A157011 Triangle T(n,k) read by rows: T(n,k)= (k-1)*T(n-1,k) + (n-k+2)*T(n-1, k-1), with T(n,1)=1, for 1 <= k <= n, n >= 1.

1, 1, 2, 1, 5, 4, 1, 9, 23, 8, 1, 14, 82, 93, 16, 1, 20, 234, 607, 343, 32, 1, 27, 588, 2991, 3800, 1189, 64, 1, 35, 1365, 12501, 30155, 21145, 3951, 128, 1, 44, 3010, 47058, 195626, 256500, 108286, 12749, 256, 1, 54, 6416, 165254, 1111910, 2456256, 1932216, 522387, 40295, 512

Offset: 1

Author

Roger L. Bagula, Feb 21 2009

Keywords

• nonn
• tabl
• easy

Comments

Row sums are apparently in A002627.

The Mathematica code gives ten sequences of which the first few are in the OEIS (see Crossrefs section). - G. C. Greubel, Feb 22 2019

Examples

			The triangle starts in row n=1 as:
  1;
  1,  2;
  1,  5,    4;
  1,  9,   23,      8;
  1, 14,   82,     93,      16;
  1, 20,  234,    607,     343,      32;
  1, 27,  588,   2991,    3800,    1189,      64;
  1, 35, 1365,  12501,   30155,   21145,    3951,    128;
  1, 44, 3010,  47058,  195626,  256500,  108286,  12749,   256;
  1, 54, 6416, 165254, 1111910, 2456256, 1932216, 522387, 40295, 512;
		

Links

• G. C. Greubel, Rows n = 1..100 of triangle, flattened

Crossrefs

Cf. A000096 (column k=1), A002627, A008517.

Cf. This sequence (m=0), A008292 (m=1), A157012 (m=2), A157013 (m=3).

Programs

Maple

A157011 := proc(n,k) if k <0 or k >= n then 0; elif k  =0 then 1; else k*procname(n-1,k)+(n-k+1)*procname(n-1,k-1) ; end if; end proc: # R. J. Mathar, Jun 18 2011

Mathematica

e[n_, 0, m_]:= 1;
e[n_, k_, m_]:= 0 /; k >= n;
e[n_, k_, m_]:= (k+m)*e[n-1, k, m] + (n-k+1-m)*e[n-1, k-1, m];
Table[Flatten[Table[Table[e[n, k, m], {k,0,n-1}], {n,1,10}]], {m,0,10}]
T[n_, 1]:= 1; T[n_, n_]:= 2^(n-1); T[n_, k_]:= T[n, k] = (k-1)*T[n-1, k] + (n-k+2)*T[n-1, k-1]; Table[T[n, k], {n, 1, 10}, {k, 1, n}]//Flatten (* G. C. Greubel, Feb 22 2019 *)

PARI

{T(n, k) = if(k==1, 1, if(k==n, 2^(n-1), (k-1)*T(n-1, k) + (n-k+2)* T(n-1, k-1)))};
for(n=1, 10, for(k=1, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Feb 22 2019

Sage

def T(n, k):
    if (k==1):
        return 1
    elif (k==n):
        return 2^(n-1)
    else: return (k-1)*T(n-1, k) + (n-k+2)* T(n-1, k-1)
[[T(n, k) for k in (1..n)] for n in (1..10)] # G. C. Greubel, Feb 22 2019