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antidiagonals.","first_index":1,"fingerprint":"f6-8de4b75c4559d58b","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1"],["1","1"]],"card_rows":[["1"],["1"],["1","1"],["1","2"]],"row_sums_preview":["1","1","2","3","5","8","13","21"],"regions":["(-inf, 0]","(-inf, 0)"],"region_exact":{},"oeis_ids":["A052553"],"derivation":{"root":"n-c8b3f39d03de798f","steps":[],"edges":[],"text":"A052553"},"headline":"A052553: Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by upward antidiagonals.","chain":null}},{"id":"l-fe6a44df1d193544","kind":"fingerprint_related","node_a_id":"n-569a801b809b6abd","node_b_id":"n-a191c1933f44029b","fingerprint":"f6-8de4b75c4559d58b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8de4b75c4559d58b"},"other_id":"n-569a801b809b6abd","other":{"id":"n-569a801b809b6abd","name":"Triangle read by rows: T(n,k) = binomial(n+1-k,k) (n >= 0, 0 <= k <= n).","first_index":0,"fingerprint":"f6-8de4b75c4559d58b","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","1"],["1","2"]],"card_rows":[["1"],["1","1"],["1","2"],["1","3","1"]],"row_sums_preview":["1","2","3","5","8","13","21","34"],"regions":["(-inf, 0]","(-inf, 0)"],"region_exact":{},"oeis_ids":["A169803"],"derivation":{"root":"n-569a801b809b6abd","steps":[],"edges":[],"text":"A169803"},"headline":"A169803: Triangle read by rows: T(n,k) = binomial(n+1-k,k) (n >= 0, 0 <= k <= n).","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:   0;\n  1:   1;\n  2:   2;\n  3:   4,    4;\n  4:   8,   16;\n  5:  16,   48,   16;\n  6:  32,  128,   96;\n  7:  64,  320,  384,   64;\n  8: 128,  768, 1280,  512;\n  9: 256, 1792, 3840, 2560, 256;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = 4*x*P(n-2,x) + 2*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[4*x*P[n - 2, x] + 2*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = 4*x*v[i - 2] + 2*v[i - 1]); 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-c8c19eba9d68881f","steps":["Veronese 2 (j=1)","basis A013611"],"edges":["e-3a4f64258abf2a3d","e-9bb075bbf20fd997"],"text":"A007318 \u2192 Veronese 2 (j=1) \u2192 basis A013611"},"headline":"A007318 \u2192 Veronese 2 (j=1) \u2192 basis A013611","chain":null}