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by rows, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k)^2/(n-k+1).","first_index":0,"fingerprint":"f6-1e5fbd8fe1b88630","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","4"],["3","15","15"]],"card_rows":[["1"],["2","4"],["3","15","15"],["4","36","84","56"]],"row_sums_preview":["1","6","33","180","985","5418","29953","166344"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-984320118b653f6b","steps":["basis A007318","reverse"],"edges":["e-2345ffa617ccabd7","e-fe7467f72472b2ce"],"text":"A132812 \u2192 basis A007318 \u2192 reverse"},"headline":"A132812 \u2192 basis A007318 \u2192 reverse","chain":null}}],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:       1;\n   2:       4,        2;\n   3:      20,       20,        4;\n   4:     112,      168,       72,        8;\n   5:     672,     1344,      896,      224,       16;\n   6:    4224,    10560,     9600,     3840,      640,       32;\n   7:   27456,    82368,    95040,    52800,    14400,     1728,      64;\n   8:  183040,   640640,   896896,   640640,   246400,    49280,    4480,    128;\n   9: 1244672,  4978688,  8200192,  7175168,  3587584,  1025024,  157696,  11264,   256;\n  10: 8599552, 38697984, 73322496, 76038144, 46964736, 17611776, 3913728, 479232, 27648, 512;\n","recurrence_sentence":"The row polynomials satisfy (n+1)*P(n,x) = -(4*n-8)*x^2*P(n-2,x) + (8*n-4+(4*n-2)*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 4 + 2*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 4 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(4*n - 8)*x^2*P[n - 2, x] + (8*n - 4 + (4*n - 2)*x)*P[n - 1, x])/(n + 1)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 4 + 2*x; for(n = 3, N, my(i = n); v[i] = (-(4*n - 8)*x^2*v[i - 2] + (8*n - 4 + (4*n - 2)*x)*v[i - 1])/(n + 1)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-0d75f6272d2a2a84","steps":["basis A007318","basis A013609"],"edges":["e-38a7985c4c1aa2e3","e-555f645e725e4322"],"text":"A114656 \u2192 basis A007318 \u2192 basis A013609"},"headline":"A114656 \u2192 basis A007318 \u2192 basis A013609","chain":null}