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A013611","chain":null}},{"id":"l-f4d86d013fb28398","kind":"fingerprint_related","node_a_id":"n-75ccd0453a9b6cec","node_b_id":"n-3a607c1aa171bfc2","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-75ccd0453a9b6cec","other":{"id":"n-75ccd0453a9b6cec","name":"Irregular triangle read by rows: T(n,k) = binomial(n-k, k), n >= 0, 0 <= k <= floor(n/2); or, coefficients of (one version of) Fibonacci polynomials.","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"]],"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":["A011973"],"derivation":{"root":"n-75ccd0453a9b6cec","steps":[],"edges":[],"text":"A011973"},"headline":"A011973: Irregular triangle read by rows: T(n,k) = binomial(n-k, k), n >= 0, 0 <= k <= floor(n/2); or, coefficients of (one version of) Fibonacci polynomials.","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:  2;\n  2:  3;\n  3:  4,  4;\n  4:  5, 10;\n  5:  6, 18,   6;\n  6:  7, 28,  21;\n  7:  8, 40,  48,   8;\n  8:  9, 54,  90,  36;\n  9: 10, 70, 150, 100, 10;\n","recurrence_sentence":"The row polynomials satisfy (n^2-n)*P(n,x) = (n^2-1+(2*n^2-2*n-4)*x)*P(n-1,x) - ((n+1)*x+(4*n+4)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[((n^2 - 1 + (2*n^2 - 2*n - 4)*x)*P[n - 1, x] - ((n + 1)*x + (4*n + 4)*x^2)*D[P[n - 1, x], x])/(n^2 - n)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((n^2 - 1 + (2*n^2 - 2*n - 4)*x)*v[i - 1] - ((n + 1)*x + (4*n + 4)*x^2)*deriv(v[i - 1], x))/(n^2 - n)); 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-87e6a919e8c0ccd0","steps":["D"],"edges":["e-ae65414b5e0c2a7c"],"text":"reversed rows \u2192 D"},"headline":"reversed rows \u2192 D","chain":null}