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v[1] = 2; v[2] = 23 + 23*x; v[3] = 2 + 1054*x + 2*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((n - 2 + (n - 2)*x)*v[i - 1] + (483*n - 966)*x*v[i - 2])/(n - 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":"A153656","target":"triangle","first_row":0,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:  2;\n   2: 23,    23;\n   3:  2,  1054,        2;\n   4:  2, 12165,    12165,           2;\n   5:  2, 13133,   533412,       13133,             2;\n   6:  2, 14101,  6422240,     6422240,         14101,             2;\n   7:  2, 15069, 12779580,   270482476,      12779580,         15069,           2;\n   8:  2, 16037, 19605432,  3385203976,    3385203976,      19605432,       16037,        2;\n   9:  2, 17005, 26899796,  9577346548,  137413443860,    9577346548,    26899796,    17005,     2;\n  10:  2, 17973, 34662672, 19073670000, 1782044310816, 1782044310816, 19073670000, 34662672, 17973, 2;\n","recurrence_sentence":"The row polynomials satisfy (n-3)*P(n,x) = (n-3+(n-3)*x)*P(n-1,x) + (483*n-1449)*x*P(n-2,x) for n >= 4, with P(1,x) = 2, P(2,x) = 23 + 23*x, P(3,x) = 2 + 1054*x + 2*x^2.","mathematica":"P[1, x_] := 2;\nP[2, x_] := 23 + 23*x;\nP[3, x_] := 2 + 1054*x + 2*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((n - 3 + (n - 3)*x)*P[n - 1, x] + (483*n - 1449)*x*P[n - 2, x])/(n - 3)]];\nTable[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 2; v[2] = 23 + 23*x; v[3] = 2 + 1054*x + 2*x^2; for(n = 4, N, my(i = n); v[i] = ((n - 3 + (n - 3)*x)*v[i - 1] + (483*n - 1449)*x*v[i - 2])/(n - 3)); v};\nrow(n, p) = {my(c = Vecrev(p)); 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":0,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":true}},{"oeis_id":"A153656","target":"reversed_triangle","first_row":0,"oeis_offset":1,"reversed":true,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:  2;\n   2: 23,    23;\n   3:  2,  1054,        2;\n   4:  2, 12165,    12165,           2;\n   5:  2, 13133,   533412,       13133,             2;\n   6:  2, 14101,  6422240,     6422240,         14101,             2;\n   7:  2, 15069, 12779580,   270482476,      12779580,         15069,           2;\n   8:  2, 16037, 19605432,  3385203976,    3385203976,      19605432,       16037,        2;\n   9:  2, 17005, 26899796,  9577346548,  137413443860,    9577346548,    26899796,    17005,     2;\n  10:  2, 17973, 34662672, 19073670000, 1782044310816, 1782044310816, 19073670000, 34662672, 17973, 2;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where (n-3)*P(n,x) = (n-3+(n-3)*x)*P(n-1,x) + (483*n-1449)*x*P(n-2,x) for n >= 4, with P(1,x) = 2, P(2,x) = 23 + 23*x, P(3,x) = 2 + 1054*x + 2*x^2.","mathematica":"P[1, x_] := 2;\nP[2, x_] := 23 + 23*x;\nP[3, x_] := 2 + 1054*x + 2*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((n - 3 + (n - 3)*x)*P[n - 1, x] + (483*n - 1449)*x*P[n - 2, x])/(n - 3)]];\nTable[Reverse[PadRight[CoefficientList[P[n, x], x], n]], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 2; v[2] = 23 + 23*x; v[3] = 2 + 1054*x + 2*x^2; for(n = 4, N, my(i = n); v[i] = ((n - 3 + (n - 3)*x)*v[i - 1] + (483*n - 1449)*x*v[i - 2])/(n - 3)); v};\nrow(n, p) = {my(c = Vecrev(p)); 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":0,"offset":1,"reversed":true,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-6e554ebca19f83a9","steps":[],"edges":[],"text":"A153656"},"headline":"A153656: Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + (2*j +3)*prime(j)*T(n-2, k-1) with j=9, read by rows.","chain":null}