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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}},{"id":"l-6370bb11e5fefc36","kind":"oeis_match","node_a_id":"n-b0d6ec97a8d172b8","node_b_id":null,"fingerprint":null,"oeis_id":"A253283","in_catalog":1,"target":"triangle","row_alignment":"tabl","rows_matched":10,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"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":"A253283","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-736d94d3fc00279e","kind":"oeis_match","node_a_id":"n-b0d6ec97a8d172b8","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":2,"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":"A253283","relation":"identification"},{"oeis_id":"A002002","relation":"match_row_sums"},{"oeis_id":"A253283","relation":"match_triangle"}],"bfiles":[{"node_id":"n-b0d6ec97a8d172b8","oeis_id":"A253283","target":"triangle","oeis_offset":0,"first_row":1,"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":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,   3;\n   4: 0, 3,  12,   10;\n   5: 0, 4,  30,   60,    35;\n   6: 0, 5,  60,  210,   280,   126;\n   7: 0, 6, 105,  560,  1260,  1260,    462;\n   8: 0, 7, 168, 1260,  4200,  6930,   5544,   1716;\n   9: 0, 8, 252, 2520, 11550, 27720,  36036,  24024,   6435;\n  10: 0, 9, 360, 4620, 27720, 90090, 168168, 180180, 102960, 24310;\n","recurrence_sentence":"The row polynomials satisfy -(n-1)*P(n,x) = -(3*n-6+(4*n-6)*x)*P(n-1,x) + (3*n-9+(4*n-14)*x)*P(n-2,x) - (n-4)*P(n-3,x) for n >= 4, with P(1,x) = 1, P(2,x) = x, P(3,x) = 2*x + 3*x^2.","mathematica":"P[1, x_] := 1;\nP[2, x_] := x;\nP[3, x_] := 2*x + 3*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(3*n - 6 + (4*n - 6)*x)*P[n - 1, x] + (3*n - 9 + (4*n - 14)*x)*P[n - 2, x] - (n - 4)*P[n - 3, x])/(-n + 1)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = x; v[3] = 2*x + 3*x^2; for(n = 4, N, my(i = n); v[i] = (-(3*n - 6 + (4*n - 6)*x)*v[i - 1] + (3*n - 9 + (4*n - 14)*x)*v[i - 2] - (n - 4)*v[i - 3])/(-n + 1)); 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":"A253283","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,   3;\n  3: 0, 3,  12,   10;\n  4: 0, 4,  30,   60,    35;\n  5: 0, 5,  60,  210,   280,   126;\n  6: 0, 6, 105,  560,  1260,  1260,    462;\n  7: 0, 7, 168, 1260,  4200,  6930,   5544,   1716;\n  8: 0, 8, 252, 2520, 11550, 27720,  36036,  24024,   6435;\n  9: 0, 9, 360, 4620, 27720, 90090, 168168, 180180, 102960, 24310;\n","recurrence_sentence":"The row polynomials satisfy -n*P(n,x) = -(3*n-3+(4*n-2)*x)*P(n-1,x) + (3*n-6+(4*n-10)*x)*P(n-2,x) - (n-3)*P(n-3,x) for n >= 3, with P(0,x) = 1, P(1,x) = x, P(2,x) = 2*x + 3*x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := x;\nP[2, x_] := 2*x + 3*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(3*n - 3 + (4*n - 2)*x)*P[n - 1, x] + (3*n - 6 + (4*n - 10)*x)*P[n - 2, x] - (n - 3)*P[n - 3, x])/(-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; v[3] = 2*x + 3*x^2; for(n = 3, N, my(i = n + 1); v[i] = (-(3*n - 3 + (4*n - 2)*x)*v[i - 1] + (3*n - 6 + (4*n - 10)*x)*v[i - 2] - (n - 3)*v[i - 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":1,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":true}}],"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}