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Also called the Catalan triangle.","first_index":0,"fingerprint":"f6-2c1fea4a558d4e6c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","2"],["10","12","3"]],"card_rows":[["1"],["3","2"],["10","12","3"],["35","60","30","4"]],"row_sums_preview":["1","5","25","129","681","3653","19825","108545"],"regions":["(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c2eaf5212a4dec13","steps":["\u03b8+1","x \u21a6 x+1"],"edges":["e-127d139330d3e4d4","e-09282c699b4099b7"],"text":"A001263 \u2192 \u03b8+1 \u2192 x \u21a6 x+1"},"headline":"A001263 \u2192 \u03b8+1 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-f54d0e1819a0a7f0","kind":"fingerprint_related","node_a_id":"n-0a5e54eacdb24124","node_b_id":"n-b0d6ec97a8d172b8","fingerprint":"f6-2c1fea4a558d4e6c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2c1fea4a558d4e6c"},"other_id":"n-b0d6ec97a8d172b8","other":{"id":"n-b0d6ec97a8d172b8","name":"Triangle read by rows: coefficients of the partial fraction decomposition of [d^n/dx^n] (x/(1-x))^n/n!.","first_index":1,"fingerprint":"f6-2c1fea4a558d4e6c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["0","1"],["0","2","3"]],"card_rows":[["1"],["0","1"],["0","2","3"],["0","3","12","10"]],"row_sums_preview":["1","1","5","25","129","681","3653","19825"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":["A253283"],"derivation":{"root":"n-b0d6ec97a8d172b8","steps":[],"edges":[],"text":"A253283"},"headline":"A253283: Triangle read by rows: coefficients of the partial fraction decomposition of [d^n/dx^n] (x/(1-x))^n/n!.","chain":null}},{"id":"l-58eede43c81e10ae","kind":"oeis_match","node_a_id":"n-0a5e54eacdb24124","node_b_id":null,"fingerprint":null,"oeis_id":"A360282","in_catalog":1,"target":"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":"triangle","terms_compared":55},"in_catalog":true,"oeis_id":"A360282","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-a818eddd49fb8156","kind":"oeis_match","node_a_id":"n-0a5e54eacdb24124","node_b_id":null,"fingerprint":null,"oeis_id":"A002002","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":1,"target":"row_sums","terms_compared":14},"in_catalog":false,"oeis_id":"A002002","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A360282","relation":"identification"},{"oeis_id":"A002002","relation":"match_row_sums"},{"oeis_id":"A360282","relation":"match_triangle"}],"bfiles":[{"node_id":"n-0a5e54eacdb24124","oeis_id":"A360282","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":20166}],"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;\n  2: 0,     3,      2;\n  3: 0,    10,     12,      3;\n  4: 0,    35,     60,     30,      4;\n  5: 0,   126,    280,    210,     60,     5;\n  6: 0,   462,   1260,   1260,    560,   105,     6;\n  7: 0,  1716,   5544,   6930,   4200,  1260,   168,    7;\n  8: 0,  6435,  24024,  36036,  27720, 11550,  2520,  252,   8;\n  9: 0, 24310, 102960, 180180, 168168, 90090, 27720, 4620, 360, 9;\n","recurrence_sentence":"The row polynomials satisfy (2*n^2-3*n)*P(n,x) = (8*n^2-16*n+6+(4*n^2-8*n+4)*x)*P(n-1,x) - (2*n^2-5*n+2)*x^2*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[Together[((8*n^2 - 16*n + 6 + (4*n^2 - 8*n + 4)*x)*P[n - 1, x] - (2*n^2 - 5*n + 2)*x^2*P[n - 2, x])/(2*n^2 - 3*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; for(n = 2, N, my(i = n + 1); v[i] = ((8*n^2 - 16*n + 6 + (4*n^2 - 8*n + 4)*x)*v[i - 1] - (2*n^2 - 5*n + 2)*x^2*v[i - 2])/(2*n^2 - 3*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":"A360282","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: 0,     1;\n  2: 0,     3,      2;\n  3: 0,    10,     12,      3;\n  4: 0,    35,     60,     30,      4;\n  5: 0,   126,    280,    210,     60,     5;\n  6: 0,   462,   1260,   1260,    560,   105,     6;\n  7: 0,  1716,   5544,   6930,   4200,  1260,   168,    7;\n  8: 0,  6435,  24024,  36036,  27720, 11550,  2520,  252,   8;\n  9: 0, 24310, 102960, 180180, 168168, 90090, 27720, 4620, 360, 9;\n","recurrence_sentence":"The row polynomials satisfy (2*n^2-3*n)*P(n,x) = (8*n^2-16*n+6+(4*n^2-8*n+4)*x)*P(n-1,x) - (2*n^2-5*n+2)*x^2*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[Together[((8*n^2 - 16*n + 6 + (4*n^2 - 8*n + 4)*x)*P[n - 1, x] - (2*n^2 - 5*n + 2)*x^2*P[n - 2, x])/(2*n^2 - 3*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; v[2] = x; for(n = 2, N, my(i = n + 1); v[i] = ((8*n^2 - 16*n + 6 + (4*n^2 - 8*n + 4)*x)*v[i - 1] - (2*n^2 - 5*n + 2)*x^2*v[i - 2])/(2*n^2 - 3*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}}],"derivation":{"root":"n-0a5e54eacdb24124","steps":[],"edges":[],"text":"A360282"},"headline":"A360282: Triangle read by rows. T(n, k) = (1/2) * binomial(2*(n - k + 1), n - k + 1) * binomial(2*n - k, k - 1) for n > 0, T(0, 0) = 1.","chain":null}