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v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (5 + (n + 4)*x)*v[i - 1] + (x - 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":"A257607","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,       60,        25;\n  3:     125,      535,       535,        125;\n  4:     625,     4210,      7490,       4210,        625;\n  5:    3125,    30885,     86110,      86110,      30885,       3125;\n  6:   15625,   216560,    880735,    1377760,     880735,     216560,      15625;\n  7:   78125,  1471235,   8330745,   18948695,   18948695,    8330745,    1471235,     78125;\n  8:  390625,  9764910,  74498800,  234897010,  341076510,  234897010,   74498800,   9764910,   390625;\n  9: 1953125, 63667585, 638670520, 2698662880, 5418658690, 5418658690, 2698662880, 638670520, 63667585, 1953125;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (5+(n+4)*x)*P(n-1,x) + (x-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 + (n + 4)*x)*P[n - 1, x] + (x - 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)); 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v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (5 + (n + 4)*x)*v[i - 1] + (x - 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-8919a5e7419ba068","steps":[],"edges":[],"text":"A257607"},"headline":"A257607: 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) = x + 5.","chain":null}