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reverse","chain":null}},{"id":"l-1fe893d5cebb5e47","kind":"oeis_match","node_a_id":"n-e4d248920a317259","node_b_id":null,"fingerprint":null,"oeis_id":"A001850","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":0,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A001850","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-8a8ce4d9df0f7212","kind":"oeis_match","node_a_id":"n-e4d248920a317259","node_b_id":null,"fingerprint":null,"oeis_id":"A063007","in_catalog":1,"target":"triangle","row_alignment":"tabl","rows_matched":10,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":true,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":10,"target":"triangle","terms_compared":55},"in_catalog":true,"oeis_id":"A063007","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-9a3189127414591b","kind":"oeis_match","node_a_id":"n-e4d248920a317259","node_b_id":null,"fingerprint":null,"oeis_id":"A104684","in_catalog":1,"target":"reversed_triangle","row_alignment":"tabl","rows_matched":9,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":true,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":9,"target":"reversed_triangle","terms_compared":50},"in_catalog":true,"oeis_id":"A104684","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-f1667c93d0d2f629","kind":"oeis_match","node_a_id":"n-e4d248920a317259","node_b_id":null,"fingerprint":null,"oeis_id":"A109983","in_catalog":1,"target":"triangle","row_alignment":"node_lengths","rows_matched":8,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"node_lengths","rows_matched":8,"target":"triangle","terms_compared":64},"in_catalog":true,"oeis_id":"A109983","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A109983","relation":"identification"},{"oeis_id":"A104684","relation":"match_reversed_triangle"},{"oeis_id":"A001850","relation":"match_row_sums"},{"oeis_id":"A063007","relation":"match_triangle"},{"oeis_id":"A109983","relation":"match_triangle"}],"bfiles":[{"node_id":"n-e4d248920a317259","oeis_id":"A001850","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":24,"oeis_terms_available":24,"term_count":200,"raw_size":15989},{"node_id":"n-e4d248920a317259","oeis_id":"A063007","target":"triangle","oeis_offset":0,"first_row":0,"rows_used":44,"terms_matched":55,"oeis_terms_agreeing":55,"oeis_terms_available":55,"term_count":990,"raw_size":20689},{"node_id":"n-e4d248920a317259","oeis_id":"A104684","target":"reversed_triangle","oeis_offset":0,"first_row":0,"rows_used":44,"terms_matched":50,"oeis_terms_agreeing":50,"oeis_terms_available":50,"term_count":990,"raw_size":20689},{"node_id":"n-e4d248920a317259","oeis_id":"A109983","target":"triangle","oeis_offset":0,"first_row":0,"rows_used":32,"terms_matched":64,"oeis_terms_agreeing":64,"oeis_terms_available":64,"term_count":1024,"raw_size":11515}],"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: 0, 1, 2;\n  2: 0, 0, 1, 6,  6;\n  3: 0, 0, 0, 1, 12, 30, 20;\n  4: 0, 0, 0, 0,  1, 20, 90, 140,  70;\n  5: 0, 0, 0, 0,  0,  1, 30, 210, 560,  630,  252;\n  6: 0, 0, 0, 0,  0,  0,  1,  42, 420, 1680, 3150,  2772,   924;\n  7: 0, 0, 0, 0,  0,  0,  0,   1,  56,  756, 4200, 11550, 16632, 12012,   3432;\n  8: 0, 0, 0, 0,  0,  0,  0,   0,   1,   72, 1260,  9240, 34650, 72072,  84084,  51480,  12870;\n  9: 0, 0, 0, 0,  0,  0,  0,   0,   0,    1,   90,  1980, 18480, 90090, 252252, 420420, 411840, 218790, 48620;\n","recurrence_sentence":"The row polynomials satisfy n*P(n,x) = ((2*n-1)*x+(4*n-2)*x^2)*P(n-1,x) - (n-1)*x^2*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = x + 2*x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := x + 2*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[(((2*n - 1)*x + (4*n - 2)*x^2)*P[n - 1, x] - (n - 1)*x^2*P[n - 2, x])/(n)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = x + 2*x^2; for(n = 2, N, my(i = n + 1); v[i] = (((2*n - 1)*x + (4*n - 2)*x^2)*v[i - 1] - (n - 1)*x^2*v[i - 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":"A063007","target":"triangle","first_row":0,"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: 1,  2;\n  2: 1,  6,    6;\n  3: 1, 12,   30,    20;\n  4: 1, 20,   90,   140,    70;\n  5: 1, 30,  210,   560,   630,    252;\n  6: 1, 42,  420,  1680,  3150,   2772,    924;\n  7: 1, 56,  756,  4200, 11550,  16632,  12012,   3432;\n  8: 1, 72, 1260,  9240, 34650,  72072,  84084,  51480,  12870;\n  9: 1, 90, 1980, 18480, 90090, 252252, 420420, 411840, 218790, 48620;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in ascending powers of x, starting at the lowest nonzero coefficient, where n*P(n,x) = ((2*n-1)*x+(4*n-2)*x^2)*P(n-1,x) - (n-1)*x^2*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = x + 2*x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := x + 2*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[(((2*n - 1)*x + (4*n - 2)*x^2)*P[n - 1, x] - (n - 1)*x^2*P[n - 2, x])/(n)]];\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] = 1; v[2] = x + 2*x^2; for(n = 2, N, my(i = n + 1); v[i] = (((2*n - 1)*x + (4*n - 2)*x^2)*v[i - 1] - (n - 1)*x^2*v[i - 2])/(n)); 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":0,"offset":0,"reversed":false,"trim_zeros":"low","regular_triangle":true}},{"oeis_id":"A104684","target":"reversed_triangle","first_row":0,"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:     2,      1;\n  2:     6,      6,      1;\n  3:    20,     30,     12,      1;\n  4:    70,    140,     90,     20,      1;\n  5:   252,    630,    560,    210,     30,     1;\n  6:   924,   2772,   3150,   1680,    420,    42,     1;\n  7:  3432,  12012,  16632,  11550,   4200,   756,    56,    1;\n  8: 12870,  51480,  84084,  72072,  34650,  9240,  1260,   72,  1;\n  9: 48620, 218790, 411840, 420420, 252252, 90090, 18480, 1980, 90, 1;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, down to the lowest nonzero coefficient, where n*P(n,x) = ((2*n-1)*x+(4*n-2)*x^2)*P(n-1,x) - (n-1)*x^2*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = x + 2*x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := x + 2*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[(((2*n - 1)*x + (4*n - 2)*x^2)*P[n - 1, x] - (n - 1)*x^2*P[n - 2, x])/(n)]];\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] = 1; v[2] = x + 2*x^2; for(n = 2, N, my(i = n + 1); v[i] = (((2*n - 1)*x + (4*n - 2)*x^2)*v[i - 1] - (n - 1)*x^2*v[i - 2])/(n)); 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":0,"offset":0,"reversed":true,"trim_zeros":"low","regular_triangle":true}},{"oeis_id":"A109983","target":"triangle","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: 0, 1, 2;\n  2: 0, 0, 1, 6,  6;\n  3: 0, 0, 0, 1, 12, 30, 20;\n  4: 0, 0, 0, 0,  1, 20, 90, 140,  70;\n  5: 0, 0, 0, 0,  0,  1, 30, 210, 560,  630,  252;\n  6: 0, 0, 0, 0,  0,  0,  1,  42, 420, 1680, 3150,  2772,   924;\n  7: 0, 0, 0, 0,  0,  0,  0,   1,  56,  756, 4200, 11550, 16632, 12012,   3432;\n  8: 0, 0, 0, 0,  0,  0,  0,   0,   1,   72, 1260,  9240, 34650, 72072,  84084,  51480,  12870;\n  9: 0, 0, 0, 0,  0,  0,  0,   0,   0,    1,   90,  1980, 18480, 90090, 252252, 420420, 411840, 218790, 48620;\n","recurrence_sentence":"The row polynomials satisfy n*P(n,x) = ((2*n-1)*x+(4*n-2)*x^2)*P(n-1,x) - (n-1)*x^2*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = x + 2*x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := x + 2*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[(((2*n - 1)*x + (4*n - 2)*x^2)*P[n - 1, x] - (n - 1)*x^2*P[n - 2, x])/(n)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = x + 2*x^2; for(n = 2, N, my(i = n + 1); v[i] = (((2*n - 1)*x + (4*n - 2)*x^2)*v[i - 1] - (n - 1)*x^2*v[i - 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}}],"derivation":{"root":"n-e4d248920a317259","steps":[],"edges":[],"text":"A109983"},"headline":"A109983: Triangle read by rows: T(n, k) (0<=k<=2n) is the number of Delannoy paths of length n, having k steps.","chain":null}