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2^k.","chain":null}},{"id":"l-363007e298d3f662","kind":"oeis_match","node_a_id":"n-b0fc620949fb5990","node_b_id":null,"fingerprint":null,"oeis_id":"A001333","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A001333","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-51676d67e7cfbfd7","kind":"oeis_match","node_a_id":"n-b0fc620949fb5990","node_b_id":null,"fingerprint":null,"oeis_id":"A078057","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":2,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"target":"row_sums","terms_compared":14},"in_catalog":false,"oeis_id":"A078057","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-699ec35da941401d","kind":"oeis_match","node_a_id":"n-b0fc620949fb5990","node_b_id":null,"fingerprint":null,"oeis_id":"A035607","in_catalog":1,"target":"reversed_triangle","row_alignment":"tabl","rows_matched":11,"oeis_nonn":1,"details":{"alignment":{"first_row":2,"low_zeros_dropped":true,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":11,"target":"reversed_triangle","terms_compared":71},"in_catalog":true,"oeis_id":"A035607","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-afb20fca09e9bbdc","kind":"oeis_match","node_a_id":"n-b0fc620949fb5990","node_b_id":null,"fingerprint":null,"oeis_id":"A266213","in_catalog":1,"target":"reversed_triangle","row_alignment":"tabl","rows_matched":11,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":11,"target":"reversed_triangle","terms_compared":66},"in_catalog":true,"oeis_id":"A266213","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-d6c10339a45fa159","kind":"oeis_match","node_a_id":"n-b0fc620949fb5990","node_b_id":null,"fingerprint":null,"oeis_id":"A113413","in_catalog":1,"target":"triangle","row_alignment":"tabl","rows_matched":11,"oeis_nonn":1,"details":{"alignment":{"first_row":2,"low_zeros_dropped":true,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":11,"target":"triangle","terms_compared":69},"in_catalog":true,"oeis_id":"A113413","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-e5bcd495bf30cfb1","kind":"oeis_match","node_a_id":"n-b0fc620949fb5990","node_b_id":null,"fingerprint":null,"oeis_id":"A122542","in_catalog":1,"target":"triangle","row_alignment":"tabl","rows_matched":11,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":11,"target":"triangle","terms_compared":66},"in_catalog":true,"oeis_id":"A122542","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A122542","relation":"identification"},{"oeis_id":"A035607","relation":"match_reversed_triangle"},{"oeis_id":"A266213","relation":"match_reversed_triangle"},{"oeis_id":"A001333","relation":"match_row_sums"},{"oeis_id":"A078057","relation":"match_row_sums"},{"oeis_id":"A113413","relation":"match_triangle"},{"oeis_id":"A122542","relation":"match_triangle"}],"bfiles":[{"node_id":"n-b0fc620949fb5990","oeis_id":"A001333","target":"row_sums","oeis_offset":0,"first_row":1,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":32,"oeis_terms_available":32,"term_count":200,"raw_size":8548},{"node_id":"n-b0fc620949fb5990","oeis_id":"A035607","target":"reversed_triangle","oeis_offset":0,"first_row":2,"rows_used":44,"terms_matched":71,"oeis_terms_agreeing":71,"oeis_terms_available":71,"term_count":990,"raw_size":12275},{"node_id":"n-b0fc620949fb5990","oeis_id":"A078057","target":"row_sums","oeis_offset":0,"first_row":2,"rows_used":200,"terms_matched":14,"oeis_terms_agreeing":32,"oeis_terms_available":32,"term_count":200,"raw_size":8624},{"node_id":"n-b0fc620949fb5990","oeis_id":"A113413","target":"triangle","oeis_offset":0,"first_row":2,"rows_used":44,"terms_matched":69,"oeis_terms_agreeing":69,"oeis_terms_available":69,"term_count":990,"raw_size":12275},{"node_id":"n-b0fc620949fb5990","oeis_id":"A122542","target":"triangle","oeis_offset":0,"first_row":1,"rows_used":44,"terms_matched":66,"oeis_terms_agreeing":66,"oeis_terms_available":66,"term_count":990,"raw_size":11823},{"node_id":"n-b0fc620949fb5990","oeis_id":"A266213","target":"reversed_triangle","oeis_offset":0,"first_row":1,"rows_used":44,"terms_matched":66,"oeis_terms_agreeing":66,"oeis_terms_available":66,"term_count":990,"raw_size":11823}],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 1;\n   2: 0, 1;\n   3: 0, 2,  1;\n   4: 0, 2,  4,   1;\n   5: 0, 2,  8,   6,   1;\n   6: 0, 2, 12,  18,   8,   1;\n   7: 0, 2, 16,  38,  32,  10,   1;\n   8: 0, 2, 20,  66,  88,  50,  12,  1;\n   9: 0, 2, 24, 102, 192, 170,  72, 14,  1;\n  10: 0, 2, 28, 146, 360, 450, 292, 98, 16, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+x)*P(n-1,x) + x*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := x;\nP[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 1, x] + x*P[n - 2, x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = x; for(n = 3, N, my(i = n); v[i] = (1 + x)*v[i - 1] + x*v[i - 2]); 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":"A035607","target":"reversed_triangle","first_row":2,"oeis_offset":0,"reversed":true,"trim_zeros":"low","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0: 1;\n  1: 1,  2;\n  2: 1,  4,   2;\n  3: 1,  6,   8,   2;\n  4: 1,  8,  18,  12,   2;\n  5: 1, 10,  32,  38,  16,    2;\n  6: 1, 12,  50,  88,  66,   20,   2;\n  7: 1, 14,  72, 170, 192,  102,  24,   2;\n  8: 1, 16,  98, 292, 450,  360, 146,  28,  2;\n  9: 1, 18, 128, 462, 912, 1002, 608, 198, 32, 2;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, down to the lowest nonzero coefficient, where P(n,x) = (1+x)*P(n-1,x) + x*P(n-2,x) for n >= 2, with P(0,x) = x, P(1,x) = 2*x + x^2.","mathematica":"P[0, x_] := x;\nP[1, x_] := 2*x + x^2;\nP[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 1, x] + x*P[n - 2, x]];\nTable[Reverse[PadRight[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x], n + 1]], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = x; v[2] = 2*x + x^2; for(n = 2, N, my(i = n + 1); v[i] = (1 + x)*v[i - 1] + x*v[i - 2]); v};\nrow(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); 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":2,"offset":0,"reversed":true,"trim_zeros":"low","regular_triangle":true}},{"oeis_id":"A113413","target":"triangle","first_row":2,"oeis_offset":0,"reversed":false,"trim_zeros":"low","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0: 1;\n  1: 2,  1;\n  2: 2,  4,   1;\n  3: 2,  8,   6,   1;\n  4: 2, 12,  18,   8,    1;\n  5: 2, 16,  38,  32,   10,   1;\n  6: 2, 20,  66,  88,   50,  12,   1;\n  7: 2, 24, 102, 192,  170,  72,  14,   1;\n  8: 2, 28, 146, 360,  450, 292,  98,  16,  1;\n  9: 2, 32, 198, 608, 1002, 912, 462, 128, 18, 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) = (1+x)*P(n-1,x) + x*P(n-2,x) for n >= 2, with P(0,x) = x, P(1,x) = 2*x + x^2.","mathematica":"P[0, x_] := x;\nP[1, x_] := 2*x + x^2;\nP[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 1, x] + x*P[n - 2, x]];\nTable[PadRight[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x], n + 1], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = x; v[2] = 2*x + x^2; for(n = 2, N, my(i = n + 1); v[i] = (1 + x)*v[i - 1] + x*v[i - 2]); v};\nrow(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); 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":2,"offset":0,"reversed":false,"trim_zeros":"low","regular_triangle":true}},{"oeis_id":"A122542","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, 2,  1;\n  3: 0, 2,  4,   1;\n  4: 0, 2,  8,   6,   1;\n  5: 0, 2, 12,  18,   8,   1;\n  6: 0, 2, 16,  38,  32,  10,   1;\n  7: 0, 2, 20,  66,  88,  50,  12,  1;\n  8: 0, 2, 24, 102, 192, 170,  72, 14,  1;\n  9: 0, 2, 28, 146, 360, 450, 292, 98, 16, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+x)*P(n-1,x) + x*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := x;\nP[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 1, x] + x*P[n - 2, x]];\nTable[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] = x; for(n = 2, N, my(i = n + 1); v[i] = (1 + x)*v[i - 1] + x*v[i - 2]); 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}},{"oeis_id":"A266213","target":"reversed_triangle","first_row":1,"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,  0;\n  2: 1,  2,  0;\n  3: 1,  4,  2,   0;\n  4: 1,  6,  8,   2,   0;\n  5: 1,  8, 18,  12,   2,   0;\n  6: 1, 10, 32,  38,  16,   2,   0;\n  7: 1, 12, 50,  88,  66,  20,   2,  0;\n  8: 1, 14, 72, 170, 192, 102,  24,  2, 0;\n  9: 1, 16, 98, 292, 450, 360, 146, 28, 2, 0;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+x)*P(n-1,x) + x*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := x;\nP[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 1, x] + x*P[n - 2, 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] = x; for(n = 2, N, my(i = n + 1); v[i] = (1 + x)*v[i - 1] + x*v[i - 2]); 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":1,"offset":0,"reversed":true,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-b0fc620949fb5990","steps":[],"edges":[],"text":"A122542"},"headline":"A122542: Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.","chain":null}