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 A216154 Triangle read by rows, T(n,k) n>=0, k>=0, generalization of A000255. 1
 1, 1, 1, 3, 4, 1, 11, 21, 9, 1, 53, 128, 78, 16, 1, 309, 905, 710, 210, 25, 1, 2119, 7284, 6975, 2680, 465, 36, 1, 16687, 65821, 74319, 35035, 7945, 903, 49, 1, 148329, 660064, 857836, 478464, 133630, 19936, 1596, 64, 1, 1468457, 7275537, 10690812, 6879684, 2279214, 419958, 44268, 2628, 81, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 LINKS FORMULA Recurrence: T(0,0)=1, T(0,k)=0 for k>0 and for n>=1 T(n,k) = T(n-1,k-1)+(1+2*k)*T(n-1,k)+(k+1)*(k+2)*T(n-1,k+1). Let Z(n, k) = Sum_{j=0..n} C(-j, -n)*L(j, k) where L denotes the unsigned Lah numbers A271703. Then T(n, k) = Z(n+1, k+1). - Peter Luschny, Apr 13 2016 EXAMPLE 1,      1,      1,      3,      4,      1,     11,     21,      9,      1,     53,    128,     78,     16,      1,    309,    905,    710,    210,     25,      1,   2119,   7284,   6975,   2680,    465,     36,      1, 16687,  65821,  74319,  35035,   7945,    903,     49,      1, 148329, 660064, 857836, 478464, 133630,  19936,   1596,     64,      1, MAPLE A216154 := proc(n, k) local L, Z; L := (n, k) -> `if`(k<0 or k>n, 0, (n-k)!*C(n, n-k)*C(n-1, n-k)): Z := (n, k) -> add(C(-j, -n)*L(j, k), j=0..n); Z(n+1, k+1) end: seq(seq(A216154(n, k), k=0..n), n=0..9); # Peter Luschny, Apr 13 2016 MATHEMATICA T[0, 0] = 1; T[0, _] = 0; T[n_, k_] /; 0 <= k <= n := T[n, k] = T[n-1, k-1] + (2k+1) T[n-1, k] + (k+1) (k+2) T[n-1, k+1]; T[_, _] = 0; Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Aug 02 2019 *) PROG (Sage) def A216154_triangle(dim):     M = matrix(ZZ, dim, dim)     for n in (0..dim-1): M[n, n] = 1     for n in (1..dim-1):         for k in (0..n-1):             M[n, k] = M[n-1, k-1]+(1+2*k)*M[n-1, k]+(k+1)*(k+2)*M[n-1, k+1]     return M A216154_triangle(9) CROSSREFS A000255 (col. 0), A110450 (diag. n,n-2). Cf. A111596, A271703. Sequence in context: A172094 A114608 A154602 * A325174 A109956 A123319 Adjacent sequences:  A216151 A216152 A216153 * A216155 A216156 A216157 KEYWORD nonn,tabl AUTHOR Peter Luschny, Sep 19 2012 STATUS approved

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Last modified August 4 00:29 EDT 2021. Contains 346441 sequences. (Running on oeis4.)