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begins:\n   1:       1;\n   2:       3,       1;\n   3:      12,       8,       1;\n   4:      55,      55,      15,       1;\n   5:     273,     364,     156,      24,       1;\n   6:    1428,    2380,    1400,     350,      35,      1;\n   7:    7752,   15504,   11628,    4080,     680,     48,     1;\n   8:   43263,  100947,   92169,   41895,    9975,   1197,    63,    1;\n   9:  246675,  657800,  708400,  396704,  123970,  21560,  1960,   80,  1;\n  10: 1430715, 4292145, 5328180, 3552120, 1381380, 318780, 42504, 3036, 99, 1;\n","recurrence_sentence":"The row polynomials satisfy (2*n^2+n)*P(n,x) = ((27/2)*n^2-(27/2)*n+3+(6*n^2-8*n+2)*x)*P(n-1,x) - (((9/2)*n-3/2)*x+(4*n-1)*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(((27/2)*n^2 - (27/2)*n + 3 + (6*n^2 - 8*n + 2)*x)*P[n - 1, x] - (((9/2)*n - 3/2)*x + (4*n - 1)*x^2)*D[P[n - 1, x], x])/(2*n^2 + n)]];\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] = (((27/2)*n^2 - (27/2)*n + 3 + (6*n^2 - 8*n + 2)*x)*v[i - 1] - (((9/2)*n - 3/2)*x + (4*n - 1)*x^2)*deriv(v[i - 1], x))/(2*n^2 + n)); v};\nrow(n, p) = {my(c = Vecrev(p)); 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":"none","regular_triangle":false}},{"oeis_id":"A102537","target":"reversed_triangle","first_row":1,"oeis_offset":1,"reversed":true,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 1;\n   2: 1,  3;\n   3: 1,  8,   12;\n   4: 1, 15,   55,    55;\n   5: 1, 24,  156,   364,    273;\n   6: 1, 35,  350,  1400,   2380,    1428;\n   7: 1, 48,  680,  4080,  11628,   15504,    7752;\n   8: 1, 63, 1197,  9975,  41895,   92169,  100947,   43263;\n   9: 1, 80, 1960, 21560, 123970,  396704,  708400,  657800,  246675;\n  10: 1, 99, 3036, 42504, 318780, 1381380, 3552120, 5328180, 4292145, 1430715;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where (2*n^2+n)*P(n,x) = ((27/2)*n^2-(27/2)*n+3+(6*n^2-8*n+2)*x)*P(n-1,x) - (((9/2)*n-3/2)*x+(4*n-1)*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(((27/2)*n^2 - (27/2)*n + 3 + (6*n^2 - 8*n + 2)*x)*P[n - 1, x] - (((9/2)*n - 3/2)*x + (4*n - 1)*x^2)*D[P[n - 1, x], x])/(2*n^2 + n)]];\nTable[Reverse[PadRight[CoefficientList[P[n, x], x], n]], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (((27/2)*n^2 - (27/2)*n + 3 + (6*n^2 - 8*n + 2)*x)*v[i - 1] - (((9/2)*n - 3/2)*x + (4*n - 1)*x^2)*deriv(v[i - 1], x))/(2*n^2 + n)); 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":1,"offset":1,"reversed":true,"trim_zeros":"none","regular_triangle":true}},{"oeis_id":"A243662","target":"triangle","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:       1;\n   2:       3,       1;\n   3:      12,       8,       1;\n   4:      55,      55,      15,       1;\n   5:     273,     364,     156,      24,       1;\n   6:    1428,    2380,    1400,     350,      35,      1;\n   7:    7752,   15504,   11628,    4080,     680,     48,     1;\n   8:   43263,  100947,   92169,   41895,    9975,   1197,    63,    1;\n   9:  246675,  657800,  708400,  396704,  123970,  21560,  1960,   80,  1;\n  10: 1430715, 4292145, 5328180, 3552120, 1381380, 318780, 42504, 3036, 99, 1;\n","recurrence_sentence":"The row polynomials satisfy (2*n^2+n)*P(n,x) = ((27/2)*n^2-(27/2)*n+3+(6*n^2-8*n+2)*x)*P(n-1,x) - (((9/2)*n-3/2)*x+(4*n-1)*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(((27/2)*n^2 - (27/2)*n + 3 + (6*n^2 - 8*n + 2)*x)*P[n - 1, x] - (((9/2)*n - 3/2)*x + (4*n - 1)*x^2)*D[P[n - 1, x], x])/(2*n^2 + n)]];\nTable[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (((27/2)*n^2 - (27/2)*n + 3 + (6*n^2 - 8*n + 2)*x)*v[i - 1] - (((9/2)*n - 3/2)*x + (4*n - 1)*x^2)*deriv(v[i - 1], x))/(2*n^2 + n)); 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":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-f00f6de2d4bbc6f5","steps":[],"edges":[],"text":"A243662"},"headline":"A243662: Triangle read by rows: the reversed x = 1+q Narayana triangle at m=2.","chain":null}