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

0,5

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

Links

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

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.

Example

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;

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 *)

Prog

(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
(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

Crossrefs

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

Sequence in context: A051425 A051469 A155494 * A166340 A157178 A174303

Adjacent sequences: A154224 A154225 A154226 * A154228 A154229 A154230

Keyword

nonn,tabl

Author

Roger L. Bagula, Jan 05 2009

Extensions

Edited by G. C. Greubel, Mar 02 2021

Status

approved