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Triangle by columns, (2^k, 0, 0, 0, ...).","first_index":0,"fingerprint":"f6-4732d0911d5a78c5","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["2"],["8","8"]],"card_rows":[["2"],["8","8"],["24","72","24"],["64","448","384","64"]],"row_sums_preview":["0","2","16","120","960","8320","77952","785792"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ae7910bbe46f8a41","steps":["D","basis A008277"],"edges":["e-6151a42dffed124f","e-f43dacdafb30f22b"],"text":"A130123 \u2192 D \u2192 basis A008277"},"headline":"A130123 \u2192 D \u2192 basis A008277","chain":null}}],"oeis":[],"bfiles":[],"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,  3,    1;\n   2: 1,  6,    7,     1;\n   3: 1, 10,   25,    15,      1;\n   4: 1, 15,   65,    90,     31,      1;\n   5: 1, 21,  140,   350,    301,     63,       1;\n   6: 1, 28,  266,  1050,   1701,    966,     127,       1;\n   7: 1, 36,  462,  2646,   6951,   7770,    3025,     255,      1;\n   8: 1, 45,  750,  5880,  22827,  42525,   34105,    9330,    511,     1;\n   9: 1, 55, 1155, 11880,  63987, 179487,  246730,  145750,  28501,  1023,    1;\n  10: 1, 66, 1705, 22275, 159027, 627396, 1323652, 1379400, 611501, 86526, 2047, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+(n+1)*x)*P(n-1,x) - x^2*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1 + 3*x + x^2.","mathematica":"P[1, x_] := 1 + 3*x + x^2;\nP[n_, x_] := P[n, x] = Expand[(1 + (n + 1)*x)*P[n - 1, x] - x^2*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1 + 3*x + x^2; for(n = 2, N, my(i = n); v[i] = (1 + (n + 1)*x)*v[i - 1] - x^2*deriv(v[i - 1], x)); 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}}],"derivation":{"root":"n-5fe240122e98ffb3","steps":["basis A008277","reverse"],"edges":["e-66762082773dd3e0","e-cfa21e99eff5f537"],"text":"basis-map coefficients of A112227 \u2192 basis A008277 \u2192 reverse"},"headline":"basis-map coefficients of A112227 \u2192 basis A008277 \u2192 reverse","chain":null}
