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T(n, k) = binomial(n - 1, k - 1)*(n + k)! / k!.","chain":null}},{"id":"l-ff521cb9471a94c8","kind":"fingerprint_related","node_a_id":"n-57b4b14604f4c0d7","node_b_id":"n-8ef19f31fced7b25","fingerprint":"f6-1e5fbd8fe1b88630","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1e5fbd8fe1b88630"},"other_id":"n-57b4b14604f4c0d7","other":{"id":"n-57b4b14604f4c0d7","name":"Triangle read by rows: T(n,k) is the number of Schroeder paths of semilength n containing exactly k peaks but no peaks at level one (n >= 1; 0 <= k <= n-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","1"],["5","5","1"]],"card_rows":[["1"],["2","1"],["5","5","1"],["14","21","9","1"]],"row_sums_preview":["1","3","11","45","197","903","4279","20793"],"regions":["(-inf, +inf)","(-inf, 0)","(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":["A126216"],"derivation":{"root":"n-57b4b14604f4c0d7","steps":[],"edges":[],"text":"A126216"},"headline":"A126216: Triangle read by rows: T(n,k) is the number of Schroeder paths of semilength n containing exactly k peaks but no peaks at level one (n >= 1; 0 <= k <= n-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;\n  2:          12,           6;\n  3:         120,         120,           24;\n  4:        1680,        2520,         1080,          120;\n  5:       30240,       60480,        40320,        10080,         720;\n  6:      665280,     1663200,      1512000,       604800,      100800,        5040;\n  7:    17297280,    51891840,     59875200,     33264000,     9072000,     1088640,      40320;\n  8:   518918400,  1816214400,   2542700160,   1816214400,   698544000,   139708800,   12700800,    362880;\n  9: 17643225600, 70572902400, 116237721600, 101708006400, 50854003200, 14529715200, 2235340800, 159667200, 3628800;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (4*n-2+(3*n-3)*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[(4*n - 2 + (3*n - 3)*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] = (4*n - 2 + (3*n - 3)*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-4cba07695b3e3f90","steps":["reverse"],"edges":["e-a9296ebc2b2330cd"],"text":"A357367 \u2192 reverse"},"headline":"A357367 \u2192 reverse","chain":null}