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begins:\n  0: 1;\n  1: 0,     2;\n  2: 0,     2,      4;\n  3: 0,     4,     12,      8;\n  4: 0,    12,     44,     48,      16;\n  5: 0,    48,    200,    280,     160,     32;\n  6: 0,   240,   1096,   1800,    1360,    480,     64;\n  7: 0,  1440,   7056,  12992,   11760,   5600,   1344,   128;\n  8: 0, 10080,  52272, 105056,  108304,  62720,  20608,  3584,  256;\n  9: 0, 80640, 438336, 944992, 1076544, 718368, 290304, 69888, 9216, 512;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (n-1+2*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(n - 1 + 2*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {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] = (n - 1 + 2*x)*v[i - 1]); 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":"A125553","target":"triangle","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"low","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:      2;\n   2:      2,       4;\n   3:      4,      12,       8;\n   4:     12,      44,      48,       16;\n   5:     48,     200,     280,      160,      32;\n   6:    240,    1096,    1800,     1360,     480,      64;\n   7:   1440,    7056,   12992,    11760,    5600,    1344,     128;\n   8:  10080,   52272,  105056,   108304,   62720,   20608,    3584,    256;\n   9:  80640,  438336,  944992,  1076544,  718368,  290304,   69888,   9216,   512;\n  10: 725760, 4106304, 9381600, 11578880, 8618400, 4049472, 1209600, 222720, 23040, 1024;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in ascending powers of x, starting at the lowest nonzero coefficient, where P(n,x) = (n-1+2*x)*P(n-1,x) for n >= 2, with P(1,x) = 2*x.","mathematica":"P[1, x_] := 2*x;\nP[n_, x_] := P[n, x] = Expand[(n - 1 + 2*x)*P[n - 1, x]];\nTable[PadRight[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x], n], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 2*x; for(n = 2, N, my(i = n); v[i] = (n - 1 + 2*x)*v[i - 1]); v};\nrow(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); 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":"low","regular_triangle":true}},{"oeis_id":"A137320","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,     2;\n  2: 0,     2,      4;\n  3: 0,     4,     12,      8;\n  4: 0,    12,     44,     48,      16;\n  5: 0,    48,    200,    280,     160,     32;\n  6: 0,   240,   1096,   1800,    1360,    480,     64;\n  7: 0,  1440,   7056,  12992,   11760,   5600,   1344,   128;\n  8: 0, 10080,  52272, 105056,  108304,  62720,  20608,  3584,  256;\n  9: 0, 80640, 438336, 944992, 1076544, 718368, 290304, 69888, 9216, 512;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (n-1+2*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(n - 1 + 2*x)*P[n - 1, 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] = (n - 1 + 2*x)*v[i - 1]); 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-60c7e36cbe887b7e","steps":[],"edges":[],"text":"A137320"},"headline":"A137320: Coefficients of raising factorial polynomials, T(n,k) = [x^k] p(x, n) where p(x, n) = (m*x + n - 1)*p(x, n - 1) with p[x, 0] = 1, p[x, -1] = 0, p[x, 1] = m*x and m = 2. 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