   23745880,    390625;\n  9: 1953125, 174424285, 3164289220, 18819820180, 44261068390, 44261068390, 18819820180, 3164289220, 174424285, 1953125;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (5+(2*n+3)*x)*P(n-1,x) + (2*x-2*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(5 + (2*n + 3)*x)*P[n - 1, x] + (2*x - 2*x^2)*D[P[n - 1, x], 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; for(n = 1, N, my(i = n + 1); v[i] = (5 + (2*n + 3)*x)*v[i - 1] + (2*x - 2*x^2)*deriv(v[i - 1], x)); 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":"A257615","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:       5,         5;\n  2:      25,        70,         25;\n  3:     125,       715,        715,         125;\n  4:     625,      6380,      12870,        6380,         625;\n  5:    3125,     52785,     186010,      186010,       52785,        3125;\n  6:   15625,    416370,    2360295,     4092220,     2360295,      416370,       15625;\n  7:   78125,   3180215,   27488205,    75698255,    75698255,    27488205,     3180215,      78125;\n  8:  390625,  23745880,  301457500,  1245003880,  1968154630,  1245003880,   301457500,   23745880,    390625;\n  9: 1953125, 174424285, 3164289220, 18819820180, 44261068390, 44261068390, 18819820180, 3164289220, 174424285, 1953125;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (5+(2*n+3)*x)*P(n-1,x) + (2*x-2*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(5 + (2*n + 3)*x)*P[n - 1, x] + (2*x - 2*x^2)*D[P[n - 1, x], 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; for(n = 1, N, my(i = n + 1); v[i] = (5 + (2*n + 3)*x)*v[i - 1] + (2*x - 2*x^2)*deriv(v[i - 1], x)); 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-3f0a1785078e99be","steps":[],"edges":[],"text":"A257615"},"headline":"A257615: Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 2*x + 5.","chain":null}