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Triangle read by rows, T(n, k) for 1 <= k <= n.","chain":null}},{"id":"l-f6039a5d87e939b4","kind":"fingerprint_related","node_a_id":"n-8ae1ed4c0b89ddf6","node_b_id":"n-654b1fe7cec95567","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-8ae1ed4c0b89ddf6","other":{"id":"n-8ae1ed4c0b89ddf6","name":"shift_x_plus_one of Triangle read by rows: The n-th derivative of the logistic function written in terms of y, where y = 1/(1 + exp(-x)).","first_index":1,"fingerprint":"f6-a83eeccfb5efd735","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":0,"preview_rows":[["1"],["0","-1"],["0","1","2"]],"card_rows":[["1"],["0","-1"],["0","1","2"],["0","-1","-6","-6"]],"row_sums_preview":["1","-1","3","-13","75","-541","4683","-47293"],"regions":["(-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2ae6a2215aff79d4","steps":["x \u21a6 x+1"],"edges":["e-455c21f397ee5eef"],"text":"A163626 \u2192 x \u21a6 x+1"},"headline":"A163626 \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:          4,          2;\n  2:         24,         24,           4;\n  3:        192,        288,         112,           8;\n  4:       1920,       3840,        2400,         480,         16;\n  5:      23040,      57600,       49920,       17280,       1984,         32;\n  6:     322560,     967680,     1075200,      537600,     115584,       8064,        64;\n  7:    5160960,   18063360,    24514560,    16128000,    5225472,     741888,     32512,      128;\n  8:   92897280,  371589120,   596090880,   487710720,  213534720,   47738880,   4646400,   130560,    256;\n  9: 1857945600, 8360755200, 15482880000, 15173222400, 8414945280, 2612736000, 419082240, 28661760, 523264, 512;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2*n+2+2*n*x)*P(n-1,x) - (2*x+2*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[(2*n + 2 + 2*n*x)*P[n - 1, x] - (2*x + 2*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] = (2*n + 2 + 2*n*x)*v[i - 1] - (2*x + 2*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-41809911e05bb4ba","steps":["x \u21a6 x+1"],"edges":["e-ee15557b740e7c6a"],"text":"A257609 \u2192 x \u21a6 x+1"},"headline":"A257609 \u2192 x \u21a6 x+1","chain":null}