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x/(x+1)","chain":null}},{"id":"l-e3d0e6e8f1ed99de","kind":"fingerprint_related","node_a_id":"n-d6d713aa669b74c1","node_b_id":"n-9d90ef1d46f325d6","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-d6d713aa669b74c1","other":{"id":"n-d6d713aa669b74c1","name":"shift_x_plus_one of Euler's triangle: triangle of Eulerian numbers T(n,k) (n>=0, 0 <= k <= n) read by rows.","first_index":1,"fingerprint":"f6-a83eeccfb5efd735","polytool_family":null,"degree_pattern":"irregular","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1"],["2","1"]],"card_rows":[["1"],["1"],["2","1"],["6","6","1"]],"row_sums_preview":["1","1","3","13","75","541","4683","47293"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-efd1fdf07e5468d8","steps":["x \u21a6 x+1"],"edges":["e-aed75c9f11c92985"],"text":"A173018 \u2192 x \u21a6 x+1"},"headline":"A173018 \u2192 x \u21a6 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:   2,      4;\n  2:   4,     24,       24;\n  3:   8,    112,      288,       192;\n  4:  16,    480,     2400,      3840,       1920;\n  5:  32,   1984,    17280,     49920,      57600,      23040;\n  6:  64,   8064,   115584,    537600,    1075200,     967680,      322560;\n  7: 128,  32512,   741888,   5225472,   16128000,   24514560,    18063360,     5160960;\n  8: 256, 130560,  4646400,  47738880,  213534720,  487710720,   596090880,   371589120,   92897280;\n  9: 512, 523264, 28661760, 419082240, 2612736000, 8414945280, 15173222400, 15482880000, 8360755200, 1857945600;\n","recurrence_sentence":"The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (2+8*x+10*x^2+4*x^3)*P(n-1,x) + (2*x+6*x^2+6*x^3+2*x^4)*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[Together[((2 + 8*x + 10*x^2 + 4*x^3)*P[n - 1, x] + (2*x + 6*x^2 + 6*x^3 + 2*x^4)*D[P[n - 1, x], x])/(1 + 2*x + x^2)]];\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] = ((2 + 8*x + 10*x^2 + 4*x^3)*v[i - 1] + (2*x + 6*x^2 + 6*x^3 + 2*x^4)*deriv(v[i - 1], x))/(1 + 2*x + x^2)); 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-41809911e05bb4ba","steps":["x \u21a6 x/(x+1)"],"edges":["e-554dfd3e11a18bac"],"text":"A257609 \u2192 x \u21a6 x/(x+1)"},"headline":"A257609 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