A154227 Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + ((n+1)*(n+2)/2)*T(n-2, k-1), read by rows.

1, 1, 1, 1, 8, 1, 1, 19, 19, 1, 1, 35, 158, 35, 1, 1, 57, 592, 592, 57, 1, 1, 86, 1629, 5608, 1629, 86, 1, 1, 123, 3767, 28549, 28549, 3767, 123, 1, 1, 169, 7760, 105621, 309458, 105621, 7760, 169, 1, 1, 225, 14694, 320566, 1985274, 1985274, 320566, 14694, 225, 1

Offset: 0

Author

Roger L. Bagula, Jan 05 2009

Keywords

• nonn
• tabl

Comments

Row sums are: {1, 2, 10, 40, 230, 1300, 9040, 64880, 536560, 4641520, ...}.

The row sums of this class of sequences (see Cf section) is given by the following. Let S(n) be the row sum then S(n) = 2*S(n-1) + f(n)*S(n-2) for a given f(n). For this sequence f(n) = binomial(n+2, 2). - G. C. Greubel, Mar 02 2021

Examples

			Triangle begins as:
  1;
  1,   1;
  1,   8,     1;
  1,  19,    19,      1;
  1,  35,   158,     35,       1;
  1,  57,   592,    592,      57,       1;
  1,  86,  1629,   5608,    1629,      86,      1;
  1, 123,  3767,  28549,   28549,    3767,    123,     1;
  1, 169,  7760, 105621,  309458,  105621,   7760,   169,   1;
  1, 225, 14694, 320566, 1985274, 1985274, 320566, 14694, 225, 1;
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Crossrefs

Cf. A154228, A154229, A154230, A154231, A154233.

Programs

Magma

f:= func< n | Binomial(n+2,2) >;
function T(n,k)
  if k eq 0 or k eq n then return 1;
  else return T(n-1, k) + T(n-1, k-1) + f(n)*T(n-2, k-1);
  end if; return T;
end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 02 2021

Maple

T:= proc(n, k) option remember;
      if k=0 or k=n then 1
    else T(n-1, k) + T(n-1, k-1) + binomial(n+2,2)*T(n-2, k-1)
      fi; end:
seq(seq(T(n, k), k=0..n), n=0..12); # G. C. Greubel, Mar 02 2021

Mathematica

T[n_, k_]:= T[n,k]= If[k==0 || k==n, 1, T[n-1, k] + T[n-1, k-1] + Binomial[n+2, 2]*T[n-2, k-1] ];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Mar 02 2021 *)

Sage

def f(n): return binomial(n+2,2)
def T(n,k):
    if (k==0 or k==n): return 1
    else: return T(n-1, k) + T(n-1, k-1) + f(n)*T(n-2, k-1)
flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 02 2021

Formula

T(n, k) = T(n-1, k) + T(n-1, k-1) + ((n+1)*(n+2)/2)*T(n-2, k-1) with T(n, 0) = T(n, n) = 1.

Extensions

Edited by G. C. Greubel, Mar 02 2021