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Triangle read by rows, T(n, k) for 1 <= k <= n.","first_index":0,"fingerprint":"f6-a83eeccfb5efd735","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","1"],["6","6","1"]],"card_rows":[["1"],["2","1"],["6","6","1"],["24","36","14","1"]],"row_sums_preview":["1","3","13","75","541","4683","47293","545835"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":["A090582"],"derivation":{"root":"n-3b2b9e31bd278a34","steps":[],"edges":[],"text":"A090582"},"headline":"A090582: T(n, k) = Sum_{j=0..n-k} (-1)^j*binomial(n - k + 1, j)*(n - k + 1 - j)^n. Triangle read by rows, T(n, k) for 1 <= k <= n.","chain":null}},{"id":"l-fe6fb042350a6a2d","kind":"fingerprint_related","node_a_id":"n-c9b2c9de3080193e","node_b_id":"n-e62f038d3232303b","fingerprint":"f6-a83eeccfb5efd735","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a83eeccfb5efd735"},"other_id":"n-e62f038d3232303b","other":{"id":"n-e62f038d3232303b","name":"mobius_x_over_x_plus_one of Triangle read by rows: T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(n) = 3*n + 3.","first_index":0,"fingerprint":"f6-a83eeccfb5efd735","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","6"],["9","54","54"]],"card_rows":[["1"],["3","6"],["9","54","54"],["27","378","972","648"]],"row_sums_preview":["1","9","117","2025","43821","1137969","34476597","1193741145"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-dfedc6a36c97eba4","steps":["x \u21a6 x/(x+1)"],"edges":["e-a57608fc217f45d7"],"text":"A257620 \u2192 x \u21a6 x/(x+1)"},"headline":"A257620 \u2192 x \u21a6 x/(x+1)","chain":null}}],"oeis":[],"bfiles":[],"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:           6,            3;\n  2:          54,           54,            9;\n  3:         648,          972,          378,           27;\n  4:        9720,        19440,        12150,         2430,           81;\n  5:      174960,       437400,       379080,       131220,        15066,          243;\n  6:     3674160,     11022480,     12247200,      6123600,      1316574,        91854,         729;\n  7:    88179840,    308629440,    418854240,    275562000,     89282088,     12675852,      555498,       2187;\n  8:  2380855680,   9523422720,  15277157280,  12499492320,   5472661320,   1223495280,   119082150,    3346110,     6561;\n  9: 71425670400, 321415516800, 595213920000, 583309641600, 323498765520, 100442349000, 16110929160, 1101854340, 20116026, 19683;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3*n+3+3*n*x)*P(n-1,x) - (3*x+3*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(3*n + 3 + 3*n*x)*P[n - 1, x] - (3*x + 3*x^2)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (3*n + 3 + 3*n*x)*v[i - 1] - (3*x + 3*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 - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-dfedc6a36c97eba4","steps":["x \u21a6 x+1"],"edges":["e-143420f2b7aa0495"],"text":"A257620 \u2192 x \u21a6 x+1"},"headline":"A257620 \u2192 x \u21a6 x+1","chain":null}