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(\u03b8+1)\u00b2","chain":null}},{"id":"l-34767d18bac5e324","kind":"oeis_match","node_a_id":"n-19ce16e7d575fc90","node_b_id":null,"fingerprint":null,"oeis_id":"A001793","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A001793","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-46db2694f429da3c","kind":"oeis_match","node_a_id":"n-19ce16e7d575fc90","node_b_id":null,"fingerprint":null,"oeis_id":"A125092","in_catalog":1,"target":"reversed_triangle","row_alignment":"tabl","rows_matched":10,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":10,"target":"reversed_triangle","terms_compared":61},"in_catalog":true,"oeis_id":"A125092","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A125092","relation":"match_reversed_triangle"},{"oeis_id":"A001793","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-19ce16e7d575fc90","oeis_id":"A001793","target":"row_sums","oeis_offset":1,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":28,"oeis_terms_available":28,"term_count":200,"raw_size":7619},{"node_id":"n-19ce16e7d575fc90","oeis_id":"A125092","target":"reversed_triangle","oeis_offset":0,"first_row":0,"rows_used":44,"terms_matched":61,"oeis_terms_agreeing":61,"oeis_terms_available":61,"term_count":990,"raw_size":12719}],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:   1;\n  1:   4,   1;\n  2:   9,   8,    1;\n  3:  16,  27,   12,    1;\n  4:  25,  64,   54,   16,    1;\n  5:  36, 125,  160,   90,   20,    1;\n  6:  49, 216,  375,  320,  135,   24,    1;\n  7:  64, 343,  756,  875,  560,  189,   28,   1;\n  8:  81, 512, 1372, 2016, 1750,  896,  252,  32,  1;\n  9: 100, 729, 2304, 4116, 4536, 3150, 1344, 324, 36, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+3*x+3*x^2+x^3)*P(n-3,x) - (3+6*x+3*x^2)*P(n-2,x) + (3+3*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 4 + x, P(2,x) = 9 + 8*x + x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4 + x;\nP[2, x_] := 9 + 8*x + x^2;\nP[n_, x_] := P[n, x] = Expand[(1 + 3*x + 3*x^2 + x^3)*P[n - 3, x] - (3 + 6*x + 3*x^2)*P[n - 2, x] + (3 + 3*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; v[2] = 4 + x; v[3] = 9 + 8*x + x^2; for(n = 3, N, my(i = n + 1); v[i] = (1 + 3*x + 3*x^2 + x^3)*v[i - 3] - (3 + 6*x + 3*x^2)*v[i - 2] + (3 + 3*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":"A125092","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,  4;\n  2: 1,  8,   9;\n  3: 1, 12,  27,   16;\n  4: 1, 16,  54,   64,   25;\n  5: 1, 20,  90,  160,  125,   36;\n  6: 1, 24, 135,  320,  375,  216,   49;\n  7: 1, 28, 189,  560,  875,  756,  343,   64;\n  8: 1, 32, 252,  896, 1750, 2016, 1372,  512,  81;\n  9: 1, 36, 324, 1344, 3150, 4536, 4116, 2304, 729, 100;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+3*x+3*x^2+x^3)*P(n-3,x) - (3+6*x+3*x^2)*P(n-2,x) + (3+3*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 4 + x, P(2,x) = 9 + 8*x + x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4 + x;\nP[2, x_] := 9 + 8*x + x^2;\nP[n_, x_] := P[n, x] = Expand[(1 + 3*x + 3*x^2 + x^3)*P[n - 3, x] - (3 + 6*x + 3*x^2)*P[n - 2, x] + (3 + 3*x)*P[n - 1, x]];\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; v[2] = 4 + x; v[3] = 9 + 8*x + x^2; for(n = 3, N, my(i = n + 1); v[i] = (1 + 3*x + 3*x^2 + x^3)*v[i - 3] - (3 + 6*x + 3*x^2)*v[i - 2] + (3 + 3*x)*v[i - 1]); 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-c8c19eba9d68881f","steps":["(\u03b8+1)\u00b2","reverse"],"edges":["e-6f35c326bbf0f62d","e-f997576344e2820e"],"text":"A007318 \u2192 (\u03b8+1)\u00b2 \u2192 reverse"},"headline":"A007318 \u2192 (\u03b8+1)\u00b2 \u2192 reverse","chain":null}