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begins:\n  0:    1;\n  1:    1,     1;\n  2:    2,     5,      3;\n  3:    5,    21,     28,      12;\n  4:   14,    84,    180,     165,      55;\n  5:   42,   330,    990,    1430,    1001,     273;\n  6:  132,  1287,   5005,   10010,   10920,    6188,    1428;\n  7:  429,  5005,  24024,   61880,   92820,   81396,   38760,    7752;\n  8: 1430, 19448, 111384,  352716,  678300,  813960,  596904,  245157,   43263;\n  9: 4862, 75582, 503880, 1899240, 4476780, 6864396, 6864396, 4326300, 1562275, 246675;\n","recurrence_sentence":"The row polynomials satisfy (4*n^2+2*n)*P(n,x) = (14*n^2-9*n+1+(18*n^2-15*n+3)*x)*P(n-1,x) + (n-1+(10*n-4)*x+(9*n-3)*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[Together[((14*n^2 - 9*n + 1 + (18*n^2 - 15*n + 3)*x)*P[n - 1, x] + (n - 1 + (10*n - 4)*x + (9*n - 3)*x^2)*D[P[n - 1, x], x])/(4*n^2 + 2*n)]];\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] = ((14*n^2 - 9*n + 1 + (18*n^2 - 15*n + 3)*x)*v[i - 1] + (n - 1 + (10*n - 4)*x + (9*n - 3)*x^2)*deriv(v[i - 1], x))/(4*n^2 + 2*n)); 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":"A104978","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:    1,     1;\n  2:    2,     5,      3;\n  3:    5,    21,     28,      12;\n  4:   14,    84,    180,     165,      55;\n  5:   42,   330,    990,    1430,    1001,     273;\n  6:  132,  1287,   5005,   10010,   10920,    6188,    1428;\n  7:  429,  5005,  24024,   61880,   92820,   81396,   38760,    7752;\n  8: 1430, 19448, 111384,  352716,  678300,  813960,  596904,  245157,   43263;\n  9: 4862, 75582, 503880, 1899240, 4476780, 6864396, 6864396, 4326300, 1562275, 246675;\n","recurrence_sentence":"The row polynomials satisfy (4*n^2+2*n)*P(n,x) = (14*n^2-9*n+1+(18*n^2-15*n+3)*x)*P(n-1,x) + (n-1+(10*n-4)*x+(9*n-3)*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[Together[((14*n^2 - 9*n + 1 + (18*n^2 - 15*n + 3)*x)*P[n - 1, x] + (n - 1 + (10*n - 4)*x + (9*n - 3)*x^2)*D[P[n - 1, x], x])/(4*n^2 + 2*n)]];\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] = ((14*n^2 - 9*n + 1 + (18*n^2 - 15*n + 3)*x)*v[i - 1] + (n - 1 + (10*n - 4)*x + (9*n - 3)*x^2)*deriv(v[i - 1], x))/(4*n^2 + 2*n)); 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":"A108426","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:      1,       1;\n  2:      3,       5,       2;\n  3:     12,      28,      21,       5;\n  4:     55,     165,     180,      84,      14;\n  5:    273,    1001,    1430,     990,     330,      42;\n  6:   1428,    6188,   10920,   10010,    5005,    1287,     132;\n  7:   7752,   38760,   81396,   92820,   61880,   24024,    5005,    429;\n  8:  43263,  245157,  596904,  813960,  678300,  352716,  111384,  19448,  1430;\n  9: 246675, 1562275, 4326300, 6864396, 6864396, 4476780, 1899240, 503880, 75582, 4862;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where (4*n^2+2*n)*P(n,x) = (14*n^2-9*n+1+(18*n^2-15*n+3)*x)*P(n-1,x) + (n-1+(10*n-4)*x+(9*n-3)*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[Together[((14*n^2 - 9*n + 1 + (18*n^2 - 15*n + 3)*x)*P[n - 1, x] + (n - 1 + (10*n - 4)*x + (9*n - 3)*x^2)*D[P[n - 1, x], x])/(4*n^2 + 2*n)]];\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] = ((14*n^2 - 9*n + 1 + (18*n^2 - 15*n + 3)*x)*v[i - 1] + (n - 1 + (10*n - 4)*x + (9*n - 3)*x^2)*deriv(v[i - 1], x))/(4*n^2 + 2*n)); 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-9ed87ca196739b1c","steps":[],"edges":[],"text":"A104978"},"headline":"A104978: Triangle read by rows, where the g.f. satisfies A(x, y) = 1 + x*A(x, y)^2 + x*y*A(x, y)^3.","chain":null}