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begins:\n   1: 1;\n   2: 0, 1;\n   3: 0, 1,     1;\n   4: 0, 1,     5,       1;\n   5: 0, 1,    21,      14,       1;\n   6: 0, 1,    85,     147,      30,       1;\n   7: 0, 1,   341,    1408,     627,      55,      1;\n   8: 0, 1,  1365,   13013,   11440,    2002,     91,     1;\n   9: 0, 1,  5461,  118482,  196053,   61490,   5278,   140,   1;\n  10: 0, 1, 21845, 1071799, 3255330, 1733303, 251498, 12138, 204, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = x*P(n-1,x) + x*d/dx P(n-1,x) + x^2*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[x*P[n - 1, x] + x*D[P[n - 1, x], x] + x^2*D[P[n - 1, x], {x, 2}]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = x*v[i - 1] + x*deriv(v[i - 1], x) + x^2*deriv(deriv(v[i - 1], x), x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, 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x^2*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = x.","mathematica":"P[1, x_] := x;\nP[n_, x_] := P[n, x] = Expand[x*P[n - 1, x] + x*D[P[n - 1, x], x] + x^2*D[P[n - 1, x], {x, 2}]];\nTable[Reverse[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] = x; for(n = 2, N, my(i = n); v[i] = x*v[i - 1] + x*deriv(v[i - 1], x) + x^2*deriv(deriv(v[i - 1], x), x)); v};\nrow(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); c = concat(c, vector(n - #c)); c = Vecrev(c); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":2,"offset":1,"reversed":true,"trim_zeros":"low","regular_triangle":true}},{"oeis_id":"A036969","target":"triangle","first_row":2,"oeis_offset":1,"reversed":false,"trim_zeros":"low","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 1;\n   2: 1,     1;\n   3: 1,     5,       1;\n   4: 1,    21,      14,        1;\n   5: 1,    85,     147,       30,        1;\n   6: 1,   341,    1408,      627,       55,        1;\n   7: 1,  1365,   13013,    11440,     2002,       91,      1;\n   8: 1,  5461,  118482,   196053,    61490,     5278,    140,     1;\n   9: 1, 21845, 1071799,  3255330,  1733303,   251498,  12138,   204,   1;\n  10: 1, 87381, 9668036, 53157079, 46587905, 10787231, 846260, 25194, 285, 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 P(n,x) = x*P(n-1,x) + x*d/dx P(n-1,x) + x^2*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = x.","mathematica":"P[1, x_] := x;\nP[n_, x_] := P[n, x] = Expand[x*P[n - 1, x] + x*D[P[n - 1, x], x] + x^2*D[P[n - 1, x], {x, 2}]];\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] = x; for(n = 2, N, my(i = n); v[i] = x*v[i - 1] + x*deriv(v[i - 1], x) + x^2*deriv(deriv(v[i - 1], x), x)); 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":2,"offset":1,"reversed":false,"trim_zeros":"low","regular_triangle":true}},{"oeis_id":"A269945","target":"triangle","first_row":1,"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,     5,       1;\n  4: 0, 1,    21,      14,       1;\n  5: 0, 1,    85,     147,      30,       1;\n  6: 0, 1,   341,    1408,     627,      55,      1;\n  7: 0, 1,  1365,   13013,   11440,    2002,     91,     1;\n  8: 0, 1,  5461,  118482,  196053,   61490,   5278,   140,   1;\n  9: 0, 1, 21845, 1071799, 3255330, 1733303, 251498, 12138, 204, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = x*P(n-1,x) + x*d/dx P(n-1,x) + x^2*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[x*P[n - 1, x] + x*D[P[n - 1, x], x] + x^2*D[P[n - 1, x], {x, 2}]];\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] = x*v[i - 1] + x*deriv(v[i - 1], x) + x^2*deriv(deriv(v[i - 1], x), 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":1,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-0164479e47952770","steps":[],"edges":[],"text":"A269945"},"headline":"A269945: Triangle read by rows. Stirling set numbers of order 2, T(n, n) = 1, T(n, k) = 0 if k < 0 or k > n, otherwise T(n, k) = T(n-1, k-1) + k^2*T(n-1, k), for 0 <= k <= n.","chain":null}